# Dynamical Systems seminars 2018

## List of seminars 2018 http://www.dynamicalsystems.cl​

### The details of each seminar (beamer or notes) will be availabe here.

​​ 14th May,  Common fixed points of set-valued mappings in hyperconvex metric spaces
Eduardo Jorquera
In this work, we establish several common fixed point theorems for famil ies of set-valued mappings defined in hyperconvex metric spaces. Then we give several applications of our results. This is a joint work with M. Balaj and M. A. Khamsi.
Contact: [email protected]

28th May, Entropías intermedias y temperatura nula en curvatura negativa

Felipe Riquelme, PUCV
Un problema bastante general en teoría ergódica consiste en estudiar al conjunto de entropías de un sistema dinámico respecto a sus medidas ergódicas. Katok conjeturó que dicho conjunto contiene al intervalo $[0,h_{top}(f))$ en el caso de difeomorfismos suaves en variedades compactas. Si bien la conjetura permanece abierta, muchos avances se han logrado a la fecha. Se conoce, por ejemplo, que el flujo geodésico en variedades compactas a curvatura negativa verifica esta propiedad. La demostración de esto último recae en la realización del flujo geodésico como un flujo de suspensión sobre un shift de Markov de tipo finito.
En esta charla mostraremos que la tesis de la conjetura sigue siendo válida para el flujo geodésico sin la hipótesis de compacidad. Ante la ausencia de una realización simbólica genérica, las herramientas de la demostración serán puramente geométricas. Estas consisten en gran parte en el estudio del formalismo termodinámico del sistema, particularmente en los estados a temperatura nula. Este trabajo es un trabajo en curso junto a Anibal Velozo.

Notes: Non-offical notes from Felipe's talk available from the next link.
Contact: [email protected]

31th May, Multidimensional continued fractions and symbolic dynamics for toral translations
Pierre Arnoux, Université Aix-Marseille
We give a dynamical, symbolic and geometric interpretation to multi-dimensional continued fractions algorithms. For some strongly convergent algorithms, the construction gives symbolic dynamics of sublinear complexity for almost all toral translations; it can be used to obtain a symbolic model of the diagonal flow on lattices in $\mathbb R^3$.
Joint work with Valérie Berthé, Milton Minvervino, Wolfgang Steiner and Jorg Thuswaldner

5th July, Norm and pointwise convergence of multiple ergodic averages and applications
Andreas Koutsogiannis, The Ohio State University
Via the study of multiple ergodic averages for a single transformation, Furstenberg, in 1977, was able to provide an ergodic theoretical proof of Szemerédi's theorem, i.e., every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions. We will present some recent developments in the area for more general averages, e.g., for multiple commuting transformations with iterates along specific classes of integer valued sequences. We will also get numerous applications of the aforementioned study to number theory, as we will present the corresponding results along prime (and shifted prime) numbers, topological dynamics and combinatorics. Finally, we will present a result to the most general, and far more difficult case of pointwise convergence along special sublinear functions. This is part of independent, as well as joint work with D. Karageorgos (norm case); and S. Donoso and W. Sun (pointwise case).

9th July, Analogies between the geodesic flow on a negatively curved manifold and countable Markov shifts
Anibal Velozo, Princeton
By the work of Bowen and Ratner it is known that the geodesic flow on a compact negatively curved manifold can be modeled as a suspension flow over a subshift of finite type. Unfortunately, a symbolic representation is not available if the manifold is non-compact. In this talk I will briefly explain some recent developments on the study of the thermodynamic formalism of the geodesic flow on non-compact negatively curved manifolds. Surprisingly some of the methods used to understand the geodesic flow have consequences to the theory of countable Markov shifts. I will explain such consequences, as well as some open problems.

Contact: [email protected]

9th July, Morse theory for the action functional and a Poincare-Birkhoff theorem for flows
Umberto Hryniewicz, Universidade Federal do Rio de Janeiro
The goal of this talk is twofold. Firstly I would like to explain how pseudo-holomorphic curves can be used to study Morse theory of the action functional from classical mechanics. Then I will move to applications, focusing on a generalization of the Poincare-Birkhoff theorem for Reeb flows on the three-sphere.

12th July, Regularity of Lyapunov Exponents
Carlos Vásquez, Pontificia Universidad Católica de Valpraíaso
In this work, we consider a $C^infty$--,one parameter family of $C^{infty}$ diffeomorphisms $f_t$, $tin I$, defined on a compact orientable Riemannian manifold $M$. If the family admits a $Df_t$--invariant subbundle $E_t$ and an invariant probability measure $mu$ for every $tin I$, then the integrated Lyapunov exponent $lambda(t)$ of $f_t$ over $E_t$ is well defined. We discuss about conditions for the differentiability of $lambda(t)$.   Work in progress joint with Radu Saghin and Pancho Valenzuela-Henríquez.

