OPEN PROBLEMS FROM THE WORKSHOP

“EMERGING APPLICATIONS OF MEASURE RIGIDITY”

HELD AT THE AMERICAN INSTITUTE OF MATHEMATICS JUNE, 2004

Contents

1. Local rigidity 1 2. Global rigidity 2 3. Measure rigidity 4 4. Equidistribution 6 5. Diophantine analysis 11 6. Quantum Chaos 15 7. Polygonal billiards 19 8. Divergent trajectories 20 9. André-Oort Conjecture 21 10. Arithmeticity 23 11. Symbolic coding 23 References 24

1. Local rigidity

1.1. It is well-known that an Anosov diffeomorphism is structurally stable: every C^{1}-diffeomorphism which is sufficiently close in C^{1}-topology to an Anosov diffeomorphism is topologically conjugate to it. However, the conjugation map is not differentiable in general. On the other hand, Anosov^{1} actions by higher rank abelian groups exhibit much more rigid behavior (see [93] for the first result of this type). It was shown in [97] that most of known algebraic^{2} Anosov ℤ^{k}- and ℝ^{k}- actions, k ≥ 2, are locally C^{∞}-rigid. Recall that a C^{∞}-action of ℤ^{k} is called locally C^{∞}-rigid if any C^{∞}-action of ℤ^{k} which is sufficiently C^{1}-close to this action is conjugate to it by a C^{∞}-map. A C^{∞}-action of ℝ^{k} is called locally C^{∞}-rigid if any C^{1}-small perturbation of this action is C^{∞}-conjugate to it up to an automorphism of ℝ^{k}.

Date: November 1, 2004; Scribe: A. Gorodnik.

1An action of a group G is called Anosov if there is an element g ∈ G that acts normally hyperbolically with respect to the orbit foliation of G.

2That is, the actions on infrahomogeneous spaces of Lie groups induced by either automorphisms or translations.

It was shown in [97] that most natural algebraic Anosov ℤ^{k}- and ℝ^{k}- actions, k ≥ 2, are locally C^{∞}-rigid provided that they do not reduce to rank one actions via some elementary constructions. We call such actions “irreducible”. See, for example, [97] for some natural conditions that guarantee that an action is “irreducible”.

Recently, local rigidity was proved in [39] for partially hyperbolic higher rank abelian actions by toral automorphisms. The method of [39] allows to construct C^{∞}-conjugacy only for C^{l}-perturbations of the original action for some large l. Another interesting example of a partially hyperbolic action is given in the following conjecture, which was communicated by R. Spatzier:

Conjecture 1. Let G be a connected semisimple Lie group, Γ an irreducible lattice in G, and A a closed subgroup of a split Cartan subgroup of G with dim A > 1. Then any C^{1}-small perturbation of the action of A on G/Γ is C^{∞}-conjugate to the action of A defined by a continuous homomorphism from A to the centralizer of A in G.

We also state one of important partial cases of Conjecture 1:

Conjecture 2. If in Conjecture 1 the group A is not contained in a wall of a Weyl chamber of the split Cartan subgroup D, then any C^{1}-small perturbation of the action of A on G/Γ is C^{∞}-conjugate to the action of A defined by a continuous homomorphism from A to D.

Conjecture 2 was proved in [97] when A is the full split Cartan subgroup. It was pointed out by A. Katok that these conjectures might be possible to solve using the method from [39].

1.2. Local rigidity for semisimple Lie groups of higher rank and their lattices (motivated by the program of R. Zimmer [209]) has been an active area of research too. First results in this direction were obtained for Anosov actions (see [85, 94, 95, 97]) and for actions with weaker hyperbolicity assumptions (see [131] and references therein). Recently, local rigidity results were established without any hyperbolicity assumptions (see [60]).

2. Global rigidity

2.1. The only known examples of Anosov diffeomorphisms are automorphisms of infranilmanifolds. Moreover, every Anosov diffeomorphism on an infranilmanifold is topologically conjugate to a hyperbolic automorphism (see [62, 127]). This motivates the following “ 100,000” folklore conjecture (stated in [130]):

Conjecture 3. Every Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism on an infranilmanifold.

Although there are some partial results on Conjecture 3 (see, for example, [61, 14, 69, 90]), it is not even known whether an Anosov automorphism is topologically transitive in general. We also mention that the conjugation map in Conjecture 3 is not necessarily smooth. There are examples of Anosov diffeomorphisms on manifolds that are homeomorphic but not diffeomorphic to a torus (see [54]).

In contrast, there exist Anosov flows that are not topologically transitive (see [63]), and it is not clear how to state a conjecture regarding classification of general Anosov flows. Such a conjecture is available in the special case when either stable or unstable foliation has dimension one (see [195, 70]).

2.2. One can also state an analog of Conjecture 3 for Anosov ℤ^{k}- and ℝ^{k}- actions, k ≥ 2. In this case, it is usually possible to show that if a continuous conjugation map exists, it is also smooth (see, for example, [85, 93, 97]). R. Spatzier communicated the following conjecture.

Conjecture 4. Every “irreducible” Anosov ℤ^{k}- and ℝ^{k}- action, k ≥ 2, is C^{∞}-conjugate to an algebraic action.

Some partial results on Conjecture 4 were obtained in [93, 142] and, recently, by F. Rodriguez-Hertz [162] and B. Kalinin, R. Spatzier. One may also hope to classify “irreducible” partially hyperbolic ℤ^{k}- and ℝ^{k}- action, k ≥ 2, and more generally, higher rank actions of commuting expanding maps.

2.3. There are also analogous conjectures for actions of connected semisimple Lie groups of higher rank and their lattices satisfying some hyperbolicity assumptions (see [85, 131]).

Conjecture 5. Every action of a connected semisimple Lie group of higher rank (i.e., all simple factors have real rank at least 2) or its lattice, which that has an element which acts non-trivially uniformly partially hyperbolicly, is C^{∞}-conjugate to an algebraic action.

Partial results on Conjecture 5 were obtained in [85, 94, 95, 158, 72, 131]. Note that without partial hyperbolicity assumption, one may only hope to classify the actions when restricted to an open dense subset. See [94, 59] for examples of nonstandard lattice actions. In general, there are conjectures originated from [209] on classification of actions satisfying some transitivity assumptions or preserving a rigid geometric structure in the sense of Gromov (see [117, 130] for up-to-date statements).

2.4. Ratner’s measure rigidity theorem has applications to the study of general properties of continuous volume preserving actions of higher rank semisimple Lie groups and their lattices on compact manifolds. In particular, Ratner’s theorem plays a key role in the construction of arithmetic quotients of such actions (see [123, 124] for connected groups, [56, 57] for lattices, and [58] for a survey). This raises the following question:

Question 6 (D. Fisher). Do the new results on measure rigidity for actions of higher rank abelian groups give rise to obstructions to smooth or continuous actions of a higher rank abelian group on a compact manifold?

Some basic obstructions for smooth volume preserving actions of higher rank abelian groups are already known, see particularly work of H. Hu and A. Katok. The question is whether one can use results on measure rigidity to obtain more information.

The results on arithmetic quotients for actions of semisimple groups and their lattices have no straightforward analogues here, since the proofs of those results use not only Ratner’s theorem but applications of the cocycle superrigidity theorems to cocycles which are necessarily only measurable. Though some cocycle superrigidity theorems are known for particular classes of actions of higher rank abelian groups, none of these apply to measurable cocycles because of the Dye theorem [43] and its generalizations [154, 37].

3. Measure rigidity

3.1. Let G be a Lie group, Γ a discrete subgroup, and H a subgroup of G generated by one-parameter unipotent subgroups. One of the prototypical examples of measure rigidity is the classification of finite ergodic H-invariant measures on G/Γ (see [159], and [144] for an accessible exposition).

Problem 7 (L. Silberman). Extend the results on measure rigidity of unipotent flows to adelic setting.

It seems natural to expect (and is known in some cases) that the set of finite ergodic invariant measures for other dynamical systems with parabolic behavior has a manageable structure, which is possible to described in algebraic terms.

Suppose that H is a connected semisimple Lie subgroup of a Lie group G, and let P be a parabolic subgroup of H. One of manifestations of the measure rigidity of unipotent flows is the fact that every finite P -invariant measure on G/Γ is H-invariant (see [146]).

Question 8 (E. Lindenstrauss). Suppose that H acts on a space X preserving some geometric structure. Under what conditions on X, every finite P - invariant measure is H-invariant? In other words, which H-actions are stiff (see [64])?

For example, one may consider an SL(2, ℝ)-action on the moduli space of quadratic differentials over complex structures on a compact surface. There are a lot of similarities between this action and SL(2, ℝ)-actions on homogeneous spaces (see [48, 199, 141]). Some partial results on topological and measure rigidity for the latter actions were obtained in [143] and [51].

3.2. A. Katok constructed an example of a Finsler metric on 2-dimensional sphere such that the corresponding geodesic flow is ergodic and has only two periodic orbits (see [92]).

Conjecture 9 (A. Katok). The only ergodic probability measures for this example are the smooth measure and the measures supported on periodic orbits.

3.3. Since a polygonal billiard is a parabolic dynamical system, one expects that invariant measures and invariant closed sets should be scarce.

Question 10 (A. Katok). Classify ergodic invariant probability measure and closed invariant subsets for polygonal billiards.

For rational polygonal billiards, the phase space decomposes into invariant subsets P_{θ} that correspond to directions θ of the flow. It was shown in [103] that the billiard flow is uniquely ergodic on P_{θ} for the set of directions θ of full measure. There is an estimate of the Hausdorff dimension of this set, which may be positive (see [140, 139, 34]). It is also known that the restriction of the billiard flow on all but countably many of the subsets P_{θ} is minimal.

3.4. Let Γ be a discrete subgroup of SL(2, ℝ). If Γ has infinite covolume, then

the only finite ergodic invariant measures for the horocyclic flow u_{t} = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} )

on SL(2, ℝ)/Γ are the the ones supported on periodic orbits. It turns out that there is a large family of infinite ergodic invariant Radon measures. Such measures can be constructed from the minimal positive Γ-invariant eigenfunction of the Laplacian (see, for example, [6]). Recently, F. Ledrappier and O. Sarig proved that if Γ is a normal subgroup of a uniform lattice in SL(2, ℝ), then every u_{t}-ergodic Radon measure on SL(2, ℝ)/Γ is of this form up to a constant (see also [7, 166] for previous classification results).

