## Rice Dynamics Workshop

Abstracts

Diophantine polygonal billiards are ergodic, Giovanni Forni (University of Maryland)
Billiards in polygons are of two fundamentally different types: rational and non-rational. Rational billiards are pseudo-integrable, that is, their phase space is foliated by invariant surfaces. It was proved by Kerckhoff, Masur and Smillie in 1986 by Teichmuleller theory that the flow on almost all invariant surface is uniquely ergodic.

The only known results on the ergodicity of non-rational billiards have been obtained by fast approximation methods based on the rational case (Kerckhoff, Masur and Smillie, Vorobets) and apply to a zero measure G-delta dense set of polygons. In this talk we will formulate a new ergodicity criterion which applies to both the rational and non-rational case. From this criterion we derive an ergodicity theorem for non-rational billiards under a full measure Diophantine condition on the angles.

The proofs are based on methods of complex analysis in one-variable and analysis on Riemannian manifolds. Ergodicity of the polygonal billiard flow follows from bounds on the degeneration of the geometry along a certain deformation of the phase space. In the rational case this idea goes back to Masur in 1982 and has been revisited more recently by Cheung-Eskin and Trevino. In this case the invariant surfaces have a locally euclidean 2-dimensional geometry and the deformation is given by the Teichmueller flow. Our work generalizes this approach to the locally solvable 3-dimensional phase space of a non-rational polygonal billiard flow.

Unipotent flows and infinite measures, Amir Mohammadi (University of Texas at Austin)The theory of unipotent flows on homogeneous spaces obtained as a quotient of a Lie group by a lattice has been central object of study for few decades which has lead to a rich theory and several applications. Unipotent flows on infinite volume spaces, however, are far less understood; in this talk we will highlight the similarities and main differences between these two settings. This is a joint work with Hee Oh.

On random groups of intermediate growth, Rostislav Grigorchuk, Texas A&M UniversityIn 1968 it became apparent that all known classes off groups have either polynomial or exponential growth and John Milnor formally asked whether groups of intermediate growth exist. In 1984, I introduced the first such examples.

This continuum can be viewed as a Cantor subset X in the space of marked groups with a natural continuous map T:X---->X which preserves many group properties. In fact the dynamical system (X,T) is topologically conjugate to the one sided shift. I will explain why, for any reasonable T-invariant probability measure μ on X, a typical (i.e. μ-almost) property of a group from the family of groups X is to have growth bounded from above by a function of the type e^{{n{α}}}, where α = constant and α < 1.

At the same time a co-meager subset Y ⊂ X of groups has a different type of behavior at infinity, namely it has the so-called oscillating growth which I will define during the talk. At the beginning of the talk we will also discuss a general approach to randomness in group theory based on the use of the space of marked groups.

Cocycle rigidity of parabolic actions in SL(n, R), n>3, Zhenqi Wang (Yale University)For traditional horocycle flow in SL(2, R), the detailed analysis of the cohomological equation Uf = g was given by Flaminio-Forni. Their result was further applied in some rank-one models to obtain cocycle rigidity for higher rank parobolic actions. In this talk I will discuss a general result about cocycle rigidity of parabolic actions and will show how to use representation tool in higher rank Lie groups.

Renormalization and rigidity of circle maps, Sasa Kocic (University of Mississippi)Rigidity theory of circle diffeomorphisms, which concerns smooth conjugacy to a rigid rotation, is a classic problem in dynamical systems initiated by Arnold and settled by Herman and Yoccoz.

In joint work with Kostya Khanin, I obtained several results concerning the renormalization and rigidity of circle maps with breaks, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity. Our main result is the proof of the renormalization conjecture for circle maps with breaks, i.e., that renormalizations of any two *C ^{2+α}*-smooth (α > 0) circle maps with breaks, with the same irrational rotation number and the same size of the break, approach each other exponentially fast. (A similar claim is still an open problem for non-analytic critical circle maps.) As a corollary, we obtained a strong rigidity statement for such maps: for almost all irrational ρ, any two circle maps with breaks, with the same rotation number ρ and the same size of the break, are

*C*-smoothly conjugate to each other. As we also proved, the latter claim cannot be extended to all irrational rotation numbers.

^{1}In this talk, I will discuss the above results which are analogous to the case of circle diffeomorphisms. I will also present a more recent result, which stands in contrast to the case of circle diffeomorphisms: *C ^{r}*-smooth circle maps with breaks are generically not

*C*rigid.

^{1+ε}An action of a countable group is called totally nonfree if, generically, all points have distinct stabilizers. Every group has a universal nonfree action, namely, the action on the space of its own subgroups by conjugation.

