Learning Seminar 11:00 AM-12:00 PM, GC 5417
Tamara Kucherenko, CUNY - City College, New York, NY
Title: Phase Transitions in Symbolic Dynamics
Abstract: We discuss various mechanisms for generating phase transitions on compact symbolic systems. We present several results, classical and recent, concerning the number and frequency of phase transitions, as well as the existence of freezing phase transitions. In the latter case we focus on the type of potentials which would trigger a freezing phase transition and the support of the resulting ground state.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Yunping Jiang, CUNY - Queens College, New York, NY
Title: Holder vs Dini in Transfer Operators
Abstract: Ruelle transfer operators are central to thermodynamic formalism, providing critical insights into phenomena such as the existence and uniqueness of Gibbs measures, calculations of pressures and Hausdorff dimensions, and estimates of correlation decay rates. These operators exhibit distinct behaviors across different function spaces. This discussion will explore the contrast between transfer operators on Holder and Dini function spaces. I will discuss how the spectral gap can be used to analyze the exponential decay of correlations in expanding dynamical systems with Holder potentials. However, the spectral gap vanishes when the dynamical systems are not fully expanding, or the potentials are not Holder. In collaboration with Yuan-Ling Ye, we established an optimal quasi-spectral gap condition for studying transfer operators in weakly expanding systems with Dini potentials. This condition enables us to derive precise estimates for the decay rate of correlations.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Jayadev Athreya, University of Washington, Seattle, WA
Title: Rational Billiards, Translation Surfaces, and Dynamics
Abstract: We will discuss how to use the theory of translation surfaces to answer dynamical questions about billiards in rational polygons. Lots of pictures and examples!

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Reserved for discussion with Jayadev Athreya.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Christian Wolf, CUNY - City College, New York, NY
Title: Computability in Complex Dynamics: Old and New
Abstract: The study of dynamically defined sets and invariants from a computable analysis point of view has seen an enormous increase in recent years. Roughly speaking, a mathematical object is computable if there exists a Turing machine that approximates the object up to any pre-described accuracy. The goal of this talk is two-fold: First, we review some striking results about the computability of Julia sets due to Braverman and Yampolsky. Second, we will discuss how the one-dimensional theory could be generalized to higher dimensions. In particular, we establish a poly-time computability result for the chain recurrent set of Axiom A skew products in two complex variables. The results presented in this talk are joint work with Suzanne Boyd.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Melkana Brakalova-Trevithick, Fordham University, New York, NY
Title: On local behavior of q.c maps
Abstract: We will discus the asymptotic behavior of a q.c map at a Lebesgue point of its complex dilatation, i.e. at a point where the complex dilatation is approximately continuous and the circular dilatation may or may not be one. A special case includes points where the mapping is approximately asymptotically conformal.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: Henon like Renormalization
Abstract: See below.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Enrique Pujals, CUNY - Graduate Center, New York, NY
Title: Henon like Renormalization
Abstract: We will start discussing renormalization of dissipative diffeos of the disk and we will focus on the particular case of dissipative Henon-like maps (no necessary close to one-dimensional unimodal maps) presenting a renormalization theory that controls the small-scale geometry. In the morning talk we will discuss how renormalization has been used for unimodal maps to decompose the dynamics in transitive pieces and in the second part in the afternoon we will focus on the 2d-case This is part of a joint work with S. Crovisier, M. Lyubich and J. Yang.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Sylvain Crovisier, Université Paris-Saclay, Orsay Cedex, France
Title: Dynamical decompositions of diffeomorphisms
Abstract: n order to describe the dynamics of a diffeomorphism, one first attempts to decompose the system into invariant elementary pieces, that can be studied separately. In the case of hyperbolic dynamics, Smale's « spectral decomposition theorem » provides such a decomposition, with a finite number of pieces. This talk deals with the decomposition of general smooth diffeomorphisms that satisfy weaker forms of hyperbolicity and discusses the notion of homoclinic class.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Sylvain Crovisier, Université Paris-Saclay, Orsay Cedex, France
Title: Strong positive recurrence of diffeomorphisms
Abstract: The ergodic theory of differentiable dynamics is deeply understood under a uniformly hyperbolic assumption: the dynamics decomposes into a finite number of basic sets, most orbits equidistribute towards natural invariant measures (physical or maximizing the entropy), and their statistical properties are described by limit theorems. Significant progress has been made in extending these results to broader classes of non-uniformly hyperbolic systems. During this lecture, I will discuss an approach, that we recently developped with J. Buzzi and O. Sarig: the strong positive recurrence property (SPR). Focusing on mixing measures of maximal entropy, I will show that the SPR property implies exponential decay of correlations and a central limit theorem. Moreover, every smooth surface diffeomorphism with positive topological entropy satisfies this property.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Amie Wilkinson, University of Chicago, Chicago, Illinois
Title: Symmetry in Smooth Dynamics