Question 11 (F. Ledrappier, O. Sarig). Let G be a noncompact semisimple Lie group of rank one, Γ a disctrete subgroup of G, and U a horospherical subgroup of G. What are the U-ergodic Radon measures on G/Γ? In particular, are they either carried by closed U-orbits or given by the harmonic function construction?

A u_{t}-invariant measure μ is called squashable if the centralizer of u_{t} contains an invertible nonsingular transformation that does not preserve μ. F. Ledrappier and O. Sarig showed recently that for a normal coabelian subgroup Γ of a uniform lattice, the only nonsquashable u_{t}-ergodic measure on SL(2, ℝ)/Γ is Haar. If Γ is conilpotent, then any u_{t}-ergodic measure on SL(2, ℝ)/Γ is squashable except possibly the Haar measure.

Question 12 (F. Ledrappier, O. Sarig). Let Γ be a normal conilpotent subgroup of a uniform lattice in SL(2, ℝ). Is the Haar measure nonsquashable?

For general discrete subgroup Γ ⊂ SL(2, ℝ), it is not known whether the Haar measure on SL(2, ℝ)/Γ is nonsquashable, or whether there exist other nonsquashable u_{t}-ergodic Radon measures.

3.5. Another important class of examples with rigid behavior is provided by the algebraic actions of higher rank abelian groups (see [121] for a survey). Although several complementary approaches have been developed for the study of invariant measures in this case (see [121]), all of them require some positive entropy assumptions. Such an assumption is not needed in the adelic setting. Let

𝔸 ⊂ ∏_{v-place} ℚ_{v}

denote the ring of adeles and D the diagonal subgroup in SL(2). E. Lindenstrauss showed that the only probability D(𝔸)-invariant measure on SL(2, 𝔸)/SL(2, ℚ) is the Haar measure (see [122]).

Question 13 (E. Lindenstrauss). Let 𝔸′ be defined as the ring of adeles, but the product is taken over a subset of places of ℚ and Γ an “irreducible” lattice in SL(2, 𝔸′). What are the finite ergodic D(𝔸′)-invariant measures on SL(2, 𝔸′)/Γ?

3.6. One may also expect measure rigidity for algebraic actions of “large” groups.

Conjecture 14 (A. Furman). Consider one of the following actions of a group Γ:

(1) Γ is a “large” subgroup of the group of automorphism of a nilmanifold of finite volume.

(2) Γ is a “large” subgroup of a Lie group acting by translations on G/Λ

where Λ is a lattice in G. Then the only ergodic Γ-invariant probability measures are the measures supported on finite Γ-orbits and the Haar measure.

There are results on the topological analog of this conjecture (see [16, 185, 148, 149, 74]).

4. Equidistribution

4.1. Let G be a Lie group, Γ a lattice in G, and U = {u(t)} ⊂ G a oneparameter Ad-unipotent subgroup. Suppose that for x ∈ G/Γ, Ux is dense in G/Γ.

Question 15 (G. Margulis [130]). Prove equidistribution of the sequence {u(t_{n})x} in G/Γ, where t_{n} is one of the following:

(1) t_{n} = [n^{α}]^{3} for α > 1,

(2) t_{n} = [P (n)], where P (x) is a polynomial,

(3) t_{n} is the n-th prime number.

A. Venkatesh suggested a proof of (1) when α is close to 1.

It is known that Česaro averages along sequences as in Question 15 converge almost everywhere for functions in L^{p}, p > 1 (see [22, 23, 24, 25, 202]). Note that there is a subtle difference between sequences t_{n} = [n^{α}] and t_{n} = n^{α} for α ∈ ℚ − ℤ. In fact, there is no general pointwise ergodic theorem possible for the latter sequence (see [18]).

4.2. Let V be a connected Ad-unipotent subgroup of the Lie group G such that V x is dense in G/Γ for some x ∈ G/Γ.

Question 16 (G. Margulis [130]). Show that for a “good” sequence of subsets A_{n} ⊂ V and every f ∈ C_{c}(G/Γ),

lim_{n→∞} \frac{1}{Vol(A_{n})} ∫_{A_{n}} f(vx) dv = ∫_{G/Γ} f dμ

with effective error term, where dv is a Haar measure on V, and μ is the probability Haar measure on G/Γ.

Such equidistribution results were proved by several authors (see [160, 132, 176]), but the methods of the proofs are not effective. In the case when V is a horospherical subgroup of G (see Section 4.3 below), one can deduce an equidistribution result with explicit error term from decay of matrix coefficient on L^{2}(G/Γ) (see [108]).

4.3. Let L be Lie group, G a closed subgroup of L, and Λ a lattice in L. For a semisimple element a ∈ G, the expanding horospherical subgroup U of G associated to a is defined by

U = {g ∈ G : a^{−n}ga^{n} → e as n → ∞}.

Suppose that for x_{0} ∈ L/Λ, the orbit Gx_{0} is dense in L/Λ.

Let μ be a measure on Ux_{0} which is the image of a probability measure on U, absolutely continuous with respect to the Haar measure on U, under the map u ↦ ux_{0}, u ∈ U. Then it is known that a^{n}μ → λ as n → ∞ where λ is the probability Haar measure on L/Λ (see [177]).

One may consider the following refinement of the above result. Take any analytic curve γ : [0, 1] → U, and let ν be the image of the Lebesgue measure on [0, 1] under the map t ↦ γ(t)x_{0}.

3Here [x] denotes the integer part of x.

Question 17 (N. Shah). Under what condition on γ, we have that a^{n}ν → λ as n → ∞?

Recently Question 17 was solved by N. Shah for L = SO(m, 1) and G = SO(n, 1), m > n. He showed that a^{n}ν → λ as n → ∞ provided γ([0, 1]) does not lie on an proper affine subspace or an (n − 2)-dimensional sphere in U.

Conjecture 18 (N. Shah). The same result holds for all Lie groups L containing G = SO(n, 1).

The above kind of questions are related to the following more general problem. Consider a representation of a semisimple Lie group G on real vector space V equipped with a norm ∥ · ∥. Take a point p ∈ V, and consider the set

R_{T} = {g ∈ G : ∥gp∥ < T }

for T > 0. Suppose that the stabilizer of p is finite, so R_{T} is compact for each T. Let Γ be be a lattice in G, and let μ_{T} denote the image of the normalized Haar measure on R_{T} projected to G/Γ.

Question 19 (N. Shah). What is the limiting distribution of the measure μ_{T} as T → ∞?

In some examples such results are known (see [53, 73]), but more general answers can be very important for understanding distribution of Γ-orbits on homogeneous spaces G/H where either Γ ∩ H is a lattice in H or ΓH is dense in G.

4.4. For irrational α, the sequence {αn^{2} (mod 1) : n ≥ 1} is equidistributed in [0, 1]. In fact, one expects that if α is badly approximable by rationals, then statistical properties of this sequence are the same as the sequence of independent uniformly distributed random variables. For [a, b] ⊂ [0, 1], we define pair correlation:

R_{2}([a, b], N, α) = \frac{1}{N} # \{1 ≤ i ≠ j ≤ N : αi^{2} − αj^{2} ∈ \frac{1}{N} [a, b] (mod 1)\} .

Conjecture 20 (Z. Rudnick, P. Sarnak). If α ∈ ℝ is badly approximable by rationals (see [165] for exact conditions), then

(1)  R_{2}([a, b], N, α) → b − a as N → ∞.

Although it was shown that (1) holds on the set of α of full measure (see [164]) and on a residual set of α in the sense of Baire category (see [165]), one does not know any explicit α for which it is true. It is expected that (1) holds for algebraic integers, and it is not hard to show that there are well approximable irrational α for which (1) fails.

It was discovered in [136] that Conjecture 20 is related to an equidistribution problem on a hyperbolic surface X = Γ\ℍ^{2}, Γ a lattice. We assume that

Γ ∩ {z ↦ z + a : a ∈ ℝ} = {z ↦ z + a : a ∈ ℤ}.

Then the curve {x + iy : x ∈ [0, 1]} corresponds to a closed horocycle {u_{y}(t) : 0 ≤ t ≤ 1} of length y^{−1} in in the unit tangent bundle T^{1}(X), and it is wellknown that it becomes equidistributed in T^{1}(X) as y → 0^{+}. Also, for irrational α, the sequence {u_{y}(αn) : n ≥ 1} is equidistributed in the horocycle. This motivates the following conjecture:

Conjecture 21 (J. Marklof, A. Strömbergsson). Let f be a continuous function on T^{1}(X) with given growth condition at the cusps (see [136]). Then for α ∈ ℝ which is badly approximable by rationals and 0 < c_{1} < c_{2},

\frac{1}{M} ∑_{m=1}^{M} f(u_{y}(αm)) → ∫_{T^{1}(X)} f dλ

uniformly as M → ∞ and c_{1}M^{−2} ≤ y ≤ c_{2}M^{−2}, where λ denotes the Liouville measure.

It was observed in [136] that Conjecture 21 implies Conjecture 20. Conjecture 21 was proved in [136] under the condition that c_{1}M^{−ν} ≤ y ≤ c_{2}M^{−ν} for some ν < 2. Furthermore, the statement of Conjecture 21 holds for almost all α with respect to Lebesgue measure [136] for any positive ν, in particular for ν = 2. Hence this gives a new proof of the main result in [164].

4.5. Let M be a compact Riemannian manifold, and ϕ_{t} : M → M is an Anosov flow, that is, ϕ_{t} is a C^{1}-flow and there exists a continuous invariant splitting

T M = E^{0} ⊕ E^{s} ⊕ E^{u}

where E^{0} is the one-dimensional bundle tangent to the flow direction, and for some C, λ > 0,

∥Dϕ_{t}v∥ ≤ Ce^{−λt}∥v∥, v ∈ E^{s}, t ≥ 0;
∥Dϕ_{t}v∥ ≥ C^{−1}e^{λt}∥v∥, v ∈ E^{u}, t ≥ 0.

The distribution E^{s} is tangent to the strong stable manifolds

W^{ss}(x) = {y ∈ M : d(ϕ_{t}x, ϕ_{t}y) → 0 as t → +∞}.

Suppose that the flow is topologically transitive. Then it is known (see [28]) that the foliation W^{ss}(x), x ∈ X, is uniquely ergodic, i.e., there is a unique holonomy invariant transverse measure.