A self-similar group is a transformation group acting on a regular rooted tree in a special way so that the action reproduces itself on subtrees. The talk is concerned with one class of self-similar groups, the branch groups, for which the natural action is totally nonfree. In particular, I will describe how the natural action can be recovered from the universal nonfree action.

Ergodic properties of m-free integers in number fields, Francesco Cellarosi (University of Illinois, Urbana-Champaign)For an arbitrary number field *K / Q* of degree *d*, we study the n-point correlations for *m*-free integers in the ring *O _{K}* and define an associated natural

*O*-action. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a

_{K}*Z*action on a compact abelian group. As a corollary, we obtain that this natural action is not weakly mixing and has zero measure-theoretical entropy. The case

^{d}*K = Q*, was studied by Ya.G. Sinai and myself, and our theorem provides a different proof to a result by P. Sarnak. This is a joint work with I. Vinogradov.

Consider a faithful Z^r -action on a nilmanifold N by automorphisms. Theorems of Berend, Katok-Spatzier and Einsiedler- Lindenstrauss assert that when N is a torus, if the action is hyperbolic, strongly irreducible and of rank at least 2, then invariant closed subsets and measures are very scarce. In this talk, we will explain how such rigidity properties pass from the base torus of a nilmanifold to the nilmanifold itself.

Global Rigidity of Higher Rank Abelian Anosov Algebraic Actions, Federico Rodriguez-Hertz (Penn State University)We proved that any algebraic Anosov action ρ on a nilmanifold without rank-1 factor is globally rigid, i.e. any action homotopic to ρ with one Anosov element is smoothly conjugated to ρ. This answer a question by A. Katok and R. Spatzier in the nilmanifold context. In this talk we plan to show some ideas of the proof, recall previous approaches, especially that of D. Fisher, B. Kalinin and R. Spatzier (JAMS, 2013) and discuss some related open problems. This is a joint work with Zhiren Wang.

The Fried entropy for smooth group actions and connections with algebraic number theory, Svetlana Katok (Penn State University) In a joint work with A. Katok and F. Rodriguez Hertz we study a meaningful numerical entropy invariant for smooth actions of higher rank abelian groups introduced by D. Fried in 1983, that we call the *Fried entropy*. We use algebraic number theory to obtain a lower bound for the Fried entropy of any Cartan action on an n-dimensional torus that goes to infinity with n, and prove that the Fried entropy is bounded away from zero by a constant independent on n. Arithmeticity of maximal rank smooth abelian actions proved by A. Katok and F. Rodriguez Hertz implies that the same estimates hold for the Fried entropy of any such action.

In this talk we will discuss some recent rate results for rigidity sequences of weakly mixing transformations. We will then discuss rate results for uniform rigidity sequences of weakly mixing homeomorphisms. Given a sequence that satisfies a given growth rate, we will show how to construct a Lebesgue measure-preserving homeorphism of the two-torus that is weakly mixing and has the prescribed uniform rigidity sequence.

Arithmeticity and topology of smooth actions of higher rank abelian groups, Anatole Katok (Penn State University)The "non-uniform measure rigidity" program is an outgrowth of the measure rigidity program for hyperbolic algebraic actions of higher rank abelian groups that includes both, the famous Furstenberg "x2, x3 problem" and the joint work with M. Einsiedler and E. Lindenstrauss that has applications to the Littlewood conjecture on Diophantine approximations. While the latter program is still not completed (and probably hopeless without major new ideas) since all the results are based on a positive entropy assumption, the methods developed in hyperbolic measure rigidity, together with the theory of non-uniform hyperbolicity (a. k. a. Pesin theory), have surprising applications way beyond the world of algebraic actions. In this talk I will discuss the most recent development in that direction.

Consider k>1 commuting diffeomorphisms of a (k+1)- dimensional manifold M and assume that they preserve a measure that is sufficiently "rich'' or "stochastic'', for example if all non-zero elements of the suspension action have positive entropy with respect to the measure.

A striking topological conclusion is that the manifold is a "slightly modified'' (finite factor of the) torus: there is homeomorphism from an open subset in finite cover of M that contains support of the lifted measure to the( k+1)-dimensional torus with a finite set removed. In particular, the fundamental group of a finite cover of M contains a free abelian subgroup or rank (k+1).

On the ergodic side we obtain precise information about the structure of the action. It is fully ``arithmetic'', i.e. exactly the same as an action by commuting hyperbolic matrices with integer entries and determinant 1 or -1 on the torus or its finite factor. There is further geometric information: the correspondence is smooth the sense of Whithey on a set whose complement has arbitrary small measure.

This is a joint work with Federico Rodriguez Hertz. The main technical ingredient is our joint result with Boris Kalinin (Annals of Math, 2011)