12th July, Existence of global cross-sections: from Schwartzman cycles to holomorphic curves
Umberto Hryniewicz, Universidade Federal do Rio de Janeiro
The notion of a global section for a flow in dimension three goes back to the work of Poincare in Celestial Mechanics. During the second half of the XX century, initiated with the work of Sol Schwartzman, the construction of global cross-sections was organized as the study of linking properties of invariant measures. Fundamental contributions were given by Fried and Sullivan. Recently Ghys introduced the quadratic linking form and the notion of right-handed vector fields, and studied Lorentz knots. He used Schwartzman-Fried-Sullivan theory to investigate knot types of periodic orbits of right-handed flows and made the following statement: any collection of such orbits is a fibered link that binds an open book decomposition whose pages are global cross-sections. In this talk I would like to explain how holomorphic curves can be used to improve the abstract results from Schwartzman-Fried-Sullivan theory to obtain statements about (large) classes of Reeb flows that require few assumptions.

12th July, Characterization of uniform hyperbolcity for fiber bunched cocycles
Renato Velozo, Ponitificia Universidad Católica de Chile
We prove a characterization of uniform hyperbolicity for fiberbunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched SLp2, Rq-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an α-Holder cocycle which has uniform gap between the Lyapunov exponents, but it is not uniformly hyperbolic.

23th July, Sensitive dependence of geometric Gibbs measures at positive temperature
Daniel Coronel, UNAB
In this talk we give the main ideas of the construction of  the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states diverge at positive temperature. These are the first examples of divergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel.

Contact: [email protected]

13th Aug, Desvíos rotacionales para mapas del toro y aplicaciones
El número de rotación de Poincaré es sin duda alguna el invariante más importante en el estudio dinámico de homeomorfismos del círculo (que preservan orientación).
En general, estos sistemas exhiben lo que llamamos "desvíos rotacionales uniformemente acotados", es decir, cualquier órbita de un homeomorfismo de este tipo siempre se mantiene a distancia uniformemente acotada de la órbita de la rotación rígida correspondiente. Esta importante propiedad tiene implicaciones
profundas en dinámica unidimensional.
En dimensiones superiores, en analogía con la teoría de Poincaré del círculo, es posible definir el "conjunto de rotación" de homeomorfismos del d-toro homotópicos a la identidad, que a diferencia del caso unidimensional, en general no se reduce a un punto.
En esta charla discutiremos varias consecuencias de la acotación uniforme de los desvíos rotacionales en dimensiones superiores, enfocándonos fundamentalmente en homeomorfismos sin puntos periódicos en dimensión 2. También presentaremos algunos resultados recientes que relacionan la geometría del conjunto de rotación con la acotación a priori de los desvíos rotacionales.

20th Aug, Phase transitions and limit laws
Mike Todd, University of St Andrews
The statistics’ of a dynamical system is the collection of statistical limit laws it satisfies.  This starts with Birkhoff’s Ergodic Theorem, which is about averages of some observable along orbits: this is a pointwise result, for typical points for a given invariant measure.  Then we can look for forms of Central Limit Theorem, Large Deviations and so on: these are about how averages fluctuate, globally, with respect to the invariant measure.   In this talk, I’ll show how the form of the pressure function´ for a dynamical system determines its statistical limit laws.  This is particularly interesting when the system has slow mixing properties, or, even more extreme, in the null recurrent case (where the relevant invariant measure is infinite). I’ll start by introducing these ideas for simple interval maps with nice Gibbs measures and then indicate how this generalizes. This is joint work with Henk Bruin and Dalia Terhesiu.

20th Aug,  Cociclos sobre dinámicas hiperbólicas, exponentes de Lyapunov y aplicaciones
Las orbitas periódicas de los sistemas uniformemente hiperbólicos concentran gran parte de la información dinámica de los mismos. De esta forma, muchas veces es posible estudiar diversas propiedades de cociclos sobre estos sistemas (e.g. exponentes de Lyapunov) observando tan sólo lo que sucede sobre las órbitas periódicas.En esta charla discutiremos los alcances y limitaciones de este enfoque y  algunas aplicaciones.

20th Aug, Cohomology Equation for isometries of Gromov Hyperbolic spaces
Alexis Moraga, PUC

27th Aug, The joint spectrum
Çağrı Sert, ETH Zürich
We will define the notion of joint spectrum of a compact subset of GLd(C) which is a multidimensional generalization of joint spectral radius. We will talk about its properties such as convexity, continuity (and discontinuity) and mention its realization and finiteness properties. Finally, we will make connections with random products of matrices. (Joint work with Emmanuel Breuillard).