In addition to the above assumptions, we suppose that there exists a continuous invariant splitting E^{s} = E_{+}^{s} + E_{−}^{s} such that for some C > 0 and μ_{+} > μ_{−} > λ,

∥Dϕ_{t}v∥ ≤ Ce^{−μ_{+}t}∥v∥, v ∈ E_{+}^{s}, t ≥ 0;
∥Dϕ_{t}v∥ ≥ C^{−1}e^{−μ_{−}t}∥v∥, v ∈ E_{−}^{u}, t ≥ 0.

A basic example of such splitting is the geodesic flow of ℂℍ^{2}. The distribution E_{+}^{s} integrates to the fast stable foliation W_{+}^{s}.

The following is a nonlinear analog of the Ragunathan’s question about classifications of measures invariant under unipotent flows:

Question 22 (F. Ledrappier). Describe the invariant ergodic measures for the fast stable foliation W_{+}^{s}.

One may ask the same question for Anosov diffeomorphisms as well.

4.6. Let X be a Riemannian locally symmetric space of noncompact type and of finite volume. A flat in X is a totally geodesic submanifold of sectional curvature zero. Note that X = Γ\G/K, where G is a connected semisimple real algebraic group, K is a maximal compact subgroups of G, and Γ is a lattices in G, and flats are ΓgAK, g ∈ G, for a Cartan subgroup A of G. It was shown (see [150]) that the number of compact flats with bounded volume is finite. Note that this number is related to the number of totally real number fields of fixed degree with bounded regulator (see [150]).

Question 23 (H. Oh). Determine the asymptotics of the number of compact flats with volume less than T as T → ∞.

This asymptotics and the rate of convergence has been determined for rank one spaces (see [128, 79, 67, 68, 205, 114, 156, 112]); however, the question about optimal rate of convergence is still open (see [88, 126, 125, 31]). When X is compact, using techniques developed in [183], one can determine the asymptotics of the sum

∑_{F - regular, systol(F)<T} Vol(F).

Here a flat is called regular if its shortest closed geodesics goes in the regular direction and systol(F) denotes the length of this geodesics. See also [40] for another analog of the prime geodesic theorem for higher rank compact symmetric spaces.

Another open question concerns the distribution of compact maximal flats on the unit tangent bundle T^{1}(X). Since the identity component G of the isometry group does not act transitively on X in the higher rank case, it is more convenient to consider the compact orbits of a Cartan subgroup A on the homogeneous space Γ\G, which we also call flats. Denote by μ_{F} the Lebesgue measure on a flat F ⊂ Γ\G and by μ̄_{F} the normalized Lebesgue measure on F. Let

ν_{T} = \frac{∑_{F: Vol(F)<T} μ_{F}}{∑_{F: Vol(F)<T} Vol(F)} and ν̄_{T} = \frac{∑_{F: Vol(F)<T} μ̄_{F}}{#\{F : Vol(F) < T \}}.

Question 24 (H. Oh). Do the measures ν_{T} and ν̄_{T} converge to the normalized Haar measure on Γ\G?

According to [113], the normalized Haar measure on Γ\G is the unique ergodic measure of maximal entropy for the geodesic flow. Thus, to resolve Question 24, it suffices to estimate the entropy of the weak^{∗} limit points of ν_{T} and ν̄_{T} and show that they do not escape to infinity.

For the rank one groups, Question 24 has been answered positively (see [26, 206, 155, 168, 104]). In higher rank case, the equidistribution of ε-seperated closed geodesics from different homotopy classes was established in [113]. As in [113], one can also prove analog of the Conjecture 24 for compact Γ\G and the measure ν_{T} as above with summation taken over regular flats F such that systol(F) < T. Another equidistribution result was obtained in a recent work of Y. Benoist and H. Oh, where the averages along Hecke orbits of maximal compact flats were considered.

5. Diophantine analysis

5.1. A vector y = (y_{1}, …, y_{n}) ∈ ℝ^{n} is called v-approximable (for v > 0) if there are infinitely many q = (q_{1}, …, q_{n}) ∈ ℤ^{n} and p ∈ ℤ such that

|y_{1}q_{1} + ⋯ + y_{n}q_{n} − p| < ∥q∥^{−v}.

Here ∥ · ∥ denotes the max-norm on ℝ^{n}. If a vector y ∈ ℝ^{n} is (n + ε)- approximable for some ε > 0, it is called very well approximable (VWA). An easy argument (using Borel-Cantelli lemma) implies that the set of VWA vectors has Lebesgue measure zero in ℝ^{n}. Therefore, it is natural to expect that a generic point on a “nondegenerate” submanifold of ℝ^{n} is not VWA. This is the Sprindžuk conjecture proved in [109] (see also [17]).

Similarly, an m × n real matrix A is called VWA if for some ε > 0 there are infinitely many q ∈ ℤ^{n} and p ∈ ℤ^{m} such that

∥Aq − p∥^{m} < ∥q∥^{−n−ε}.

It is easy to see that the set of VWA matrices has measure zero in ℝ^{m×n}, and one hopes that an analog of the Sprindžuk conjecture holds in this set-up as well.

The definition of nondegenerate submanifold of ℝ^{n} in [109], which is wellsuited for the case of vectors, is a manifold with smooth coordinate charts f : U(⊂ ℝ^{k}) → ℝ^{n} such that the spaces spanned by the partial derivatives f at points of U have dimension n. It is not quite clear what is the right definition of “nondegenerate” submanifold for the case of matrices.

Question 25 (D. Kleinbock, G. Margulis (see [109], Sec. 6.2)). Find reasonable and checkable conditions for a smooth map f : U(⊂ ℝ^{k}) → M_{m×n}(ℝ) which generalizes nondegeneracy of vector-valued maps and implies that almost every point of f(U) is not VWA.

Such conditions were obtained in some cases in [115, 116, 105].

5.2. A far-reaching generalization of the Sprindžuk conjecture was suggested in [106]. Let μ be a measure on ℝ^{k} and and f : supp(μ) → ℝ^{n} that satisfy some reasonable conditions. What are the Diophantine properties of the generic points in ℝ^{n} with respect to the measure f_{∗}μ? Several results in this direction were obtained in [198, 107, 111] for locally finite measure. It would be interesting to consider the case of Hausdorff measures:

Question 26 (D. Kleinbock). Give estimates on Hausdorff dimension of vapproximable vectors in a nondegenerate submanifold of ℝ^{n} using dynamics.

See [41] for a discussion of what is currently known about the Hausdorff dimension and for a related result.

5.3. For α ∈ ℝ, let ⟨α⟩ = dist(α, ℤ). It is not hard to show that the set of (α, β) ∈ ℝ^{2} such that

lim inf_{q→∞} q(log q)^{2+ε} ⟨qα⟩ ⟨qβ⟩ > 0

for every ε > 0 has full Lebesgue measure. In fact, it was shown in [184] that

lim_{q→∞} q(log q)^{2+ε} ⟨qα⟩ ⟨qβ⟩ = ∞

on a set of (α, β) ∈ ℝ^{2} of full measure. On the other hand, the following question remains open:

Question 27 (A. Pollington). Are there α, β ∈ ℝ such that for every ε > 0,

lim inf_{q→∞} q(log q)^{2−ε} ⟨qα⟩ ⟨qβ⟩ > 0?

It follows from [65] that

lim inf_{q→∞} q(log q)^{2} ⟨qα⟩ ⟨qβ⟩ = 0

for almost all (α, β). Thus, the set of (α, β) in Question 27 has measure zero. Question 27 is related to the well-known conjecture of Littlewood:

Conjecture 28 (Littlewood). For any α, β ∈ ℝ,

lim inf_{q→∞} q ⟨qα⟩ ⟨qβ⟩ = 0.

The best result on Conjecture 28 is [47], which shows that the set of exceptions (α, β) for the Littlewood conjecture is a countable union of sets of box dimension zero. The proof in [47] uses dynamics on the homogeneous space SL(3, ℝ)/SL(3, ℤ). It was observed some time ago that Conjecture 28 is implied by the following conjecture:

Conjecture 29 (G. Margulis [130]). Let A be the group of all diagonal matrices in SL(3, ℝ). Then every bounded A-orbit in SL(3, ℝ)/SL(3, ℤ) is closed.

Conjecture 29 is a very special case of the general conjecture describing closed invariant subsets for actions of Cartan subgroups on general homogeneous spaces (see [130]).

G. Margulis suggested the following conjecture, which might be easier to handle than Conjecture 29:

Conjecture 30 (G. Margulis). For every compact set K of SL(3, ℝ)/SL(3, ℤ), there are only finitely many closed A-orbits contained in K.

This conjecture can be reformulated in terms of the Markov spectrum of forms

F (x) = ∏_{i=1}^{3} (∑_{j=1}^{3} a_{ij}x_{j}) , a_{ij} ∈ ℝ.

Let

Δ(F) = det(a_{ij}) and m(F) = inf \{|\frac{F (x)}{Δ(F)}| : x ∈ ℤ^{3} − 0\} .

Then Question 30 is equivalent to the following question:

Question 31 (G. Margulis). Show that for every ε > 0, the set [ε, ∞)∩{m(F)} is finite.

5.4. For 0 ≤ s ≤ 1, define

C_{s} = {(α, β) ∈ ℝ^{2} : inf_{q≥1} max{q^{s} ⟨qα⟩ , q^{1−s} ⟨qβ⟩} > 0} .

In particular, C_{1/2} is the set of badly approximable vectors. Since Conjecture 28 holds for all (α, β) ∉ C_{s}, one may naively hope to prove it by showing that intersection of the sets C_{s}, 0 ≤ s ≤ 1, is empty. In this regard, we mention the following conjecture:

Conjecture 32 (W. Schmidt [170]). For any s, t ∈ [0, 1], we have C_{s} ∩ C_{t} ≠ ∅.

Note that W. Schmidt stated this conjecture in [170] only for s = 1/3 and t = 2/3.

It is known that each of the sets C_{s} has zero measure and full Hausdorff dimension. It was shown that the set C_{s} ∩ C_{0} ∩ C_{1} has full Hausdorff dimension as well (see [157]). Conjecture 32 is related to the following conjecture:

Conjecture 33. Let A be the group of all diagonal matrices in SL(3, ℝ), and A_{1}, A_{2} ⊂ A are rays in A. Then there exists x ∈ SL(3, ℝ)/SL(3, ℤ) such that A_{1}x and A_{2}x are bounded, but Ax is not bounded in SL(3, ℝ)/SL(3, ℤ).

