Monotone families of circle diffeomorphisms driven by expandingcircle maps

Abstract: We consider monotone families of circle diffeomorphisms forced
by strongly expanding circle maps. We obtain estimates of the fibered
Lyapunov exponents for such systems and show that in the limit as the
expansion tends to infinity, they approach the values of the Lyapunov
exponents for the corresponding random case. The estimates are based on a
control of the distribution of the iterates of almost every point, up to a
fixed (small) scale, depending on the expansion.
This is joint work with Raphaël Krikorian.

Nonconmutative coboundary equations over integrable dynamics

I will present the result of a joint work with Rafael de la Llave. We prove an analog of Livshits theorem for real analytic families of cocycles over an integrable system with values
in a Banach algebra G or a Lie group. Namely, we consider an integrable dynamical system f: M ≡T^d × [−1, 1]^d → M, f(θ, I) = (θ + I, I), and a real-analytic family of cocycles η_ϵ : M → G, indexed by a complex parameter ϵ in an open ball E_ρ ⊂ C. We show that if η_ϵ is close to identity and has trivial periodic data, i.e., η_ϵ(f^{n−1}(p)). . . η_ϵ(f(p)) · η_ϵ(p) = Id for each periodic point p = f_{n}p and each ϵ ∈ E_ρ, then there exists a real-analytic family of maps ϕ_ϵ: M → G satisfying the coboundary equation η_ϵ(θ, I) = (ϕ_ϵ ◦ f(θ, I))^{−1}· ϕϵ(θ, I) for all (θ, I) ∈ M and ϵ ∈ E_{ρ/2}. We also show that if the coboundary equation above with an analytic left-hand side ηϵ has a solution in the sense of formal power series in ϵ, then it has an analytic solution. In a work in progress with Bassam Fayad and Rafael de la Llave we show that analogous results hold for individual maps, i.e., without the use of families.

Elliptic Islands in the Planar Circular Restricted 3-Body Problem

We consider the planar circular restricted three body problem, modeling the motion of a massless asteroid in the plane undergoing gravitational attraction toward two bodies, each with a circular orbit around their center of mass. For small mass ratios, this is approximated by the Kepler problem as long as the asteroid remains far from the smaller body. The existence of hyperbolic sets containing orbits in which the asteroid undergoes repeated close interactions with the smaller body was proven independently by Bolotin and MacKay and by Font, Nunes, and Simó. My conjecture is that there are elliptic periodic orbits with repeated close interactions, where the asteroid remains close to a Kepler ellipse intersecting the orbit of the smaller body.

Stationary probability measures on projective spaces

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.

Chaotic properties of smooth dynamical systems

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.

Random walks in quasiperiodic environment

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.

Wojtkowski’s ergodic hypothesis: a conjecture for decades?

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),

Sparse equidistribution problems in dynamical systems

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.

Oscillations and topology

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.

Tales of symplectic maps

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