2019 Maryland Dynamics Workshop Abstracts

Snir Ben-Ovadia: Generalized SRB Measures Abstract: SRB measures on Riemannian manifolds are an important object with many interesting properties. Hyperbolic ergodic SRB measures are especially interesting, since they admit the property of physicality- a positive Riemannian volume set of generic points, w.r.t Birkhoffs Ergodic Theorem. On the other hand, Hu and Young have introduced a large family of “well behaved hyperbolic systems which dont admit an SRB measure. We introduce the notion of GSRB measures (Generalized SRB), which are invariant, hyperbolic, conservative (perhaps infinite) measures with absolutely continuous conditional measures on unstable leaves, and which are finite on level sets. We show that GSRB measures are physical, w.r.t Hopfs Ergodic Theorem. We show that a GSRB measure exists if and only if the leaf condition is satisfied (i.e. an existence of one unstable leaf with a positive leaf volume for the hyperbolic points). In addition, we show uniqueness of GSRB measures on ergodic homoclinic classes. GSRB measures (analogously to SRB measures) can also be characterzied as equilibrium states of the geometric potential. We analyze an example by Hu and Young of a two-dimensional smooth hyperbolic system with no SRB measures, and show it admits a GSRB measure by the relatively-easy-to-check leaf condition.

Valerie´ Berthe´: Topological methods for symbolic discrepancy Abstract: We discuss in this lecture the notion of bounded symbolic discrepancy for subshifts from a topological dynamics viewpoint. Bounded discrepancy provides particularly strong convergence properties of ergodic sums toward frequencies. It is also closely related to the notions of balance in word combinatorics and of bounded remainder set. We focus on three families of shifts, namely hypercubic, substitutive and dendric shifts. For this latter family, we study and rely on their dimension group, providing necessary and suﬃcient conditions for two dendric subshifts to be (strong) orbit equivalent.

Brian Chung: Stationary measure and orbit closure classiﬁcation for random walks on surfaces Abstract: We study the problem of classifying stationary measures and orbit closures for non- abelian action on surfaces. In particular we show that under a certain average growth condition, the orbit closures are either ﬁnite or dense. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting. We then apply this theorem to two concrete set- tings, namely discrete perturbation of the standard map and Out(F2)-action on a certain character variety.

Vaughn Climenhaga: The measure of maximal entropy for geodesic flows without conjugate points Abstract: If M is a closed negatively curved Riemannian manifold, the geodesic flow has a unique measure of maximal entropy, which was described by Bowen and Margulis. When negative curvature is weakened to rank one nonpositive curvature, uniqueness of the measure of maximal entropy was proved by Knieper, using a Patterson-Sullivan construction. I will discuss joint work with Gerhard Knieper and Khadim War, in which the curvature condition is weakened even further, and describe a class of manifolds without conjugate points for which uniqueness of the measure of maximal entropy can be established using an approach based on the specification property. In

1 particular, our result applies to every surface without conjugate points whose genus is at least 2.

Maria Correia: Chaos and stability in a three-parameter family of billiard tables Abstract: In this work, we investigated a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric and convex lemon billiards, in which the latter classes comprise instances where the new parameter is zero. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. We characterize the periodic points and investigate the transition to ergodicity in the three-parameter umbrella families.

Diana Davis: Periodic paths on the pentagon Abstract: Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I’ll explain my joint work with Samuel Lelivre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising ”dense but not equidistributed” behavior.

Jacopo De Simoi: Spectral determination of analytic dispersing billiards Abstract: In this talk we will see how it is generically possible, in the analytic category, to recover the geometry of some class of dispersing billiards with some symmetries (three obstacles systems) from the purely dynamical data encoded in their Marked Length Spectrum. This is part of an ongoing project with V. Kaloshin, M. Leguil and P. Balint.

Jason Duvall: Schmidt’s game for Nonuniformly expanding circle maps Abstract: Schmidt’s game is a set-theoretic tool useful in analyzing certain sets arising from uniformly expanding self-maps of compact manifolds. Speciﬁcally, it is known that certain ”ex- ceptional” sets of points with nondense orbits under such maps are winning for Schmidt’s game; this implies that they have full Hausdorﬀ dimension despite being Lebesgue-null. We show how these results can be extended to Manneville-Pomeau maps, a class of nonuniformly expanding circle transformations.