Note that for rays A_{1} and A_{2} which lie in the cone

{diag(e^{u}, e^{v}, e^{−u−v}) : u, v ≥ 0} ⊂ A,

Conjecture 33 follows from Conjectures 32. On the other hand, it was pointed out by D. Kleinbock that in the case when A_{1} and A_{2} lie in the opposite Weyl chambers, Conjecture 33 can be proved using the argument from [108], and moreover, the set of x which satisfy Conjecture 33 has full Hausdorff dimension.

5.5. Let Q be a nondegenerate positive definite quadratic form of dimension d ≥ 3.

Conjecture 34 (Davenport-Lewis). Suppose that Q is not a multiple of a rational form. Then the gaps between consecutive elements of the set {Q(x) : x ∈ ℤ^{d}} go to zero as Q(x) → ∞.

Conjecture 34 was proved in [15] for d ≥ 9, and recently the method in [15] was extended to d ≥ 5 as well. The case d = 3, 4 is still open.

When Q is a nondegenerate indefinite definite quadratic form of dimension d ≥ 3 which is not a multiple of a rational quadratic form, the set {Q(x) : x ∈ ℤ^{d}} is dense in ℝ. This is the Oppenheim conjecture proved by Margulis in [129]. However, the proof in [129] is not effective.

Question 35 (G. Margulis [130]). Give an effective estimate on T = T (ε) such that there exists x ∈ ℤ^{d} with

0 < |Q(x)| < ε and ∥x∥ < T.

This question is especially difficult since the estimate on T should depend on the Diophantine properties of coefficients of the quadratic form Q. An easier question with x satisfying conditions

|Q(x)| < ε and ∥x∥ < T

is treated for d ≥ 5 in an upcoming work of G. Margulis and F. Götze.

5.6. Let Q(x) = ax_{1}^{2} + bx_{1}x_{2} + cx_{2}^{2} be a nondegenerate indefinite quadratic form with rational coefficients that does not represent zero over ℚ. For x ∈ ℝ^{2}, define

m(Q, x) = inf_{z∈ℤ^{2}} |Q(x + z)| and m(Q) = sup_{x∈ℝ^{2}} m(Q, x).

If the supremum m(Q) is isolated, we also define

m_{2}(Q) = sup{m(Q, x) : x ∈ ℝ^{2}, m(Q, x) < m(Q)}.

The interest in the quantity m(Q) was motivated by the study of existence of a Euclidean algorithm in quadratic fields ℚ(\sqrt{m}), m > 0. If Q represents the norm of ℚ(\sqrt{m}) computed with respect to an integral basis, then Euclidean algorithm exists iff m(Q) < 1.

The following conjecture was communicated by A. Pollington:

Conjecture 36 (E. Barnes, H. Swinnerton-Dyer [11]). For any quadratic form Q as above, the supremum m(Q) is rational and isolated. Both m(Q) and m_{2}(Q) are attained at points with coordinates in the the splitting field of Q.

Conjecture 36 is based on numerous computations performed in [10, 11]. The supremum m_{2}(Q) need not be isolated (see [71]).

5.7. The following question was communicated by D. Kleinbock:

Question 37 (Y. Bugeaud). Let

B_{s} = {(α, β) ∈ ℝ^{2} : inf_{n≥1} n(min{∥nα∥, ∥nβ∥})^{s}(max{∥nα∥, ∥nβ∥})^{2−s} > 0} .

Compute the Hausdorff dimension of the set B_{s}, 0 < s < 1.

Note that B_{0} is the set of badly approximable vectors and its Hausdorff dimension is 2. On the other hand, B_{1} is the set of exceptions of the Littlewood conjecture, and its Hausdorff dimension is 0.

5.8. Some other interesting open problems on Diophantine approximation are stated in [110], Section 13.

6. Quantum Chaos

6.1. The term “quantum chaos” refers to the study of quantizations of Hamiltonian systems whose dynamics is chaotic. We concentrate on the case of the geodesic flow on compact (or, more generally, finite volume) Riemannian manifold X possibly with piecewise smooth boundary (e.g. billiards in ℝ^{2}). The geodesic flow on the boundary is defined as elastic reflection. Denote by Δ the Laplace-Beltrami operator on X and by dV the normalized Riemannian volume on X. Let 0 = λ_{0} < λ_{1} ≤ λ_{2} … be the eigenvalues of −Δ and ϕ_{i}, i ≥ 0, the corresponding eigenfunctions with the Dirichlet boundary condition such that ∥ϕ_{i}∥_{2} = 1:

−Δϕ_{i} = λ_{i}ϕ_{i}, ϕ_{i}|_{∂X} = 0.

One is interested in the semiclassical limit of this system, i.e., in the behavior of the eigenvalues and the eigenfunctions as i → ∞. According to the correspondence principle in quantum mechanics, certain properties of the classical dynamical system are inherited by the semiclassical limit of its quantization.

Consider the probability measures

dμ_{i}(x) = |ϕ_{i}(x)|^{2}dV (x)

on X. One of the fundamental questions is to describe all possible weak^{∗} limits of the sequence {μ_{i}} as i → ∞, which are called quantum limits. It was shown (see also [182, 204, 36, 207]) that if the geodesic flow is ergodic on X, then μ_{i_{k}} → dV in the weak^{∗} topology as i_{k} → ∞ along a subsequence {i_{k}} of density one. This property is referred as quantum ergodicity.

In general, it might be possible that some of the quantum limits are not absolutely continuous and even assign positive measure to an unstable periodic orbit (this is called a scar) or to a family of marginally stable periodic orbits (this is called a bouncing ball mode). However, it seems that no rigorous proof of this phenomena has been given. For example, for the stadium billiard there are substantial numerical and heuristic evidences of the existence of scars and bouncing ball modes (see, for example, [82, 91, 119, 8, 187] and references therein). On the other hand, the numerical data in [12] suggest that no scarring occurs for some dispersive billiards.

Some numerical experiments were performed for X = Γ\ℍ^{2}, where Γ is an arithmetic lattice, and no scars were observed (see [81, 80, 4]). Z. Rudnick and P. Sarnak [163] formulated the following conjecture:

Conjecture 38 (Quantum unique ergodicity). Suppose that X has negative sectional curvature. Then

μ_{i} → dV as i → ∞.

The current research on this conjecture is concentrated on the case of arithmetic manifolds X = Γ\X̃, where X̃ is a symmetric space of noncompact type and Γ an arithmetic lattice. The arithmeticity assumption implies that there is an infinite set of Hecke operators acting on X, which commute with the left invariant differential operators (in rank one the Laplacian is the only such operator). We assume that ϕ_{i}, i ≥ 0, are joint eigenfunctions of the invariant differential operators and Hecke operators. Then the weak^{∗} limits of the sequence of measures {μ_{i}} are called arithmetic quantum limits. It is believed (see [32]) that the Laplace-Beltrami operator on X = SL_{2}(ℤ)\ℍ^{2} has simple cuspidal spectrum; then the assumption on Hecke operators is automatic.

Conjecture 39 (Arithmetic quantum unique ergodicity). The Riemannian volume is the only arithmetic quantum limit.

Positive results towards this conjecture were obtained for the case X = Γ\ℍ^{2}, where Γ is a congruence subgroup in either SL_{2}(ℤ) or in the group of quaternions of norm one. In this case, T. Watson [197] proved Conjecture 39 assuming the generalized Riemann hypothesis. His proof also implies the optimal rate of convergence. Unconditionally, Conjecture 39 for this case was proved by E. Lindenstrauss [120]. The only issue that was not handled in [120] is the escape of the limit measure to the cusp in noncompact case. To handle this difficulty, E. Lindenstrauss suggested the following intermediate problem:

Problem 40 (E. Lindenstrauss). Let X = Γ\ℍ^{2} where Γ is a noncocompact arithmetic lattice. Show that for all f, g ∈ C_{c}(X),

\frac{∫_{X} f dμ_{i}}{∫_{X} g dμ_{i}} → \frac{∫_{X} f dV}{∫_{X} g dV} as i → ∞.

The analog of Problem 40 for continuous spectrum was proved in [89, 126]. Results toward Conjecture 39 for some higher-rank symmetric spaces were

recently proved by L. Silberman and A. Venkatesh [178].

6.2. M. Berry [19] conjectured that eigenfunctions of a typical chaotic systems behave like a superposition of plane waves with random amplitude, phase and direction. This model predicts that the eigenfunctions ϕ_{i} behave like independent Gaussian random variables as i → ∞. In particular, the following conjecture should hold for generic negatively curved compact Riemannian manifolds:

Conjecture 41 (J. Marklof).

Vol(\{x ∈ X : a ≤ ϕ_{i}(x) ≤ b\}) → \frac{1}{\sqrt{2π}} ∫_{a}^{b} e^{−t^{2}/2}dt

as i → ∞.

Conjecture 41 is supported by numerical experiments (see [81, 82]).

The random wave model also predicts a central limit theorem for the convergence in Conjecture 38 (see [55, 44]). To formulate this, we need some notations. For a smooth function a on the unit cotangent bundle S^{∗}X of X, we denote by Op(a) a pseododifferential operator of order zero with principal symbol a. For example, when a is a function on X, then Op(a) is a multiplication by a. Let λ be the Liouville measure on S^{∗}X and g^{t} is the geodesic flow. It is expected that for generic negatively curved compact Riemannian manifolds,we have the following. (Suppose w.l.o.g. the surface has area 4π so that Weyl’s law reads N(λ) = #{i : λ_{i} ≤ λ} ∼ λ.)

Conjecture 42 (J. Marklof (after Feingold-Peres [55])). Suppose X is “generic”. For a ∈ C_{c}^{∞}(S^{∗}X, ℝ) with

∫_{S^{∗}X} a dλ = 0,

put

ξ_{i}(a) = λ_{i}^{1/4} |⟨Op(a)ϕ_{i}, ϕ_{i}⟩| .

Then the sequence ξ_{i}(a) has a Gaussian limit distribution whose variance is given by the classical autocorrelation function

V (a) =^{def} ∫_{−∞}^{∞} ∫_{S^{∗}X} a(xg^{t})a(x) dλ(x)dt.

