Learning Seminar on Dynamical Systems and Ergodic Theory and Related Topics

Spring Semester of 2023

Organized by Yunping Jiang and Enrique Pujals

Friday, 11:30 pm – 1:00 pm, Room 3212



February 3, Canceled due to cold and windy weather

February 10, Organization meeting (half an hour) and Enrique Pujals,

TitleOn Stability Conjecture in Smooth Dynamical Systems

February 17, Yushan Jiang

Title: Dynamics on Metric Spaces —— Some Rigidity Phenomena

Abstract: I will begin with two phenomena on metric spaces and prove them by

using dynamical systems and ergodic theory. At first, all distance non-decreasing maps

and distance non-increasing surjections on a compact metric space will be isometries.

Secondly, all distance non-increasing surjections on a metric space

with finite "volume" (e.g., Riemannian manifold with finite volume) will be isometries.

After these two, if time permit, I will introduce a property related to the existence of a metric

with "good symmetry" (i.e., with infinite isometry group). Then I'll use some tools related to

the topological entropy of geodesic flow to show the non-existence of such metric

with "good symmetry" on many manifolds (e.g., closed hyperbolic manifolds).


February 24, Bryce Gollobit

Title: Easy Examples of Stability in Smooth Dynamical Systems


March 3, Tamara Kucherenko

Title: Properties of the Pressure Function

Abstract: We discuss the properties of the pressure function for continuous potentials on shift spaces with a finite alphabet. The classical theory of thermodynamic formalism shows that such pressure functions are convex, Lipschitz and have slant asymptotes. It turns out that in the case of continuous potentials no other properties are present. On the other hand, one can say much more about the pressure under the restriction that the potential is Holder and the shift is mixing and of finite type. It was shown by Ruelle in 1970s that in this case the pressure function analytic and strictly convex. We present the first additional properties of the pressure obtained in more than 40 years: asymptotic bounds and quantitative convexity estimates.


March 10, Tao Chen

Title: On Ergodicity of Transcendental Meromorphic Functions as Dynamical Systems

Abstract: Due to Lyubich and McMullen, the typical orbits of rational maps are well-understood. For any rational map, we have either

(1) the Julia set is the whole sphere, and the map is ergodic or (2) for almost every point, its omega limit is contained in the post-critical set. 

However, given a map, it is not clear which of these two cases applies.  In this talk, I will present some results in this aspect: 

Theorem 1 (Keen-Kotus). If the post singular set of a meromorphic function is a compact repeller, then Case (1) applies. 

Theorem 2 (Hemke). For any entire map of finite type, if all its asymptotic values are escaping to infinity exponentially and critical points are strictly pre-periodic, then Case (2) applies.  


March 17, Jessica (Yuan) Liu

Title: nonstandard analysis proof of the ergodic theorem


March 24, Lucas Furtado

Title: Motivating the quadratic family

Abstract:  The motivation will come from: (1) attempting to reconcile the definition of a group action with the fact that the map f(x)=ax(1-x) restricted to [0,1] is not injective.  (2) Attempting to make sense of the "increase in complexity" when passing from the study of homeomorphisms of the interval to endomorphisms of the interval or homeomorphisms of the circle or piecewise continuous maps of the interval.  

March 31, Andrey Gogolev

Title: Obstructions to smooth equivalence of hyperbolic dynamical systems.
Anosov structural stability asserts that under certain assumptions, such as in the perturbative setting, two hyperbolic dynamical systems are equivalent. In general, from the point of view of smooth ergodic theory, this equivalence is rather weak. Still the equivalence could happen to be the smooth. We will discuss various obstructions to smoothness and present several results which give smooth equivalence from vanishing of obstructions. The talk will be accessible for anyone who is familiar with basics on manifolds and flows.


