• Fall 2023
• Spring 2023

Upcoming Talks

• December 8, 2023
  Axel Kodat, CUNY - Graduate Center, New York, NY
  Title: An introduction to the entropy conjecture
  Abstract: Let f be a smooth endomorphism of a closed manifold M. Shub conjectured that the spectral radius of the induced map of f on homology should give a lower bound for the topological entropy of f. This was proved by Yomdin when f is infinitely differentiable, but remains widely open in finite regularity. We give a brief historical overview of this conjecture, and sketch some known results.

Past Talks

• September 22, 2023
  Tamara Kucherenko, CUNY - City College, New York, NY
  Title: Ergodic Theory on Coded Shifts
  Abstract: We discuss ergodic properties of coded shift spaces. A coded shift is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. It turns out that many well-known classes of shifts are coded including transitive subshifts of finite type, S-gap shifts, generalized gap shifts, transitive Sofic shifts, Beta shifts, and many more. We derive sufficient conditions for the uniqueness of measures of maximal entropy based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We will also outline some flexibility results for the entropy on the sequential and the residual set. (Joint work with M. Schmoll and C. Wolf)

• September 29, 2023
  Yunping Jiang, CUNY - City College, New York, NY
  Title: Decay Rate of Correlations for weakly expanding dynamical systems with Dini potentials and an optimal quasi-gap.
  Abstract: The decay rate of correlations is an essential problem in studying dynamical systems. In this talk, I will review a study of the decay rate of correlations using transfer operator's spectra. The spectrum gap implies the exponential decay of correlations for locally expanding dynamical systems with Holder potentials. However, there is no spectrum gap for weakly expanding dynamical systems with Dini potentials. My current research with Yuan-Ling Ye finds an optimal quasi-gap for weakly expanding dynamical systems with Dini potentials. This optimal quasi-gap implies the Ruelle theorem holds and provides a nice decay rate estimation. Our estimation also verifies the central limit theorem for some Dini potentials.

• October 6, 2023
  Alejandro Passeggi, Universidad de La República Oriental del Uruguay
  Title: Rotation sets in low dimensional dynamics
  Abstract: Topological dynamical systems are naturally classified under the conjugacy relation. Then, the natural question is whether two dynamical systems are conjugated or not. To work this problem out, the theory of invariants rises: An invariant is any kind of information associated to each dynamic (maps, flows,...) which is constant over the conjugacy classes. A classical and natural invariant is the rotation number or rotation set, which has shown to be a very powerful tool for relevant classes of dynamical systems in dimension 1 (Poincaré theory) and in dimension 2. In this talk I will comment about this invariant in different contexts and how it is related to important problems, such as the detection of chaos, bounded homological displacements, and in a more algebraic setting the description of weak conjugacy classes.

• October 20, 2023
  Mark Demers, Fairfield University, Fairfield, CT
  Title: An Introduction to Anisotropic Banach Spaces
  Abstract: There has been an abundance of work over the past 20 years implementing the transfer operator approach, originally developed for expanding maps, to the hyperbolic setting. This requires constructing anisotropic spaces of distributions on which the relevant transfer operator has good spectral properties. This talk will introduce some of the key ideas needed to implement this approach and present some simple examples by way of illustration.

• October 27, 2023
  Sven Sandfeldt, KTH Royal Institute of Technology
  Title: Local centralizer rigidity for hyperbolic automorphisms of the 3-torus
  Abstract: I will discuss local centralizer rigidity of some hyperbolic toral automorphisms. An Anosov diffeomorphism \(f\) on the torus is always topologically conjugated to some hyperbolic automorphism \(L\). In this talk I will focus on how the size of the smooth centralizer of \(f\) can be used to improve the regularity of the conjugacy between \(f\) and \(L\).
  

• November 10, 2023
  Yushan Jiang, CUNY - Graduate Center, New York, NY
  Title: Rigidity of quasiconformal/quasisymmetric group actions on sphere \(S^n\)
  Abstract: This topic called quasi-isometric rigidity which brings complex analysis, dynamics, and geometric group theory together. At first, I will introduce quasiconformal/quasisymmetric group actions on the sphere \(S^n\). Then for \(n\ge 2\), if we add some essential requirements, due to Tukia and Sullivan, this kind of quasiconformal action will conjugate to a Möbius conformal action by a quasiconformal map, thus the original action will be ergodic w.r.t. Lebesgue measure. And for \(n=1\), due to many works (by Tukia, Casson-Jungreis, Gabai and so on), a quasisymmetric action will conjugate to a Möbius action by a homeomorphism. Unfortunately, we can't say a lot if the conjugacy is only a homeomorphism. However, due to Markovic and Navas, we can still say something more about it.

• November 17, 2023
  Elliot Kimbrough-Perry, CUNY - City College, New York, NY
  Title: Equilibrium states of one-sided symbolic systems.
  Abstract: In this talk, I will discuss the construction of a potential on a one-sided shift on two symbols exhibiting multiple phase transitions. Put precisely: after some preliminary definitions from ergodic theory, I will sketch the existence proof for a potential whose topological pressure has a finite number of points of non-differentiability—these points may be chosen as one desires. I will also show that the corresponding measures of maximal entropy may be chosen as to be supported on any subshift of finite type.

• December 1, 2023
  Lasse Rempe , University of Liverpool, United Kingdom
  Title: A counterexample to Eremenko's conjecture.
  Abstract: Let f be a transcendental entire self-map of the complex plane. The *escaping set* of f consists of those points that tend to infinity under iteration of f. (For example, all real numbers belong to the escaping set of the exponential map, since they tend to infinity under repeated exponentiation.) In 1989, Eremenko conjectured that every connected component of the escaping set is unbounded. Eremenko's conjecture has been a central problem in transcendental dynamics in the past decade. A number of stronger versions of the conjecture have been disproved, while weaker ones has been established, and the conjecture has also been shown to hold for a number of classes of functions. I will describe joint work with David Martí-Pete and James Waterman in which we construct a counterexample to Eremenko's conjecture. The talk should be accessible to a general mathematical audience, including postgraduate students.