That is, as λ → ∞, (1)

\frac{1}{λ} ∑_{λ_{i}≤λ} ξ_{i}(a)^{2} → V (a),

(2) for any interval I ∈ ℝ

\frac{1}{λ}# \{λ_{i} ≤ λ : V (a)^{−1/2}ξ_{i}(a) ∈ I\} → \frac{1}{\sqrt{2π}} ∫_{I} e^{−t^{2}/2}dt.

It was proved by W. Luo and P. Sarnak that for the modular surface X = SL_{2}(ℤ)\ℍ^{2},

∑_{λ_{i}≤λ} |⟨Op(a)ϕ_{i}, ϕ_{i}⟩|^{2} ∼ \sqrt{λ}B(a) as λ → ∞,

where B is a quadratic form on C_{c}^{∞}(X) which is closely related to but distinct from the form V defined above (see [169]). In this respect the modular and other arithmetic surfaces are ruled out as “generic” examples for the above conjecture.

6.3. A. Katok suggested polygonal billiards as a promising model for quantum chaos.

Question 43 (A. Katok). Do periodic orbits in a triangular billiard correspond to scars? More precisely, are there quantum limits supported on periodic orbits?

Based on the investigation [21], it seems likely that the answer to this question is ‘yes’ for rational billiards.

Problem 44 (J. Marklof). Classify all quantum limits of the eigenfunctions of a polygonal billiard.

6.4. According to the Berry-Tabor conjecture (see [133]), the eigenvalues of the Laplacian for generic integrable dynamical system have the same statistical properties as a Poisson process. For 2-dimensional torus, the set of eigenvalues is {Q(x) : x ∈ ℤ^{2}} where Q is a positive definite quadratic form.

Question 45 (J. Marklof). What is the distribution of the set

{(Q(x_{1}) − Q(x_{2}), Q(x_{2}) − Q(x_{3})) : x_{i} ∈ ℤ^{2}, x_{i} ≠ x_{j} for i ≠ j} ⊂ ℝ^{2}?

More precisely, determine the asymptotics of

N_{T} ((a, b), (c, d)) =^{def} # \{(x_{1}, x_{2}, x_{3}) ∈ (ℤ^{2})^{3} : \begin{matrix}a < Q(x_{1}) − Q(x_{2}) < b, \\ c < Q(x_{2}) − Q(x_{3}) < d, \\ x_{i} ≠ x_{j} for i ≠ j, ∥x_{i}∥ < T \end{matrix} \} .

The distribution of the set {Q(x) − Q(y) : x, y ∈ ℤ^{2}} was studied in [167, 49, 50], and in [134, 135] in the case of rational forms over shifted lattice points. It depends on Diophantine properties of coefficients of the quadratic form. The Berry-Tabor conjecture predicts that the set in Question 45 should be equidistributed in ℝ^{2} for a generic quadratic form Q, i.e, after a suitable normalization, N_{T} ((a, b), (c, d)) converges to (b − a)(d − c). However, it is not even known whether the set in Conjecture 45 is dense in ℝ^{2}.

7. Polygonal billiards

Question 46 (A. Katok). Construct periodic orbits for triangular billiards.

Every acute triangle has one obvious periodic orbit, but it is not known whether a general acute triangle has other periodic orbits. It is also not known whether a general obtuse triangle has at least one periodic orbit. Periodic orbits were constructed for some special classes of triangles (see [66, 83, 35, 77, 171]). Recently, a computer aided proof, which uses the program McBilliards [84], was found that shows that every triangle with all angles less than 100 degrees has a periodic orbit (see [172]).

The situation is much better for rational triangles and polygons (i.e., if the angles are rational multiples of π). Unfolding the billiard table, one can construct a compact Riemannian surface with a flat structure so that billiard trajectories correspond to geodesics on this surface (see [141] for a survey). Using this technique, it was shown that the number N(T) of periodic orbits of length at most T is bounded from above and below by quadratic polynomials in T (see [137, 138]). Moreover, for some billiard table this number has quadratic asymptotics (see [193, 194, 52, 51]), but it seems unknown whether the quadratic asymptotics holds for rational polygons in general. Note that the convergence N(T) → ∞ cannot be uniform even on a compact set of triangles. In fact, it was announced by R. E. Schwatz that for any given any ε > 0 there exists a triangle, within ε of the 30–60–90 triangle, which has no periodic paths of length less than 1/ε.

Question 47 (A. Katok). Prove ergodicity for triangular billiards with irrational angles.

It was proved that the set of ergodic (topologically transitive) triangular billiard table is residual in the sense of Baire category (see [103, 99]). However, it seems unknown whether the set of ergodic billiard tables has positive measure. It is easy to see that rational billiards cannot be ergodic. On the other hand, it was shown in [196] that if an irrational billiard table is very well approximable by a rational one, then it is ergodic. In the case when one of the angles of a triangles is rational, there is a useful unfolding procedure (see [189]) that may lead to a proof of ergodicity.

Also, it seems unknown whether there is a weakly mixing polygonal billiard (see [76] for a positive result in this direction). It was shown in [5] that typical interval exchange transformation which is not defined by a cyclic permutation is weakly mixing (see also [98, 192]).

8. Divergent trajectories

Let G be a semisimple real algebraic group, Γ a noncocompact arithmetic lattice in G, and D a closed subgroup of a maximal ℝ-split torus A. An orbit Dx of D in G/Γ is called divergent is the map d ↦ dx, d ∈ D, is proper. One can construct a divergent orbits using the following observation. Suppose that D is the union of open subsemigroups D_{1}, …, D_{l} such that for every i there exists a representation ρ_{i} : G → GL(V_{i}), defined over ℚ, and v_{i} ∈ V_{i}, such that ρ_{i}(dx)v_{i} → 0 as d ∈ D_{i} goes to ∞. Then Dx is divergent. Such divergent orbits are called obvious.

Conjecture 48 (B. Weiss). (1) If dim D > rank_{ℚ}G, then there are no

divergent orbits of D.

(2) If dim D = rank_{ℚ}G, then the only divergent trajectories are obvious ones.

(3) If dim D < rank_{ℚ}G, then there are non-obvious divergent trajectories.

Conjecture 48 was formulated in [200] where several special cases of it were checked, and it was shown in particular that Conjecture 48 holds when dim D = 1 (see also [38]). The case D = A was settled in [188]. Recently, Conjecture 48(1) was proved in [33] when rank_{ℚ}G = 2 and in [201] in complete generality. We also mention that Conjecture 48(3) was checked in [200] for G = SL(4, ℝ) for all diagonal subgroups D except

D = {(s, s^{−1}, t, t^{−1}) : s, t > 0}.

Next, we discuss a similar problem when D is a cone in A. There are examples of cones D that admit non-obvious divergent trajectories (e.g., a Weyl chamber) as well as an example of cones that admit only obvious divergent trajectories (see [200]). The latter example was constructed for G = SL(3, ℝ) and the argument used essentially that dim D = 2.

Question 49 (B. Weiss). Construct examples of cones D in A with dim D ≥ 3 and no non-obvious divergent trajectories.

Let A_{T} denote the ball of radius T in A and λ a Haar measure on A.

Question 50 (B. Weiss). Suppose that dim A ≥ 2 and for some x ∈ G/Γ and every one-parameter subgroup D of A, the orbits Dx is not divergent in G/Γ. Is it true that there exists a compact set K ⊂ G/Γ such that

lim sup_{T →∞} \frac{1}{λ(A_{T})} λ(\{a ∈ A_{T} : ax ∈ K\}) > 0?

9. André-Oort Conjecture

A Shimura datum is a pair (G, X) where G is a reductive algebraic group defined over ℚ and X is a G(ℝ)-conjugacy class of homomorphisms h : ℂ^{×} → G(ℝ) such that

(1) The adjoint action of h(ℂ^{×}) on Lie(G^{ad}(ℝ))^{4} decomposes as a direct sum of eigenspaces with characters z/z̄, 1, z̄/z.

(2) ad h(i) acts as a Cartan involution on G^{ad}(ℝ).

(3) G^{ad}(ℝ) has no factors on which the adjoint action of h(ℂ^{×}) is trivial.

Morphisms (G̃, X̃) → (G, X) of Shimura datums are induced by morphisms G̃ → G of algebraic groups in obvious way. Note that X has a natural structure of complex manifold such that its connected components are Hermitian symmetric domains, G(ℝ) acts on X by holomorphic automorphisms, and morphisms are equivariant holomorphic maps.

Let 𝔸_{f} denote the ring of finite adeles, and K is an open compact subgroup in G(𝔸_{f}). Define

Sh_{K}(G, X) = G(ℚ)\(X × G(𝔸_{f}))/K

One can show that Sh_{K}(G, X) is a finite disjoint union of Hermitian locally symmetric domains. In particular, by the Baily-Borel theorem, Sh_{K}(G, X) has a natural structure of an algebraic variety.

Example: Let G = GL_{2}, h(a + bi) = ( \begin{matrix}a & b \\ −b & a \end{matrix} ) and K = GL_{2}(ℤ̂). Then

Sh_{K}(G, X) ≃ SL_{2}(ℤ)\ℍ^{2} parametrizes isomorphism classes of elliptic curves over ℂ.

The Shimura variety associated to (G, X) is the projective limit of Sh_{K}(G, X) where K runs over open compact subgroups of G(𝔸_{f}). A point h ∈ X is called special if there exists a torus T of G defined over ℚ such that h(ℂ^{×}) ⊂ T (ℝ). One can check that in the above example, the special points are imaginary quadratic irrationals that correspond to elliptic curves with complex multiplication.

4G^{ad} is the adjoint group which is the factor of G by its center.

For g ∈ G(𝔸_{f}), we have natural projection maps

π_{1} : Sh_{K∩gKg^{−1}}(G, X) → Sh_{K}(G, X)
π_{2} : Sh_{K∩gKg^{−1}}(G, X) → Sh_{gKg^{−1}}(G, X).

with finite fibers. This defines Hecke correspondence

T_{g}(x) = π_{2}(π_{1}^{−1}(x))g : Sh_{K}(G, X) → Sh_{K}(G, X).