April 7 (Spring Recess, no class)

April 14, Axel Kodat (Cancel due to the Spring break)

April 21, Anthony Quas

Title: Introduction to the sub-additive ergodic theorem
Abstract: The sub-additive ergodic theorem is a highly versatile tool with applications in probability, differentiable dynamical systems, geometric group theory and more. I will introduce it, give some examples of its applications and and give an indication of its relationship to the multiplicative ergodic theorem.​ 


April 28, Yakov Pesin

      Note that all talks today are in the Math Lounge 4214

11:30 - 12:30 PM   Informal talk for the graduate students during the Learning Seminar on Dynamical Systems and Ergodic Theory and Related Topics (Math Lounge 4214)

Title: “Caratheodory structures and dynamics”
Abstract: I will introduce and discuss the notion of a Caratheodory structure, which is a far reaching generalization of the classical Hausdorff dimension structure. A presence of a Caratheodory structure allows one to introduce Caratheodory dimension-like characteristics of sets and measures. I will illustrate this by showing that given a continuous map f of a compact metric space X, the classical topological pressure corresponding to a given potential function function is the Caratheodory dimension generated by an appropriately chosen Caratheodory structure on X. In particular, such important invariants in dynamics as metric and topological entropies can also be viewed as dimensions this revealing their dimension nature.

12:45 - 1:30 PM     Refreshments and discussion (Pizza at 1pm)


2:00 - 3:00 PM       Colloquium presentation (Math Lounge 4214)

Title: "Equilibrium measures in hyperbolic dynamics via geometric measure theory”
Abstract: Thermodynamic formalism, i.e., the formalism of equilibrium statistical physics, originated in the work of Boltzman and Gibbs and was later adapted to the theory of dynamical systems in the classical works of Sinai, Ruelle, and Bowen. It is aimed at constructing and studying uniqueness and ergodic properties of equilibrium measures corresponding to some potential functions. This formalism was fully developed for classical uniformly hyperbolic systems using their symbolic representation via subshifts of finite type. However, for systems which are partially or non-uniformly hyperbolic very few results are known. I will describe a new approach to thermodynamics formalism which utilizes Caratheodory structures from geometric measure theory and allows one to construct and study equilibrium measures for some partially hyperbolic systems including time-1 map of Anosov flows and frame flows.


May 5, Melkana Brakalova-Trevithick

            Title: On the integrable Teichmuller spaces

           Abstract: The universal Teichmuller space T can be identified with the space of all quasi-symmetric homeomorphisms on the unit circle fixing -1,1,i (or one   

could consider the respective case of the real line and the upper half-plane.) The subspaces of T whose elements have a quasi-conformal extension to the unit disk with a p-integrable (p>0) complex dilatation with respect to the Poincare metric are called the p-integrableTeichmuller spaces. These spaces have been actively investigated for p ≥ 1, e.g. the Weil-Petersson Teichmuller space, when p = 2. Using quasiconformal techniques, we will discuss symmetry, smoothness properties, and complex Banach manifold structure  for p > 0, 0 < p ≤ 1, and p = 1 respectively, and some open questions.         

            Reference: M. Brakalova--Trevithick, Properties of quasisymmetric homeomorphisms in the Weil--Petersson class, Teichmuller theory and 

                                                Grothendieck-Teichmuller theory, ALM 49, Ch. 3 (2022) 51–63, Higher Education Press and International Press, Beijing-Boston.

                               V. Alberge, M. Brakalova, On smoothness of the elements of some integrable Teichmuller spaces, Math. Rep. (Bucur.) {\bf 23(73)} (2021),

                                                no. 1-2, 95–105.

                               M. Brakalova, Symmetric properties of the $p$-integrable Teichmuller spaces}, Analysis and Mathematical Physics, 8 (2018) 541–549,

       Springer Verlag.

                               Y. Jiang,  Kobayashi's and Teichmuller metrics and Bers complex manifold structure on circle diffeomorphisms, Acta Math. Sin. (Engl. Ser.) 3

        (2020), no. 3, 245–272.

                               Q. Li, Y. Shen, Some notes on integrable Teichmuller space on the real line, Published by Faculty of Sciences and Mathematics, University of

       Nis, Serbia, Filomat 37:8 (2023), 2633–2645, http://www.pmf.ni.ac.rs/filomat.

                               H. Wei, K. Matzusaki, The $p$-integrable Teichmuller space for  p=>1, https://arxiv.org/abs/2210.04720v1


May 12, Elliot Kimbrough-Perry

            Title: Cohomology and functions between a two-sided and one-sided shift space