3th Sep, Random products of matrices and large deviations
Çağrı Sert, ETH Zürich
We will start by surveying classical results of Furstenberg, Kesten, Guivarc’h, Le Page, Bougerol, Benoist-Quint and others, on random products of matrices such as a the non-commutative law of large numbers, properties of Lyapunov exponents, central limit theorem etc. In a second part, we will turn to large deviations and talk about the recent result on the existence of large deviation principle for random matrix products. Finally, we will make connections with the recently introduced notion of joint spectrum.

1st Oct, Conos de Markov en grupos finitamente generados
Cristobal Rivas, USACH
Un subconjunto de un grupo se dice cono si es cerrado bajo multiplicación y disjunto de su inverso. En esta charla nos interesamos en estudiar conos que pueden ser descritos por un lenguaje regular (i.e. por un automata). Veremos ejemplos, algunas obstrucciones geométricas generales y finalmente nos enfocaremos en el caso en que el grupo ambiente es hyperbólico.

22nd Oct, Exponentes de Lyapunov y rigidez para difeomorfismos hiperbólicos y parcialmente hiperbólicos
Voy a presentar unos resultados de rigidez en termino de exponentes de Lyapunov para difeomorfismos hiperbólicos y parcialmente hiperbólicos. Si un difeomorfismo hiperbólico (o parcialmente hiperbólico) es cerca a un automorfismo lineal (o un skew product sobre un automorfismo lineal), preserva el volumen, y tiene los mismos exponentes de Lyapunov (estables e inestable), entonces es suavemente conjugado al automorfismo lineal (o a un skew product sobre el automorfismo lineal). En el caso de difeomorfismos hiperbólicos el resultado puede ser visto como un análogo a la conjetura de rigidez de la entropía de Katok.

29th Oct, Almost everywhere convergence of ergodic averages
Zoltan Buczolich, Department of Analysis, Eötvös Loránd
In this talk I would like to discuss some of my results concerning almost everywhere convergence of non-conventional ergodic averages of L1 functions.
These topics include:
• divergence of ergodic averages along the squares;
• convergence along some sequences of zero Banach density;
• convergence for arithmetic weights: the prime divisor functions ω and Ω.

Contact: [email protected]

5th Nov, Critical exponents for normal subgroups via a twisted Bowen-Margulis current and ergodicity
Rhiannon Dougall, Bristol
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.

12th Nov, TBA
Yuki Yayama, UBIOBIO
TBA

19th Nov,  Optimal lower bounds for multiple recurrence
Sebastián Donoso, UOH

Multiple recurrence concerns the study of the largeness of sets of the form
\label{0}
\begin{split}
\big\{n\in\mathbb{N}\colon\mu(A\capT^{f_1(n)}A\capT^{f_2(n)}A\cap\cdots\capT^{f_k(n)}A)>C\mu(A)^{k+1}\big\}
\end{split}

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 $f_i:\mathbb{N}\to\mathbb{Z}$.
In this talk I will present recent work on this problem for different functions $f_i$. For instance, if $k\leq 3$ and $f_{i}(n)=if(n), 1\leq i\leq k$, 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 $f_{i}(n)=a_{i}n, 1\leq i\leq k$, where $a_{i}$ are distinct integers, we show that \eqref{0} can be empty for $k\geq 4$, 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 $f_{i}, 1\leq i\leq k$ are polynomials.
This is joint work with Ahn Le, Joel Moreira and Wenbo Sun.

19th Nov,  Decidability of the isomorphism and the factorization for
minimal substitution subshifts

Fabien Durand , U. Picardie
Classification is a central problem in the study of dynamical systems, in particular for families of systems that arise in a wide range of topics. Hence it is important to have algorithms deciding wether a dynamical
system have some given property.

Let us mention subshifts of finite type that appear, for example, in information theory, hyperbolic dynamics, $C^*$-algebra, statistical mechanics and thermodynamic formalism. The most important and longstanding open problem for this family originates in [Williams:1973] and is stated in [Boyle:2008] as follows : Classify subshifts of finite type up to topological isomorphism. In particular, give a procedure which decides when two non-negative
integer matrices define topologically conjugate subshifts of finite type.

Another well-known family of subshifts, that is also defined through matrices, with a wide range of interests is the family of substitution subshifts. These subshifts are concerned, for example, with automata theory, first order logic, combinatorics on words, quasicrystallography, fractal geometry, group theory and number theory.
In this talk we will show that not only the existence of isomorphism between such subshifts is decidable but also the factorization.

26th Nov,  TBA
Michael Schraudner, CMM
TBA

3rd Dec, TBA
Felipe Riquelme, PUCV
TBA

3rd Dec, TBA
Jairo Bochi, PUC

TBA

3rd Dec, TBA
Natalia Jurga, University of Surrey
TBA