Let (G̃, X̃) → (G, X) be morphism of Shimura datums that induces map ShK̃(G̃, X̃) → Sh_{K}(G, X). The special subvarieties (also called subvarieties of Hodge type) are the irreducible components of the image

ShK̃(G̃, X̃) → Sh_{K}(G, X) ⟶^{T_{g}} Sh_{K}(G, X).

Using Hecke correspondences, one shows that the set of special points in a special subvariety is dense with respect to Zariski (or even analytic) topology. The following conjecture is the converse of this fact.

Conjecture 51 (Y. André-F. Oort [3, 153]). Zariski closure of a set of special point is a finite union of special subvarieties.

See [45, 46] and references therein for partial results on this conjecture. We also mention that Conjecture 51 was proved in the case when the dimension of the Zariski closure is one assuming generalized Riemannian hypothesis (see [203]).

Conjecture 51 was partially motivated by an analogy with the theory of abelian varieties, according to which special points correspond to torsion points and special subvarieties correspond to translates of abelian subvarieties by torsion points. Analogous conjectures for abelian varieties is due to S. Lang, Yu. Manin, and D. Mumford. These conjectures were settled (see [190] for a survey). One of the proofs (see [191, 208]) is based on equidistribution of Galois orbits of “generic” sequences of points, which was established in [186] (see also [20]). This approach may also lead to a proof of Conjecture 51.

Conjecture 52. Let {x_{n}} be a sequence of special points on a Shimura variety. Suppose that x_{n} lies outside of any special subvariety for sufficiently large n. Then the (finite) Galois orbits of x_{n} become equidistributed as n → ∞ with respect to the normalized Haar measure.

For some partial results on this conjecture, see [78] and references therein. In particular, for the above example, Conjecture 52 was established in [42].

Question 53 (L. Silberman). Give an ergodic-theoretic proof of the equidistribution of special points on SL_{2}(ℤ)\ℍ^{2}.

10. Arithmeticity

Let G be the direct product of k copies of SL(2, ℝ), k ≥ 2, and U^{+} and U^{−} the upper and lower unipotent subgroups G respectively. Let Γ^{+} and Γ^{−} be lattices in U^{+} and U^{−}. We assume that these lattices are “irreducible” in the sense that the projection maps from G to its components are injective on Γ^{+} and Γ^{−}.

The following conjecture was communicated by H. Oh:

Conjecture 54 (G. Margulis, A. Selberg). If the group ⟨Γ^{+}, Γ^{−}⟩ is discrete, then it an arithmetic lattice in G.

It was observed in [151] that Conjecture 54 for k ≥ 3 follows from Conjecture 29. In particular, one can show that the Hausdorff dimension of the set of irreducible lattices Γ^{+} ⊂ U^{+} for which ⟨Γ^{+}, Γ^{−}⟩ is discrete for some irreducible lattice Γ^{−} ⊂ U^{−} is exactly k (see [152]).

11. Symbolic coding

Symbolic dynamics plays important role in the study of Anosov flows (see, for example, [27]). In the case of surfaces of constant negative curvature, a symbolic representation of the geodesic flow in terms of a Markov chain can be given quite explicitly. Such constructions go back to M. Morse, E. Artin, and G. Hedlund. More recently, these constructions were generalized and improved by several authors (see [30, 173, 174, 175], [1, 2], [100, 75, 101, 102]). The geometric code of a geodesic is a biinfinite sequence of symbols that obtained by fixing a fundamental domain and recording which sides the geodesic hits along its pass. The arithmetic code of a geodesic is obtained by expanding the coordinates of the endpoints of the geodesic into a continued fraction expansion.

Question 55 (S. Katok). Construct analogs of the geometric coding and the arithmetic codings for the Weyl chamber flow (i.e., the action the diagonal group) on SL(n, ℝ)/SL(n, ℤ).

It was pointed out by B. Weiss that an interesting symbolic coding for the Weyl chamber flow was used in [181], where a (wrong) proof of the Littlewood conjecture was given (see also [179, 180, 9]).

One should mention that symbolic representations in higher dimensions are usually quite involved and not explicit. For example, any Markov partition of a hyperbolic toral automorphism on the 3-dimensional torus must consist of fractal sets (see [29]). F. Ledrappier and S. Mozes suggested to look for a convenient symbolic representation of the Weyl chamber flow using fractal tilings. This approach was successfully applied to construct explicit symbolic representations for some automorphisms and shifts on higher-dimensional tori (see [161, 13, 86, 87]) and for the Cartan action on GL(2, ℚ_{p}) × GL(2, ℚ_{q})/Γ, Γ a irreducible lattice (see [145, 147]).

References

[1] R. Adler and L. Flatto, Cross section maps for geodesic flows I (The Modular surface). Birkhäuser, Progress in Mathematics (ed. A. Katok) (1982), 103–161.

[2] R. Adler and L. Flatto, Cross section map for geodesic flow on the modular surface. Contemp. Math. 26 (1984), 9–23.

[3] Y. André, G-functions and geometry. Aspects of Mathematics, E13. Friedr. Vieweg and Sohn, Braunschweig, 1989.

[4] R. Aurich and F. Steiner, Quantum eigenstates of a strongly chaotic system and the scar phenomenon. Decidability and predictability in the theory of dynamical systems. Chaos Solitons Fractals 5 (1995), no. 2, 229–255.

[5] A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows. Preprint, 2004.

[6] M. Babillot, On the classification of invariant measures for horospherical foliations on nilpotent covers of negatively curved manifolds. Preprint, 2002.

[7] M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on abelian covers. Lie groups and ergodic theory (Mumbai, 1996), 1–32, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.

[8] A. Bäcker, R. Schubert and P. Stifter, On the number of bouncing ball modes in billiards. J. Phys. A 30 (1997), no. 19, 6783–6795.

[9] R. P. Bambah and A. C. Woods, Minkowski’s conjecture for n = 5; a theorem of Skubenko. J. Number Theory 12 (1980), no. 1, 27–48.

[10] E. S. Barnes and H. P. F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms I. Acta Math. 87 (1952). 259–323.

[11] E. S. Barnes and H. P. F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms II. Acta Math. 88 (1952). 279–316.

[12] A. Barnett, Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards. Preprint, 2004.

[13] T. Bedford, Generating special Markov partitions for hyperbolic toral automorphisms using fractals. Ergodic Theory Dynam. Systems 6 (1986), no. 3, 325–333.

[14] Y. Benoist and F. Labourie, Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables. Invent. Math. 111 (1993), no. 2, 285–308.

[15] V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms. Ann. of Math. (2) 150 (1999), no. 3, 977–1027.

[16] D. Berend, Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532.

[17] V. Beresnevich, A Groshev type theorem for convergence on manifolds. Acta Math. Hungar. 94 (2002), no. 1-2, 99–130.

[18] V. Bergelson, M. Boshernitzan and J. Bourgain, Some results on nonlinear recurrence. J. Anal. Math. 62 (1994), 29–46.

[19] M. V. Berry, Regular and irregular semiclassical wave functions. J. Phys. A 10 (1977), no. 12, 2083–2091.

[20] Y. Bilu, Limit distribution of small points on algebraic tori. Duke Math. J. 89 (1997), no. 3, 465–476.

[21] E. Bogomolny and C. Schmit, Superscars. Preprint, 2004.

[22] J. Bourgain, An approach to pointwise ergodic theorems. Geometric aspects of functional analysis (1986/87), 204–223, Lecture Notes in Math., 1317, Springer, Berlin, 1988.

[23] J. Bourgain, On the pointwise ergodic theorem on L^{p} for arithmetic sets. Israel J. Math. 61 (1988), no. 1, 73–84.

[24] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 (1988), no. 1, 39–72.

[25] J. Bourgain, Pointwise ergodic theorems for arithmetic sets. With an appendix by H. Furstenberg, Y. Katznelson and D. Ornstein. Inst. Hautes Études Sci. Publ. Math. No. 69 (1989), 5–45.

[26] R. Bowen, The equidistribution of closed geodesics. Amer. J. Math. 94 (1972), 413– 423.

[27] R. Bowen, Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.

[28] R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations. Israel J. Math. 26 (1977), no. 1, 43–67.

[29] R. Bowen, Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978), no. 1, 130–132.

[30] R. Bowen and C. Series, Markov maps associated with Fuchsian groups. Inst. Hautes Études Sci. Publ. Math. No. 50 (1979), 153–170.

[31] Y. Cai, Prime geodesic theorem. J. Théor. Nombres Bordeaux 14 (2002), no. 1, 59–72.

[32] P. Cartier, Some numerical computations relating to automorphic functions. pp. 37– 48, Computers in number theory. Proceedings of the Science Research Council Atlas Symposium No. 2 held at Oxford, Edited by A. O. L. Atkin and B. J. Birch. Academic Press, London-New York, 1971.

[33] P. Chatterjee and D. Morris, Divergent torus orbits in homogeneous spaces of ℚ-rank two. Preprint, 2004.

[34] Y. Cheung, Hausdorff dimension of the set of nonergodic directions. With an appendix by M. Boshernitzan. Ann. of Math. (2) 158 (2003), no. 2, 661–678.

[35] B. Cipra, R. Hanson and A. Kolan, Periodic trajectories in right-triangle billiards. Phys. Rev. E (3) 52 (1995), no. 2, 2066–2071.

[36] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien. Comm. Math. Phys. 102 (1985), 497–502.

[37] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems 1 (1981), no. 4, 431–450.

[38] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359 (1985), 55–89.

[39] D. Damjanovich and A. Katok, Local rigidity of partially hyperbolic actions on the torus. Preprint, 2004.

[40] A. Deitmar, The prime geodesic theorem for higher rank spaces. Preprint, 2003.

[41] C. Drutu, Diophantine approximation on rational quadrics. Preprint, 2003.

[42] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92 (1988), 73-90.

[43] H. A. Dye, On groups of measure preserving transformation. I. Amer. J. Math. 81 (1959), 119-159.

[44] B. Eckhardt, S. Fishman, J. Keating, O. Agam, J. Main and K. Müller, Approach to ergodicity in quantum wave functions. Phys. Rev. E 52 (1995), 5893–5903.

[45] S. J. Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces. pp. 133–155, Progress in Mathematics 195, Birkhauser, 2001.

[46] S. J. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties. Ann. Math. 157 (2003), 1–25.

[47] M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture. Preprint, 2004.

[48] A. Eskin, Counting problems and semisimple groups. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 539–552.