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Amie Wilkinson, University of Chicago, Chicago, Illinois
Title: The Dynamics of Expanding Foliations

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Axel Kodat, CUNY - Graduate Center, New York, NY
Title: Generalized entropy of density subshifts
Abstract: Correa and Pujals have recently defined a modification of the topological entropy called the generalized entropy of a dynamical system. This invariant takes values in the lattice of orders of growth and thus—in contrast to the classical topological entropy and even other existing variants adapted to the study of zero entropy systems (e.g. polynomial entropy, slow entropy)—can detect differences in orbit growth rate at arbitrarily fine scales. Unfortunately, this dramatic increase in information density comes at a cost, in part because the space of orders of growth is difficult to fully describe and thus the generalized entropy of a system is not always easy to locate precisely in this space. In particular, the realization question—which orders of growth are obtainable as generalized entropies of (some class of) dynamical systems?—becomes interesting. In this talk we answer a question of Correa and Pujals by defining a simple class of subshifts within which it is easy to produce examples having strictly superpolynomial and subexponential order of growth. Via a general construction, we also show how to realize these examples as smooth diffeomorphisms of \(\mathbb{S}^3\). Finally, we comment on how these examples relate to the problem of formulating a variational principle applicable to generalized entropy.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Alena Erchenko, Dartmouth College, Hanover, NH
Title: Local product structure of equilibrium states for geodesic flows.
Abstract: How do we define geodesic flows on CAT(0) spaces? In this talk we will concentrate on two settings: rank 1 nonpositively curved manifolds and flat surfaces with cone singularities of angles larger than 2π. We will discuss known results on the existence and uniqueness of equilibrium states for "nice" potentials for geodesic flows in those settings and their properties. Then, we will show how to obtain a local product structure in these settings using the non-uniform Gibbs property following an idea of Vaughn Climenhaga for the uniformly hyperbolic diffeomorphisms. This is based on the joint work with Benjamin Call, David Constantine, Noelle Sawyer, and Grace Work.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Pat Hooper, CUNY - City College, New York, NY
Title: Pseudo-Anosov Homeomorphisms and Flat Surfaces
Abstract: I'll describe some of the basic objects and dynamical systems related to my afternoon talk. I'll describe some results about translation surfaces, and their hyperbolic affine automorphisms. I'll explain how these affine automorphisms renormalize the straight-line flow in eigendirections.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Pat Hooper, CUNY - City College, New York, NY
Title: How to tune your heptagonal television
Abstract: Pseudo-Anosov mapping classes have canonical representatives as hyperbolic affine automorphisms of flat surfaces. It is well known that the straight-line flow in an eigendirection of such an automorphism is uniquely ergodic. Nonetheless, the time-one map of such a flow may not be ergodic because of the existence of eigenfunctions for the flow. I will explain that when the expansion factor of the Pseudo-Anosov is a Pisot number (an algebraic real number greater than one with all conjugates in the interior of the unit circle), then there are such eigenfunctions. I hope to explain how to compute the eigenvalues, give an elementary proof that there are eigenfunctions, and explain how to draw them. (So I'm promising pictures!) One consequence of our work is that given a pseudo-Anosov homeomorphism of a translation surface with Pisot expansion factor of algebraic degree d≥3, there is a continuous surjective map from the surface to the d-dimensional torus that simultaneously semi-conjugates the pseudo-Anosov to a linear automorphism of the torus and semi-conjugates the straight-line flow in the expanding eigendirection to a linear flow on the torus. This is joint work with Jayadev Athreya, Nicolas Bedaride, and Pascal Hubert in a well-trodden area of math.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Gabriel Lacerda, Universidade Federal do Rio de Janeiro, UFRJ
Title: Complexity and Dimension of Induced Dynamical Systems
Abstract: In order to study the statistical properties of a collection of points in phase space under the action of a dynamical system, one can consider the measure-induced map given by the push-forward of a probability measure. In this talk, we will explore different approaches to studying collective dynamics, both in the measure-theoretic and topological sense, a line of research that began with the work of W. Bauer and K. Sigmund in the 1970s. Specifically, these maps, known as induced dynamical systems, act on an infinite-dimensional compact metric space. The goal is to discuss some notions of complexity and dimensionality, such as generalized entropy and mean dimension, in the context of these induced maps and, through some original results, conclude that the current tools for describing their properties are not yet the best possible.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Cancelled

