**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, **

**Title****： **On 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.

Abstract: 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

**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