Xueming Hui: Topological pressure for conservative C1-diffeomorphisms with no dominated splitting Abstract: We prove three formulas for computing topological pressure of C1-generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there is no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there is no equilibrium states. For C1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and φm, we show that there exist equilibrium states with zero entropy and there exists a transition point t0 for the family {tφm}t ≥ 0, such that there is no equilibrium states for t ∈ [0, t0) and there is an equilibrium state for t ∈ [t0, +∞).

2 Svetlana Jitomirskaya: The almost Mathieu cocycle: a study of (the boundary of) non-uniform hyperbolicity Abstract: The almost Mathieu cocycle with λ > 1 represents a prototypical hyperbolic analytic cocycle, and at λ = 1 a prototypical critical one. Hyperbolic Schrodinger cocycles, at energies in the spectrum are non-uniformly hyperbolic (in fact, spectral measures are supported on points where, at a given phase, the Oseledets multiplicative ergodic theorem does not hold coherently in both directions). The critical case represents a boundary between hyperbolicity and reducibility. It is also a model heavily studied in physics literature and linked to several Nobel prizes (in addition to one Fields medal). We will describe several results on this model, that resolve some long-standing conjectures pertaining to both its non-uniformly hyperbolic and critical cases, and represent the ﬁrst explicit study of several unusual dynamical phenomena in a natural setting.

Matthew Smith: Unique Ergodicity for Measured Foliations Abstract: Measured foliations on surfaces are a family of dynamical systems with invariant measures. These foliations are typically not orientable, but it is always possible to ﬁnd a canonical cover of the original surface with an orientable foliation. The orientation cover is useful, since more tools are available in the orientable case. However, the orientation cover can increase the number of measures. In this talk we will construct examples where the orientation cover is minimal but not ergodic, even if the original foliation was uniquely ergodic. We will also discuss some related constructions and questions.

Claire Merriman: Cutting sequences of Lehner and Farey expansions Abstract: I will connect Lehner and Farey continued fractions with digits (1, +1) and (2, −1) to cutting sequences of the geodesic ﬂows on the upper half plane. The connection between geodesics on the modular surface PSL(2,Z)\H and regular continued fractions was established by Series. The analogous construction for the Lehner and Farey expansions requires a subgroup of GL(2,Z) instead of SL(2,Z), but still provides a geometric proof of ergodicity of the continued fraction map.

Wenyu Pan: Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay d Abstract: We will start with the notion of Fourier dimension of a subset of R . We will then focus on the particular case of the limit sets of Kleinian Schottky groups and the asymptotic behavior of the Fourier transform of the Patterson-Sullivan measures. We will discuss the connection with the essential spectrum of the Selberg zeta functions for the hyperbolic manifolds and random walks on SL2(C). This is a joint work with Jialun Li and Frdric Naud.

Kiho Park: Quasi-multiplicativity of typical cocycles Abstract: We will discuss the notion of quasi-multiplicativity for ﬁber-bunched GLd(R) cocycles over subshifts of ﬁnite type, and show that typical (in the sense of Bonatti-Viana, Avila-Viana) cocycles are quasi-multiplicative. We will discuss a few corollaries including the structure of the pointwise Lyapunov spectrum and subadditive thermodynamics formalism of such cocycles.

Ronnie Pavlov: Minimal subsystems and ergodic measures for subshifts of linear complexity Abstract: A subshift X is said to have linear complexity if there exists a constant C so that

3 the complexity function cn(X) (the number of n-letter words in X) is bounded from above by Cn. We will discuss several recent results about this class of subshifts. The ﬁrst (with Nic Ormes) states that every non-degenerate transitive non-minimal subshift must have complexity growing more quickly than 1.5n along a sequence. The others (with Andrew Dykstra and Nic Ormes) give bounds on the number of minimal subsystems and generic measures of a transitive subshift X of linear complexity in terms of the associated constant C. The results about generic measures are related to recent work of Cyr and Kra and older work of Boshernitzan.

