## Rice Dynamics Workshop

Abstracts

Continued fraction digit averages and MacLaurin's Inequalities, Francesco Cellarosi (University of Illinois, Urbana-Champaign)
A classical result of Kinchin says that for almost every real number x, the geometric mean of the first n digits in the continued fraction expansion of x converges to a number K=2.685... as n tends to infinity. On the other hand, for almost every x, the arithmetic mean of the first n digits tends to infinity. There is a sequence of refinements of the classical Arithmetic Mean - Geometric Mean inequality (called MacLaurin's inequalities) involving the k-th root of the k-th elementary symmetric mean, where k ranges from 1 (arithmetic mean) to n (geometric mean). We analyze what happens to these means for typical real numbers, when k is a function of n. We obtain sufficient conditions to ensure convergence / divergence of such means. Joint work with Steven J. Miller and Jake L. Wellens.

The set of uniquely ergodic 4 (or more) IETs is path connected, Jon Chaika (University of Utah)Let pi be an irreducible non-degenerate permutation on d symbols. If d is at least 4 then the set of IETs with permutation pi is path connected. This is joint work with S. Hensel.

Topological mixing for some residual sets of interval exchange transformations, Jon Fickenscher (Princeton University)We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing. This is joint work with Jon Chaika.

Anosov bundles, Andrey Gogolev (Binghamton University and Stony Brook University)Consider a non-trivial fiber bundle M -> E -> B whose total space E is compact and whose base B is simply connected. Can one equip E with a diffeomorphism or a flow which preserves each fiber and whose restriction to each fiber is Anosov? In a joint work with Tom Farrell we answer this question negatively and give an application to geometry of negatively curved bundles. However, in a joint work with Pedro Ontaneda and Federico Rodriguez Hertz, we construct non-trivial bundles that admit fiberwise Anosov diffeomorphisms that permute the fibers.

Entropy type invariants for actions of higher rank abelian groups, Anatole Katok (Penn State University)It is well-known that the standard notion of Kolmogorov (measure-theoretic) entropy as well as topological entropy, are useless for smooth actions of any compact-generated group that is not a compact extension of a cyclic group: those invariants simply vanish for any smooth action. Proper information is provided by the entropy function that in the case of Z^k or R^k actions has the form of piece-wise linear norm or a semi-norm. Out of this function two useful numerical invariants are fashioned : Fried average entropy and slow entropy. They are invariant under group automorphisms in the discrete case and under volume-preserving group automorphisms in the continuous case.

In this talk I will survey recent work on those entropies including Pesin and Ledrappier--Young type formulas for the slow entropy (C. Dong, 2014), rigidity results and connection between two entropies for maximal rank actions (A.K--S. Katok-- F. Rodriguez Hertz, GAFA, 2014), some observations on slow entropy for actions with individual elements of zero entropy, where no entropy function is available, and connections with Kakutani equivalence (work in progress)

Bernoulli actions and sofic entropy, David Kerr (Texas A&M University)By the work of Ornstein and Weiss, every factor of a Bernoulli action of a countable amenable group is again Bernoulli. In particular, every such Bernoulli action has completely positive entropy, meaning that every nontrivial factor has positive entropy. On the other hand, Popa's deformation-rigidity theory has demonstrated that many nonamenable groups, including those with property (T), have Bernoulli actions with non-Bernoulli factors. Nevertheless, we show that every Bernoulli action of a sofic group, independently of whether it admits a non-Bernoulli factor, has completely positive entropy.

Mean dimension and von Neumann-Lueck rank, Hanfeng Li (University of Buffalo, SUNY)Mean dimension is a numerical invariant for continuous actions of countable amenable groups, coming up in topological dynamics and related to entropy. The von Neumann-Lueck rank is an invariant for modules of the integral group ring of countable groups, coming up in L^{2}-invariants and related to L^{2}-Betti numbers. I will discuss the relation between the von Neumann-Lueck rank of a module of the integral group ring of a countable amenable group and the mean dimension of the associated algebraic action. This is joint work with Bingbing Liang.

We construct Wang tilings of the real Heisenberg group. We use Heisenberg Wang tilings to show that the discrete Heisenberg group admits a strongly aperiodic shift of finite type. This is joint work with Michael Schraudner and Ilie Ugarcovici.

Limiting distribution of expanding translates of sparse set of points on certain horosphere, Nimish Shah (Ohio State University)The orbit of certain expanding horosphere of dimension *mn* from the identity coset in *SL*(*m* + *n, R*) /* SL*(*n* + *m, Z*) intersects the orbit of its opposite and contracting horospherical in a dense set of rational points. We consider certain finite primitive collection of points with denominator *k* from this intersection and translate them by a diagonal elements expanding these points at certain rate depending on *k*, and describe their limit distributions, as *k* tends to infinity. This a joint work with M. Einsiedler, S. Mozes and U. Shapira.

Let *f* : ℂ → ℂ be an arbitrary
holomorphic endomorphism of degree larger than 1 of the Riemann
sphere ℂ. Denote by *J*(*f*) its Julia set. Let φ : *J*(*f*)
→ ℝ be a Hölder continuous function whose topological
pressure exceeds its supremum. It is known that then
there exists a unique equilibrium measure *μ _{φ}* for this
potential. I will discuss a special inducing scheme with fine
recurrence properties. This construction allows us to prove three
results.

(1) Dimension rigidity, i.e. a characterization of all maps and
potentials for which *HD*(*μ _{φ}*) =

*HD*(

*J*(

*f*)).

(2) Real analyticity of topological pressure *P*(*tφ*) as a function of
*t*.

(3) Exponential decay of correlations, and, as its consequence, the Central
Limit Theorem and the Law of Iterated Logarithm for Hölder
continuous observables. Finally, the Law of Iterated Logarithm for all
linear combinations of Hölder continuous observables and the
function log | *f'* |.

Geometric consequences of the Law of Iterated Logarithm lead to
comparison of equilibrium states with appropriately generalized
Hausdorff measures on the Julia set *J*(*f*).

We will discuss several recent joint works with Aaron Brown and Federico Rodriguez Hertz on actions by higher rank lattices on nilmanifolds. If the action lifts to the covering nilpotent Lie group and contains an element that acts hyperbolically on the fundamental group, then it allows a continuous semiconjugacy to an action by automorphisms. Under certain Anosov properties, the semiconjugacy is a smooth conjugacy.