[49] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann. of Math. 147 (1998) 93-141.

[50] A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori. Preprint.

[51] A. Eskin, J. Marklof and D. Witte Moris, Unipotent flows on the space of branched covers of Veech surfaces. Preprint, 2004.

[52] A. Eskin, H. Masur and M. Schmoll, Billiards in rectangles with barriers. Duke Math. J. 118 (2003), no. 3, 427–463.

[53] A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties. Ann. of Math. (2) 143 (1996), no. 2, 253–299.

[54] F. T. Farrell and L. E. Jones, Anosov diffeomorphisms constructed from π_{1}(Diff(S_{n})). Topology 17 (1978), no. 3, 273–282.

[55] M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems. Phys. Rev. A 34 (1986), 591–595.

[56] D. Fisher, A canonical arithmetic quotient for actions of lattices in simple groups. Israel J. Math. 124 (2001), 143–155.

[57] D. Fisher, On the arithmetic structure of lattice actions on compact spaces, Ergodic Theory and Dynamical Systems, 22 (2002), no. 4, 1141–1168.

[58] D. Fisher, Rigid geometric structures and representations of fundamental groups, in Rigidity in Geometry and Dynamics, proceedings of a Newton Institute conference, 135-148, (eds. M.Burger and A.Iozzi) Springer-Verlag, NY, 2002.

[59] D.Fisher, Deformations of group actions, Preprint, 2004.

[60] D. Fisher and G. Margulis, Local rigidity of partially hyperbolic standard actions of higher rank groups and lattices. Preprint, 2004.

[61] L. Flaminio and A. Katok, Rigidity of symplectic Anosov diffeomorphisms on lowdimensional tori. Ergodic Theory Dynam. Systems 11 (1991), no. 3, 427–441.

[62] J. Franks, Anosov diffeomorphisms. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) p. 61–93 Amer. Math. Soc., Providence, R.I., 1970.

[63] J. Franks and R. Williams, Anomalous Anosov flows. Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 158– 174, Lecture Notes in Math., 819, Springer, Berlin, 1980.

[64] H. Furstenberg, Stiffness of group actions. Lie groups and ergodic theory (Mumbai, 1996), 105–117, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.

[65] P. Gallagher, Metric simultaneous diophantine approximation. J. London Math. Soc. 37 (1962), 387–390.

[66] G. Galperin, A. Stepin and Ya. Vorobets, Periodic billiard trajectories in polygons: generation mechanisms. Russian Math. Surveys 47 (1992), no. 3, 5–80.

[67] R. Gangolli, The length spectrum of some compact manifolds of negative curvature. J. Diff. Geom. 12 (1977), 403–426.

[68] R. Gangolli and G. Warner, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one. Nagoya Math. J. 78 (1980), 1–44.

[69] É. Ghys, Holomorphic Anosov systems. Invent. Math. 119 (1995), 585–614.

[70] É. Ghys, Codimension one Anosov flows and suspensions. Dynamical systems, Valparaiso 1986, 59–72, Lecture Notes in Math., 1331, Springer, Berlin, 1988.

[71] H. J. Godwin, On a conjecture of Barnes and Swinnerton-Dyer. Proc. Cambridge Philos. Soc. 59 (1963), 519–522.

[72] E. Goetze and R. Spatzier, Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices. Ann. Math. 150(3) (1999), 743–773.

[73] A. Gorodnik and B. Weiss, Distribution of orbits of lattices on homogeneous varieties. Preprint, 2004.

[74] Y. Guivarc’h and A. Starkov, Orbits of linear group actions, random walks on homogeneous spaces, and toral automorphisms. To appear in Ergodic Theory Dynam. Systems.

[75] B. Gurevich and S. Katok, Arithmetic coding and entropy for the positive geodesic flow on the modular surface. Mosc. Math. J. 1 (2001), no. 4, 569–582.

[76] E. Gutkin and A. Katok, Weakly mixing billiards. Holomorphic dynamics (Mexico, 1986), 163–176, Lecture Notes in Math., 1345, Springer, Berlin, 1988.

[77] L. Halbeisen and N. Hungerbühler, On periodic billiard trajectories in obtuse triangles. SIAM Rev. 42 (2000), no. 4, 657–670.

[78] G. Harcos and P. Michel, The subconvexity problem for Rankin-Selberg L functions and Equidistribution of Heegner points II. Preprint, 2004.

[79] D. Hejhal, The Selberg trace formula for PSL_{2}(ℝ) I. Springer Lecture Notes 548, 1976.

[80] D. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups. Emerging applications of number theory (Minneapolis, MN, 1996), 291–315, IMA Vol. Math. Appl., 109, Springer, New York, 1999.

[81] D. Hejhal and B. Rackner, On the topography of Maass waveforms for PSL(2, ℤ). Experiment. Math. 1 (1992), no. 4, 275–305.

[82] E. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53 (1984), no. 16, 1515–1518.

[83] F. Holt, Periodic reflecting paths in right triangles. Geom. Dedicata 46 (1993), no. 1, 73–90.

[84] W. P. Hooper and R. E. Schwartz, Introduction to McBilliards. Preprint, 2004; http://www.math.umd.edu/ res/Billiards/index.html.

[85] S. Hurder, Rigidity for Anosov actions of higher rank lattices. Ann. of Math. (2) 135 (1992), no. 2, 361–410.

[86] S. Ito and M. Kimura, On Rauzy fractal. Japan J. Indust. Appl. Math. 8 (1991), no. 3, 461–486.

[87] S. Ito and M. Ohtsuki, Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms. Tokyo J. Math. 16 (1993), 441-472.

[88] H. Iwaniec, Prime geodesic theorem. J. Reine Angew. Math. 349 (1984), 136–159.

[89] D. Jakobson, Quantum unique ergodicity for Eisenstein series on PSL_{2}(ℤ)\PSL_{2}(ℝ). Ann. Inst. Fourier (Grenoble) 44 (1994), no. 5, 1477–1504.

[90] B. Kalinin and V. Sadovskaya, On local and global rigidity of quasi-conformal Anosov diffeomorphisms. J. Inst. Math. Jussieu 2 (2003), no. 4, 567–582.

[91] L. Kaplan and E. Heller, Linear and nonlinear theory of eigenfunction scars. Ann. Physics 264 (1998), no. 2, 171–206.

[92] A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576.

[93] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms. Israel J. Math. 75 (1991), no. 2-3, 203–241.

[94] A. Katok and J. Lewis, Global rigidity for lattice actions on tori and new examples of volume preserving actions. Israel J. Math. 93 (1996), 253–281.

[95] A. Katok, J. Lewis and R. Zimmer, Cocycle superrigidity and rigidity for lattice actions on tori. Topology 35 (1996), no. 1, 27–38.

[96] A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 131–156.

[97] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Proc. Steklov Inst. Math. 1997, no. 1 (216), 287– 314.

[98] A. Katok and A. Stepin, Approximations in ergodic theory. Uspehi Mat. Nauk 22 (1967), no. 5, 81–106.

[99] A. Katok and A. Zemljakov, Topological transitivity of billiards in polygons. Mat. Zametki 18 (1975), no. 2, 291–300.

[100] S. Katok, Coding of closed geodesics after Gauss and Morse. Geom. Dedicata 63 (1996), no. 2, 123–145.

[101] S. Katok and I. Ugarcovici, Geometrically Markov geodesics on the modular surface, to appear in Moscow Mathematical Journal.

[102] S. Katok and I. Ugarcovici, Arithmetic coding of geodesics on the modular surface via continued fractions, Preprint.

[103] S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124 (1986), no. 2, 293–311.

[104] K. Kimoto and M. Wakayama, Equidistribution of holonomy restricted to a homology class about closed geodesics. Forum Math. 14 (2002), no. 3, 383–403.

[105] D. Kleinbock, Baker-Sprindžuk conjectures for complex analytic manifolds. Algebraic Groups and Arithmetic (TIFR, India, 2004), 539–553, Amer Mathematical Society, 2004.

[106] D. Kleinbock, Diophantine exponents of measures: a dynamical approach. Preprint, 2004.

[107] D. Kleinbock, E. Lindenstrauss and B. Weiss, On fractal measures and Diophantine approximation. To appear in Selecta Math.

[108] D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces. Sinaĭ’s Moscow Seminar on Dynamical Systems, 141–172, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996.

[109] D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), no. 1, 339–360.

[110] D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Preprint, 2004.

[111] D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Preprint, 2004.

[112] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7 (1997), 755–782.

[113] G. Knieper, The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank. Preprint, 2004.

[114] S. Koyama, Prime geodesic theorem for arithmetic compact surfaces. Internat. Math. Res. Notices 8 (1998), 383–388.

[115] É. I. Kovalevskaya, Strongly jointly extremal manifolds. Vestī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 1987, no. 6, 16–19.

[116] É. I. Kovalevskaya, Simultaneously extremal manifolds. Mat. Zametki 41 (1987), no. 1, 3–8.

[117] F. Labourie, Large groups actions on manifolds. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 371– 380.

[118] J. Lewis, Infinitesimal rigidity for the action of SL(n, ℤ) on 𝕋^{n}. Trans. Amer. Math. Soc. 324 (1991), no. 1, 421–445.

[119] B. Li and B. Hu, Statistical analysis of scars in stadium billiard. J. Phys. A 31 (1998), no. 2, 483–504.

[120] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity. to appear in Annals of Math.

[121] E. Lindenstaruss, Rigidity of multiparameter actions. Preprint, 2004.

[122] E. Lindenstrauss, Arithmetic quantum unique ergodicity and adelic dynamics. In preparation.

[123] A. Lubotzky and R. Zimmer, A canonical arithmetic quotient for simple Lie group actions. Lie groups and ergodic theory (Mumbai, 1996), 131–142, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998.

[124] A. Lubotzky and R. Zimmer, Arithmetic structure of fundamental groups and actions of semisimple Lie groups. Topology 40 (2001), no. 4, 851–869.

[125] W. Luo, Z. Rudnick and P. Sarnak, On Selberg’s eigenvalue conjecture. Geom. Funct. Anal. 5 (1995), no. 2, 387–401.

[126] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL(2, ℤ)\ℍ^{2}. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 207–237.

[127] A. Manning, There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96 (1974), 422–429.

[128] G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature. (Russian) Funkcional. Anal. i Priložen. 3 (1969), no. 4, 89–90.