Anurag Rao: Diophantine approximation in Euclidean norm Abstract: We state a norm-sensitive Diophantine approximation problem arising from the work of Kleinbock-Wadleigh. While the case of the supremum norm was studied there extensively using techniques of continued fractions, we study the case of the Euclidean norm using a result of Maucourant on geodesic ﬂows for hyperbolic surfaces. Speciﬁcally, for a given approximation func- 2 αq−p q 2 tion ψ we establish a zero-one law on the set of α ∈ R for which the inequality ψ(t) + t < 1 has non-trivial integer solutions for all large enough t. Based on joint work with D. Kleinbock.

Victoria Sadovskaya: Boundedness and invariant metrics for diﬀeomorphism cocycles over hyperbolic systems Abstract: We consider a Holder continuous cocycle A over a hyperbolic dynamical system with values in the group of diﬀeomorphisms of a compact manifold M. We show that if the periodic data of A, i.e. the set of its return values along the periodic orbits in the base, is bounded in Cq, q > 1, then the set of values of the cocycle is bounded in Cr for each r less than q. Moreover, such a cocycle is isometric with respect to a Holder continuous family of Riemannian metrics on M.

Shrey Sanadhya: Cocycles of Borel automorphism groups Abstract: Cocycles of dynamical systems are an important tool for classiﬁcation of groups of transformations. In this talk we discuss the theory of cocycles for automorphism groups of standard Borel space. We introduce an invariant of cocyles called the set of essential values and discuss its properties. We provide conditions when a cocycle could be a coboundary. We also present a Borel version of Gottschalk-Hedlund theorem.

Scott Schmieding: The stabilized automorphism group of a subshift Abstract: The automorphism group Aut(σ) of a subshift (X, σ) consists of all homeomor- phisms φ: X → X such that φσ = σφ. When (X, σ) is a shift of ﬁnite type, Aut(σ) is known to have a rich group structure, and we’ll discuss some background and problems related to the study of Aut(σ). Finally, we’ll introduce a certain stabilized automorphism group, and outline results which, among other things, allow us to distinguish (up to isomorphism) the stabilized groups of various full shifts. This is joint work with Yair Hartman and Bryna Kra.

Brandon Seward: Positive entropy actions of countable groups factor onto Bernoulli shifts Abstract: I will discuss extending the classical factor theorem of Sinai from actions of countable amenable groups to actions of general countable groups. Speciﬁcally, I will show that if a free ergodic action of a countable group has positive Rokhlin entropy (or, less generally, positive soﬁc entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy.

4 Longmei Shu: Attractors of Isospectral Reductions Abstract: Isospectral reductions based on specific characteristics on networks form a dynamical system. Such a dynamical system starts at a finite network and converges to an attractor. This attractor is a smaller network where the chosen characteristic has the same value for all nodes/edges. We demonstrate that isospectral reductions of the same network by different characteristics may converge to the same as well as different attractors. We also show that networks spectrally equivalent with respect to one characteristic could be not spectrally equivalent for another characteristic.

Diaaeldin Taha: Symmetric Farey and Gauss Maps for the Hecke Triangle Groups Gq, and Cross Sections to the Geodesic Flow on SL(2, R) Abstract: In previous work, we defined Stern-Brocot trees for the discrete orbits Λq = T 2 Gq(1, 0) of the linear action of Gq on the plane R , and used them to derive an explicit cross section to the horocycle flow on SL(2, R)/Gq. In this short talk, we explore a continued fraction algorithm related to the discrete sets Λq, deriving symmetric Farey and Gauss maps from it, and using them to derive an explicit cross section to the geodesic flow on SL(2, R)/Gq.

Nattalie Tamam: Diagonalizable groups with non-obvious divergent trajectories Abstract: Singular vectors are the ones for which Dirichlets theorem can be inﬁnitely improved. For example, any rational vector is singular. The sequence of approximations for any rational vector q is ’obvious’; the tail of this sequence contains only q. In dimension one, the rational numbers are the only singulars. However, in higher dimensions there are additional singular vectors. By Dani’s correspondence, the singular vectors are related to divergent trajectories in Homogeneous dynamical systems. A corresponding ’obvious’ divergent trajectories can also be deﬁned. We will discuss the existence of non-obvious divergent trajectories for the actions of diagonalizable groups and their relation to Diophantine properties.

