# Dynamical Day

5 November 2018

Facultad de Matemáticas, UC, Santiago

The aim of this meeting is to gather a group of mathematicians working on dynamical systems to discuss new developments in the area.

Jairo Bochi

Italo Cipriano

Godofredo Iommi

Jan Kiwi

Mario Ponce

14:30-15:30 Sebastián Donoso

15:45-16:45 Rhiannon Dougall

16:50-17:50 Mario Ponce

**Rhiannon Dougall, **Bristol, UK.

Title: Critical exponents for normal subgroups via a twisted Bowen-Margulis current and ergodicity**Abstract:** For a discrete group $Gamma$ of isometries of a negatively curved space $X$, the critical exponent $delta(Gamma)$ measures the exponential growth rate of the orbit of a point. When $X$ is a manifold, this can be rephrased in terms the growth of periodic orbits for the geodesic flow in the $Gamma$-quotient. We fix a group $Gamma_0$ with good dynamical properties, and ask for $Gamma < Gamma_0$, when does $delta(Gamma)=delta(Gamma_0)$? We will motivate this problem, and discuss what is new: the construction of a twisted Bowen-Margulis current on the double-boundary, which highlights a feature of ergodicity, and extends the class for which the result is known. This is joint work with R. Coulon, B. Schapira and S. Tapie.

**Sebastián Donoso**, UOH, Rancagua.**Title: Optimal lower bounds for multiple recurrence****Abstract:** Multiple recurrence concerns the study of the largeness of sets of the form egin{equation}label{0} egin{split} ig{ninmathbb{N}colonmu(AcapT^{f_1(n)}AcapT^{f_2(n)}AcapcdotscapT^{f_k(n)}A)>Cmu(A)^{k+1}ig} end{split} end{equation} where $A$ is a measurable set of the invertible measure preserving system $(X,{mathcal B},mu,T)$, $C>0$. and $(f_1,dots,f_k)$ are functions $f_i:mathbb{N} omathbb{Z}$. In this talk I will present recent work on this problem for different functions $f_i$. For instance, if $kleq 3$ and $f_{i}(n)=if(n), 1leq ileq k$, we show that eqref{0} has positive density when $f$ is a polynomial along primes with $f(1)=0$, or a Hardy field function away from polynomials, and eqref{0} is syndetic when $f$ is a Beatty sequence. For $f_{i}(n)=a_{i}n, 1leq ileq k$, where $a_{i}$ are distinct integers, we show that eqref{0} can be empty for $kgeq 4$, and that the largeness of eqref{0} is equivalent to a solution counting problem for certain linear equations when $k=3$. We also provide partial results on the largeness of eqref{0} when $f_{i}, 1leq ileq k$ are polynomials. This is joint work with Ahn Le, Joel Moreira and Wenbo Sun. 5th

**Mario Ponce, **UC Santiago**Title: A geometric approach to the cohomological equation for cocycles of isometries ****Abstract:** We present some geometrical tools in order to obtain solutions to cohomological equations that arise in the reducibility problem of cocycles by isometries of negatively curved metric spaces. The main ingredient is the relation between the solution to the corresponding equation of reducibility for the boundary action and the solution in the metric space. This is a joint work with A. Moraga.

This event is supported by CONICYT through the project AnilloACT172001 «New Trends in Ergodic Theory» and by the Facultad de Matemáticas, Pontifcia Universidad Católica de Chile.