[129] G. A. Margulis, Discrete subgroups and ergodic theory. Number theory, trace formulas and discrete groups (Oslo, 1987), 377–398, Academic Press, Boston, MA, 1989.

[130] G. A. Margulis, Problems and conjectures in rigidity theory. Mathematics: frontiers and perspectives, 161–174, Amer. Math. Soc., Providence, RI, 2000.

[131] G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices. Ergodic Theory Dynam. Systems 21 (2001), no. 1, 121–164.

[132] G. A. Margulis and G. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116 (1994), no. 1-3, 347–392.

[133] J. Marklof, The Berry-Tabor conjecture. European Congress of Mathematics, Vol. II (Barcelona, 2000), 421–427, Progr. Math., 202, Birkhäuser, Basel, 2001.

[134] J. Marklof, Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math. (2) 158 (2003), no. 2, 419–471

[135] J. Marklof, Pair correlation densities of inhomogeneous quadratic forms. II. Duke Math. J. 115 (2002), no. 3, 409–434; correction 120 (2003), no. 1, 227–228

[136] J. Marklof and A. Strömbergsson, Equidistribution of Kronecker sequences along closed horocycles. Geom. Funct. Anal. 13 (2003), no. 6, 1239–1280.

[137] H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 215–228, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.

[138] H. Masur, The growth rate of trajectories of a quadratic differential. Ergodic Theory Dynam. Systems 10 (1990), no. 1, 151–176.

[139] H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66 (1992), no. 3, 387-442.

[140] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2) 134 (1991), no. 3, 455-543.

[141] Masur and S. Tabachnikov, Rational billiards and flat structures. Handbook of dynamical systems, Vol. 1A, 1015–1089, North-Holland, Amsterdam, 2002.

[142] S. Matsumoto, On the global rigidity of split Anosov ℝ^{n}-actions. J. Math. Soc. Japan 55 (2003), no. 1, 39–46.

[143] C. McMullen, Dynamics of SL(2, ℝ) over moduli space in genus two. Preprint, 2003.

[144] D. Witte Morris, Ratner s Theorem on Unipotent Flows, to appear in Chicago Lectures in Mathematics Series of the University of Chicago Press.

[145] S. Mozes, On closures of orbits and arithmetic of quaternions. Israel J. Math. 86 (1994), no. 1-3, 195–209.

[146] S. Mozes, Epimorphic subgroups and invariant measures. Ergodic Theory Dynam. Systems 15 (1995), no. 6, 1207–1210.

[147] S. Mozes, Actions of Cartan subgroups. Israel J. Math. 90 (1995), no. 1-3, 253–294.

[148] R. Muchnik, Orbits of Zariski dense semigroups of SL(n, ℤ). To appear in Ergodic Theory Dynam. Systems.

[149] R. Muchnik, Semigroup actions on 𝕋^{n}. To appear in Geom. Dedicata.

[150] H. Oh, Finiteness of compact maximal flats of bounded voume. Ergodic Theory Dynam. Systems 24 (2004), 217–225.

[151] H. Oh, On a problem concerning arithmeticity of discrete groups acting on ℍ^{2} × ⋯ × ℍ^{2}. To appear in the proceedings of conference ”Algebraic groups and Arithmetic”.

[152] H. Oh, Application of the paper of Einsiedler, Katok and Lindenstrauss on the arithmeticity of some discrete subgroups. Preprint, 2004.

[153] F. Oort, Canonical liftings and dense sets of CM-points. pp. 229–234, Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997.

[154] D. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164.

[155] M. Pollicott, Closed geodesic distribution for manifolds of non-positive curvature. Discrete Contin. Dynam. Systems 2 (1996), no. 2, 153–161.

[156] M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces. Amer. J. Math. 120 (1998), no. 5, 1019–1042.

[157] A. Pollington and S. Velani, On simultaneously badly approximable numbers. J. London Math. Soc. (2) 66 (2002), no. 1, 29–40.

[158] N. Qian, Tangential flatness and global rigidity of higher rank lattice actions. Trans. Amer. Math. Soc. 349(2) (1997), 657–673.

[159] M. Ratner, On Raghunathan’s measure conjecture. Ann. of Math. (2) 134 (1991), no. 3, 545–607.

[160] M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63 (1991), no. 1, 235–280.

[161] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), no. 2, 147–178.

[162] F. Rodriguez Hertz, Global rigidity of ℤ^{2} actions on 𝕋^{3}. Preprint, 2004.

[163] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys. 161 (1994), no. 1, 195–213.

[164] Z. Rudnick and P. Sarnak, The pair correlation function of fractional parts of polynomials. Comm. Math. Phys. 194 (1998), 61–70.

[165] Z. Rudnick, P. Sarnak and A. Zaharescu, The distribution of spacings between the fractional parts of n^{2}α. Invent. Math. 145 (2001), 37–57.

[166] O. Sarig, Invariant Radon measures for horocycle flows on Abelian covers. To appear in Inv. Math.

[167] P. Sarnak, Values at integers of binary quadratic forms. Harmonic Analysis and Number Theory (Montreal, PQ, 1996), 181-203, CMS Conf. Proc. 21, Amer. Math. Soc., Providence, RI, 1997.

[168] P. Sarnak and M. Wakayama, Equidistribution of holonomy about closed geodesics. Duke Math. J. 100 (1999), no. 1, 1–57.

[169] P. Sarnak, Spectra of hyperbolic surfaces. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441–478.

[170] W. M. Schmidt, Open problems in Diophantine approximation. Diophantine approximations and transcendental numbers (Luminy, 1982), 271–287, Progr. Math., 31, Birkhäuser Boston, Boston, MA, 1983.

[171] R. E. Schwartz, A very short proof that every right triangle has a periodic billiard path. Preprint, 2004; http://www.math.umd.edu/ res/Java/App46/test1.html.

[172] R. E. Schwartz, Slightly obtuse triangles have periodic billiard paths. In preparation.

[173] C. Series, Symbolic dynamics for geodesic flows. Acta Math. 146 (1981), no. 1-2, 103– 128.

[174] C. Series, The modular surface and continued fractions. J. London Math. Soc. (2) 31 (1985), no. 1, 69–80.

[175] C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergodic Theory Dynam. Systems 6 (1986), no. 4, 601–625.

[176] N. Shah, Limit distributions of polynomial trajectories on homogeneous spaces. Duke Math. J. 75 (1994), no. 3, 711–732.

[177] N. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 2, 105–125.

[178] L. Silberman and A. Venkatesh, Quantum unique ergodicity on locally symmetric spaces I. Preprint, 2004.

[179] B. F. Skubenko, A proof of Minkowski’s conjecture on the product of n linear inhomogeneous forms in n variables for n ≤ 5. Investigations in number theory, 2. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 33 (1973), 6–36.

[180] B. F. Skubenko, A new variant of the proof of the inhomogeneous Minkowski conjecture for n = 5. Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 142 (1976), 240–253.

[181] B. F. Skubenko, Minima of a decomposable cubic form in three variables. J. Soviet Math. 53 (1991), no. 3, 302–310.

[182] A. I. Šnirel’man, Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29 (1974), 181–182.

[183] R. Spatzier, PhD thesis, Warwick University, 1983.

[184] D. C. Spencer, The lattice points of tetrahedra. J. Math. Phys. Mass. Inst. Tech. 21 (1942). 189–197.

[185] A. Starkov, Orbit closures of toral automorphism groups I-II. Preprint, 2000.

[186] L. Szpiro, E. Ullmo, and S. Zhang, Équirépartition des petits points. Invent. Math. 127 (1997), no. 2, 337–347.

[187] G. Tanner, How chaotic is the stadium billiard? A semiclassical analysis. J. Phys. A 30 (1997), no. 8, 2863–2888.

[188] G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces. Duke Math. J. 119 (2003), no. 2, 367–392.

[189] S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons. Regular and Chaotic Dynamics 9 (2004), 1–12.

[190] P. Tzermias, The Manin-Mumford conjecture: a brief survey. Bull. London Math. Soc. 32 (2000), no. 6, 641–652.

[191] E. Ullmo, Positivité et discrétion des points algébriques des courbes. Ann. of Math. (2) 147 (1998), no. 1, 167–179.

[192] W. Veech, The metric theory of interval exchange transformations I. Generic spectral properties. Amer. J. Math. 106 (1984), no. 6, 1331–1359.

[193] W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97 (1989), no. 3, 553–583.

[194] W. Veech, The billiard in a regular polygon. Geom. Funct. Anal. 2 (1992), no. 3, 341–379.

[195] A. Verjovsky, Codimension one Anosov flows. Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 49–77.

[196] Ya. Vorobets, Ergodicity of billiards in polygons. Sb. Math. 188 (1997), no. 3, 389–434.

[197] T. Watson, Rankin triple products and quantum chaos. Ph.D. thesis, Princeton University, 2001.

[198] B. Weiss, Almost no points on a Cantor set are very well approximable. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2008, 949–952.

[199] B. Weiss, Dynamics on parameter spaces: submanifold and fractal subset questions. Rigidity in dynamics and geometry (Cambridge, 2000), 425–440, Springer, Berlin, 2002.

[200] B. Weiss, Divergent trajectories on noncompact parameter spaces. Geom. Funct. Anal. 14 (2004), no. 1, 94–149.

[201] B. Weiss, Divergent trajectories and ℚ-rank. Preprint, 2004.

[202] M. Wierdl, Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64 (1988), no. 3, 315–336.

[203] A. Yafaev, A conjecture of Yves André. Preprint, 2004.

[204] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55 (1987), 919–941.

[205] S. Zelditch, Trace formula for compact Γ\PSL_{2}(ℝ) and the equidistribution theory of closed geodesics. Duke Math. J. 59 (1989), no. 1, 27–81.

[206] S. Zelditch, Selberg trace formulae and equidistribution theorems for closed geodesics and Laplace eigenfunctions: finite area surfaces. Mem. Amer. Math. Soc. 96 (1992).

[207] S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards. Comm. Math. Phys., 175 (1996), 673–682.

[208] S. Zhang, Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, 159–165.

OPEN PROBLEMS 33

[209] R. Zimmer. Actions of semisimple groups and discrete subgroups. Proc. Int. Congress of Mathematicians (Berkeley, CA, 1986), p. 1247–1258.