Dan Thompson: The K-property for equilibrium states in non-positive curvature. Abstract: Equilibrium states for geodesic ﬂows over compact rank 1 manifolds and suﬃciently regular potential functions were studied recently by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I will describe new joint work with Ben Call, which shows that these equilibrium states have the Kolmogorov property. When the manifold has dimension at least 3 (for example, the interesting case of the Gromov example of a graph manifold) this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy.

Dominic Veconi: Equilibrium states for almost-Anosov diffeomorphisms Abstract: Almost-Anosov diffeomorphisms form a mild generalization of Anosov diffeomorphisms, and thus make for a good class of maps to investigate when studying nonuniformly hyperbolic behavior. In this talk, I will discuss existence and uniqueness of equilibrium states for non-Hlder geometric potentials using Young towers. I will additionally discuss how one can use thermodynamics of hyperbolic towers to prove exponential decay of correlations and the central limit theorem for these equilibrium measures.

Andrew Warren: Fluctuation bounds for amenable ergodic averages

5 Abstract: It is known that for the classical ergodic theorems of von Neumann and Birkhoff, no uniform rate of convergence exists. Nonetheless it turns out to be possible to explicitly find other kinds of uniform convergence information, such as uniform bounds on the number of fluctuations or uniform bounds on the rate of metastability. Here, we show that a similar phenomena holds generally in the ergodic theory of amenable groups actions, by deducing an explicit fluctuation bound for the amenable mean ergodic theorem along Flner sequences which satisfy an analog of Lindenstrauss’s temperedness condition. We will also discuss some partial results which are known in the pointwise case.

Jinxin Xue: Rigidity of a class of Abelian-by-cyclic groups acting on torus n Abstract: For any matrix B ∈ SL(n, Z), one can associate the semi-direct product Z nB Z , where Z acts on Zn by the natural action of B on Zn. This is a solvable group called Abelian-by- cyclic group. It has a natural action on torus, where B acts via multiplying a linear Anosov element and Zn acts by translations. This is a setting where hyperbolic and elliptic dynamics coexist. In this talk, we explain our work on classifying this class of group actions on torus. The local rigidity can be obtained by KAM method. In the two dimensional case, we obtain a complete classiﬁcation using a version of Tits alternative for the diﬀeomorphism group. This is a joint work with A. Wilkinson and S. Hurtado.

Yun Yang: Non-stationary almost sure invariance principle for hyperbolic systems with singularities Abstract: In this talk, we will investigate a wide class of two-dimensional uniformly hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for bounded dy- namically H older observables, with a sharp rate. These results apply to Sinai dispersing billiards and their perturbations. This is a joint work with Jianyu Chen and Hongkun Zhang.

Han Zhang: Limit distributions of polynomial trajectories on homogeneous spaces averaging on expanding boxes Abstract: Let G be a semisimple linear algebraic real Lie group and Γ be a closed subgroup of k G such that G/Γ admits a G-invariant probability measure. Let Θ : R → G be a regular algebraic k map deﬁned over R and {Bn}n∈N be a sequence of expanding boxes in R . Under certain conditions on {Bn}n∈N and the map Θ, we generalize Shah’s result on limit distribution of polynomial trajectories on G/Γ averaging on Bn as n → ∞. The main idea is applying Ratner’s measure clas- siﬁcation Theorem through rescaling boxes and modifying (C, α)-good maps by continuous maps. This is a joint work with Nimish Shah.

Runlin Zhang: Translates of some infinite homogeneous measures in finite volume homogeneous spaces Abstract: Let X be G quotient by an arithmetic lattice and m be a homogeneous measure supported on a closed H orbit for a reductive subgroup H. We are interested in studying the translates of such a measure. Using Ratner’s rigidity theorem and the linearization technique of Dani-Margulis, it was shown by Eskin-Mozes-Shah that if the measure m is finite, then the limit measure is either a homogeneous measure or 0. Recently, Shapira-Zheng shows that similar things hold when G is the standard SLN and T is a maximal Q-split torus. We will discuss some generalizations of this result.