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.

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.

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.

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

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.

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.

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.

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

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.

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.

13th Aug,

Alejandro Kocsard, Universidade Federal Fluminense

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.

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,

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

20th Aug,

27th Aug,

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.

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**

**Radu Saghin, **PUCV

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.

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.

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.

5th Nov, **Optimal lower bounds for multiple recurrence**

**Sebastián Donoso**, UOH

Multiple recurrence concerns the study of the largeness of sets of the form

\begin{equation}\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}

\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 $f_i:\mathbb{N}\to\mathbb{Z}$.

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.

Multiple recurrence concerns the study of the largeness of sets of the form

\begin{equation}\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}

\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 $f_i:\mathbb{N}\to\mathbb{Z}$.

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.

5th Nov, **A geometric approach to the cohomological equation for cocycles of isometries**

**Mario Ponce**, PUC

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.

12th Nov, **Hidden Gibbs measures on shift spaces over countable alphabets**

**Yuki Yayama**, UBIOBIO

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets. This is a joint work with Godofredo Iommi and Camilo Lacalle.

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets. This is a joint work with Godofredo Iommi and Camilo Lacalle.

19th Nov, **Decidability of the isomorphism and the factorization for **

minimal substitution subshifts

**Fabien Durand **, U. Picardie

minimal substitution subshifts

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.

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.

19th Nov, **On the action of the semigroup of non singular integral matrices on $\R^n$**

PDF abstract.

26th Nov, **Restrictions on the group of automorphisms preserving a minimal subshift**

**Samuel Petite, **U de Picardie

A subshift is a closed shift invariant set of sequences over a finite alphabet. An automorphism is an homeomorphism of the space commuting with the shift map. The set of automorphisms is a countable group generally hard to describe. We will present in this talk a survey of various restrictions on these groups for zero entropy minimal subshifts.

A subshift is a closed shift invariant set of sequences over a finite alphabet. An automorphism is an homeomorphism of the space commuting with the shift map. The set of automorphisms is a countable group generally hard to describe. We will present in this talk a survey of various restrictions on these groups for zero entropy minimal subshifts.

29th Nov, **Computing the entropy of multidimensional subshifts of finite type**

**Gangloff ****Silvere**, ENS de Lyon

PDF abstract.

3rd Dec,**Ground states at zero temperature in negative curvature**

**Felipe Riquelme**, PUCV

PDF abstract.

3rd Dec,

Let $X$ be the unit tangent bundle of a complete negatively curved Riemannian manifold and let $(g_t):X\to X$ be its associated geodesic flow. After the work of R. Bowen and D. Ruelle, it is well know that, if $X$ is compact, then any H\"older-continuous potential $F:X\to\mathbb{R}$ admits an unique equilibrium measure. Moreover, there is a fair enough description of some properties of the pressure map $t\mapsto P(tF)$ such as its regularity and its asymptotic behavior. For non-compact situations, the existence of equilibrium measures has been successfully studied over the last years. Moreover, regularity properties of the pressure map have been established in recent works by G. Iommi, F. Riquelme and A. Velozo.

In this talk we will be interested on the study of ground states at zero temperature for positive H\"older-continuous potentials. More precisely, for $F:X\to\mathbb{R}$ a positive potential going to 0 through infinity, we will study the asymptotic behavior of the equilibrium state $m_{tF}$ for the potential $tF$ as $t\to+\infty$. Indeed, we will show precise constructions of potentials having convergence/divergence to ergodic/non-ergodic ground states. This is a joint work with Anibal Velozo.

In this talk we will be interested on the study of ground states at zero temperature for positive H\"older-continuous potentials. More precisely, for $F:X\to\mathbb{R}$ a positive potential going to 0 through infinity, we will study the asymptotic behavior of the equilibrium state $m_{tF}$ for the potential $tF$ as $t\to+\infty$. Indeed, we will show precise constructions of potentials having convergence/divergence to ergodic/non-ergodic ground states. This is a joint work with Anibal Velozo.

3rd Dec, **Emergence**

**Jairo Bochi**, PUC

I will talk about ongoing work with Pierre Berger.

Topological entropy is a way of quantifying the complexity of a dynamical system. It involves counting how many segments of orbit of some length $t$ can be distinguished up to some fine resolution $\epsilon$. If we are allowed to disregard a set of orbits of small measure, then we are led to the concept of metric entropy. Now suppose we don't care \emph{when} a piece of orbit visits a certain region of the space, but only \emph{how often}. Pursuing this idea, we are led to fundamentally new ways of quantifying dynamical complexity. This program was initiated by Berger a couple of years ago.

The first new concept that I'll explain is \emph{topological emergence} of a dynamical system: the bigger it is, the more different statistical behaviors are allowed by the system. We will explain how topological emergence is bounded from above in terms of the dimension of the ambient space. I'll also present examples of dynamical systems where this bound is essentially attained.

