Author: Klaudiusz CZUDEK

Abstract:
We give a description of stationary probability measures
on projective spaces for an iid random walk on $\GL_d(\R)$
without any algebraic assumptions. This is done in two parts.
In a first part, we study the case (non-critical or
block-dominated case) where the random walk has distinct
deterministic exponents in the sense of Furstenberg–Kifer–Hennion.
In a second part (critical case), we show that if the random
walk has only one deterministic exponent, then any stationary
probability measure on the projective space lives on a subspace
on which the ambient group of the random walk acts semisimply.
This connects the critical setting with the work of Guivarc’h–Raugi
and Benoist–Quint. Combination of all these works allow to get
a complete description. Joint works with Richard Aoun.

One of the central discoveries in the theory of dynamical systems was that differentiable (or smooth) systems can display strongly chaotic behavior and in many ways behave like a sequence of random coin tosses. In this talk we will describe the appearance and interactions of chaotic properties in smooth dynamics. We will highlight main developments, describe the state of the art and discuss some open problems in the field.

We consider the environment viewed by the particle process in random walk in quasiperiodic environment. This process appears to be certain simply defined random walk on the circle. We prove the central limit theorem and establish the rate of mixing in the Diophantine case. This is based on a joint work with D. Dolgopyat.

In the 1980s M. P. Wojtkowski introduced an interesting dynamical system of 1D balls
moving in a vertical half-line, colliding with each other and the hard floor elastically,
and falling down under constant gravitation. To avoid the existence of linearly stable
periodic orbits, one assumes that the masses of the particles are decreasing as we
go up in the half line. He conjectured that all these systems are completely hyperbolic
and ergodic.

Complete hyperbolicity of all such systems was shown by N. S. in 1996. Here we
describe a brand new algebraic approach to such systems that enable us to verify all
the conditions of the Local Ergodic Theorem for Dynamical Systems With Invariant
Cone Fields (by Liverani and Wojtkowski) for almost all such falling ball systems, thus
proving their ergodicity, a famous, so far unsolved conjecture of Maciej P. Wojtkowski
from the mid 1980’s.

In the talk special emphasis will be given to some interesting new aspects of the
exploited algebraic approach that made it possible to prove the annoying
transversality condition (the equivalent of the Chernov-Sinai Ansatz for billiards)
assumed in my conditional result [S2022].

[S2022] Simanyi, N.: “Conditional Proof of the Ergodic Conjecture for Falling Ball
Systems”. To appear in Contemporary Mathematics (2022),
https://arxiv.org/abs/2211.10874

Abstract: Let (a_n) be a sequence of natural numbers. For a dynamical system (X,T) we will be interested in orbits of points sampled at times (a_n). 

More precisely for f\in C(X) and x\in X one is interested in  lim_{N\to \infty} \frac{1}{N}\sum_{n\leq N}f(T^{a_n}x). We will focus on three types of sequences that naturally arise in number theory, (i) prime numbers (ii) polynomials and (iii) products of bounded number of primes. 

We will recall some known results and discuss some recent developments.

Abstract: I’ll discuss an approach to studying oscillations of functions based on ideas of topological data analysis. Applications include generalizations of two classical results, Courant’s nodal domain theorem in spectral geometry and Bezout’s theorem in algebraic geometry. Joint with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin, and Vukašin Stojisavljević.

Abstract: I’ll discuss a number of rigidity phenomenaof algebraic, geometric, and dynamical nature exhibited by symplectic diffeomorphisms. 

We consider a particle moving within some convex planar billiard according to a modified reflection law, where collisions become inelastic; more precisely, at each (non-orthogonal) collision with the boundary, the (unoriented) outgoing angle of reflection is strictly smaller than the incoming angle of incidence, both being measured with respect to the normal. The resulting dissipative billiard map has a global attractor. In a joint work with A. Florio and O. Bernardi, we study the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and more recently, the work of Le Calvez. We show that for a convex billiard with pinched curvature, if the dissipation is strong enough, then the Birkhoff attractor is « simple » (a normally contracted manifold) and the dynamics is of Morse-Smale type; on the contrary, we show that if the dissipation is mild, then the Birkhoff attractor is topologically « complicated » (an indecomposable continuum) and has rich dynamics (rotation set with non-empty interior, presence of horseshoes…).