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Marco Lopez, CUNY - Graduate Center, New York, NY
Title: Computability of MMEs
Abstract: In this seminar we will introduce notions of computability for various mathematical objects, focusing on computability of measures. In particular we will present some results in the setting of coded shift systems.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Zhenqi Wang, Michigan State University, East Lansing, MI
Title: Global smooth rigidity for toral automorphisms 
Abstract: Let \(f:\mathbb{T}^d\to\mathbb{T}^d\) be a \(C^\infty\) diffeomorphism whose linearization \(L\in GL(d,\mathbb{Z})\) is very weakly irreducible. Let \(H\) be a conjugacy between \(f\) and \(L\). We show that if \(H\) is \(C^{1+\text{H\"older}}\), then \(f\) is \(C^\infty\) conjugate to \(L\). In particular, If \(L\) hyperbolic and \(H\) is \(C^1\) then \(f\) is \(C^\infty\) conjugate to \(L\). As an application, we improve regularity of the conjugacy to \(C^\infty\) in prior local and global rigidity results. This is a joint work with B. Kalinin, V Sadovskaya.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Yushan Jiang, CUNY - Graduate Center, New York, NY
Title: Expanding maps, Thermodynamics, and Conformal Dynamics on Circle
Abstract: Expanding maps are studied a lot by many people from different view of points, especially for circle expanding maps. For example, M. Shub and D. Sullivan observed a rigidity phenomenon: for two analytic circle orientation preserving expanding endomorphisms which are topological conjugate, then the conjugacy is also analytic if and only if it’s absolute continuous (later, people generalized to lower regularity conditions). In the meanwhile, Mostow studied the conformal dynamics on \(S^n\) which come from closed \((n+1)\)-hyperbolic manifolds, and he obtained Mostow Rigidity for \(n\ge 2\): topological conjugacy will always be Möbius transformation. Interestingly, this rigidity fails for \(n=1\): circle conformal dynamics come from closed hyperbolic surfaces have full flexibility and we have the Teichmüller theory to describe the deformation. However, Mostow noticed that, even for circle there still exists rigidity: the topological conjugacy is Möbius if and only if it’s absolutely continuous. These two rigidity phenomena look similar and actually, both of them can be explained via thermodynamics. I will illustrate how the thermodynamical ideas apply for expanding maps and explain how we connect conformal dynamics and expanding maps on circle.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC 6417
Huyi Hu, Michigan State University, East Lansing, MI
Title: Spectral gap of transfer operators for torus extensions over expanding maps
Abstract: We study the spectral gaps of transfer operators and the mixing property for the skew product \(F:T^d\times T^l\to T^d\times T^l\) given by \(F(x,y) = (fx, y + \tau(x))\), where \(f : T^d\to T^d\) is a \(C^\infty\) uniformly expanding endomorphism, and the fiber map \(\tau : T^d \to T^l\) is a smooth map. We construct a Hilbert space that contains Hölder functions. We apply the semiclassical analysis approach to establish a dichotomy: either the transfer operator has a spectral gap on this Hilbert space, or τ is an essential coboundary. In the former case, \(F\) exhibits exponential mixing for Hölder observables; in the latter case, either F fails to be weakly mixing, or it can be approximated by non-mixing skew products that are semiconjugate to circle rotations. This is a joint work with Jianyu Chen.

Learning Seminar 11:00 AM-12:00 PM, GC 5417
Mariusz Urbański, University of North Texas, Denton, TX
Title: Conformal Measures for Conformal Dynamics
Abstract: Conformal measures were introduced in the early 1970's by Samuel Patterson in the context of Fuchsian groups and in the late 1970s. In the early 1980s, Dennis Sullivan extended this concept to Kleinian groups and rational functions. In the middle of the 1990's conformal measures were introduced to the setting of conformal countable alphabet iterated function systems by Dan Mauldin and the speaker. Their primary goal is to understand geometric measures, such as Hausdorff or packing, and metric dimensions, such as Hausdorff, packing, and box, of dynamically self-conformal sets that include limit sets of Kleinian groups and conformal IFSs, and Julia sets of rational functions and transcendental meromorphic functions. I will discuss relations between conformal and geometric measures and will provide more applications of the latter to fractal geometry.

Ergodic Theory and Dynamical Systems Seminar 3:00-4:00 PM, GC Math Lounge
Mariusz Urbański, University of North Texas, Denton, TX
Title: Ruelle's Operator and Conformal Measures with Applications in Fractal Geometry and Number Theory
Abstract: Probabilistic invariant measures provide a central tool to describe asymptotic behavior of dynamical systems and their ergodic and stochastic properties. Starting with natural examples, I will present a method of constructing such measures. It consists of finding fixed points of Ruelle’s operator and is applicable to dynamical systems that include smooth expanding maps, rational functions on the Riemann sphere, and holomorphic endomorphisms of complex projective spaces. I will tell how spectral properties of Ruelle’s operator entail stochastic properties of a given dynamical system, especially the exponential decay of correlations, the Central Limit Theorem, and the Law of Iterated Logarithm. I will also show how these spectral properties lead to the asymptotic of the number of circles in Apollonian packings. Next, I will focus on fractal properties of Julia sets of rational functions and transcendental meromorphic functions. Finally, I will talk about geometric properties of continued fractions.