Then we'll come to another key concept: \emph{metric emergence} of a dynamical system with respect to a reference measure. Roughly speaking, it quantifies how far from ergodic our system is. (To draw a comparison, topological emergence quantifies how far from uniquely ergodic the system is.) KAM theory reveals that non-ergodicity is somewhat typical among conservative dynamical systems, and metric emergence provides a way of measuring the complexity of the KAM picture. I'll present examples and questions.

3rd Dec,**Rigorous estimates on the top Lyapunov exponent for random matrix products **

**Natalia Jurga**, University of Surrey

We study the Lyapunov exponent of random matrix products of positive $2 \times 2$ matrices and describe an efficient algorithm for its computation, which is based on the Fredholm theory of determinants of trace-class linear operators. Moreover, we obtain rigorous bounds on the error term in terms of two constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the average amount of projective contraction of the positive cone under the action of the matrices. This is joint work with Ian Morris from the University of Surrey.

Topological entropy is a way of quantifying the complexity of a dynamical system. It involves counting how many segments of orbit of some length $t$ can be distinguished up to some fine resolution $\epsilon$. If we are allowed to disregard a set of orbits of small measure, then we are led to the concept of metric entropy. Now suppose we don't care \emph{when} a piece of orbit visits a certain region of the space, but only \emph{how often}. Pursuing this idea, we are led to fundamentally new ways of quantifying dynamical complexity. This program was initiated by Berger a couple of years ago.

The first new concept that I'll explain is \emph{topological emergence} of a dynamical system: the bigger it is, the more different statistical behaviors are allowed by the system. We will explain how topological emergence is bounded from above in terms of the dimension of the ambient space. I'll also present examples of dynamical systems where this bound is essentially attained.

Then we'll come to another key concept: \emph{metric emergence} of a dynamical system with respect to a reference measure. Roughly speaking, it quantifies how far from ergodic our system is. (To draw a comparison, topological emergence quantifies how far from uniquely ergodic the system is.) KAM theory reveals that non-ergodicity is somewhat typical among conservative dynamical systems, and metric emergence provides a way of measuring the complexity of the KAM picture. I'll present examples and questions.

3rd Dec,

We study the Lyapunov exponent of random matrix products of positive $2 \times 2$ matrices and describe an efficient algorithm for its computation, which is based on the Fredholm theory of determinants of trace-class linear operators. Moreover, we obtain rigorous bounds on the error term in terms of two constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the average amount of projective contraction of the positive cone under the action of the matrices. This is joint work with Ian Morris from the University of Surrey.

18th Dec,

Determinantal point processes arise in a wide range of problems.

How does the determinantal property behave under conditioning? The talk will first address this question for specific examples such as the sine-process, where one can explicitly write the analogue of the Gibbs condition in our situation. We will then consider the general case, where, in joint work with Yanqi Qiu and Alexander Shamov, proof is given of the Lyons-Peres conjecture on completeness of random kernels. The talk is based on the preprint arXiv:1605.01400 as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexander Shamov.

How does the determinantal property behave under conditioning? The talk will first address this question for specific examples such as the sine-process, where one can explicitly write the analogue of the Gibbs condition in our situation. We will then consider the general case, where, in joint work with Yanqi Qiu and Alexander Shamov, proof is given of the Lyons-Peres conjecture on completeness of random kernels. The talk is based on the preprint arXiv:1605.01400 as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexander Shamov.

10th Jan,

TBA

14th Jan, **Piecewice**** chaotic maps**

**Alberto Pinto**

TBA

Dynamical Systems seminars 2019

Italo Cipriano

I am in charge of the
__seminar of Dynamical Systems in Santiago __
. Details of each seminar 2019 are available
__here__.
Seminars 2018 are available
__here__
.

I am a postdoctoral fellow supported by CONICYT PIA ACT172001 at Pontificia Universidad Católica de Chile. My research is in Dynamical Systems and I am primarily interested in Thermodynamic Formalism.

Time change for flows and thermodynamic formalism (with Godofredo Iommi). (2019) (accepted in Nonlinearity).

Stationary measures associated to analytic iterated function schemes (with Mark Pollicott). Math. Nachr. 291 (2018), no. 7, 1049–1054.

Entry time statistics to different shrinking sets, Stoch. Dyn., 17 (2017), no. 3, 314–323.

**Preprints**

Entry time statistics to different shrinking sets, Stoch. Dyn., 17 (2017), no. 3, 314–323.

Continuous coboundaries of the product of smooth functions (with Ryo Moore). (arXiv).

The Wasserstein distance between stationary measures associated to iterated function schemes on the unit interval. (arXiv).

The smoothness of the stationary measure. (arXiv).

A Large deviation and an escape rate result for special semi-flows. (arXiv).

Escape rate for special semi-flows over non-invertible subshifts of finite type. (arXiv).

Fall and Spring 2017. Differentiation and Integration, Universidad Técnica Federico Santa María.

Fall and Spring 2017. Introduction to Calculus, Universidad Técnica Federico Santa María.

Fall 2017. Foundations, Universidad Adolfo Ibáñez.

Spring 2016. Linear Algebra, Universidad Adolfo Ibáñez.