Seminario Local de Sistemas Dinámicos

El Seminario Local de Sistemas Dinámicos, se realiza todos los martes de 15:30 a 16:30 en la Sala 1
2019-06-18
15:30 - 16:30hrs.
Ariel Reyes. Puc-Chile
Bilateral Mañé Lemma
Sala 1, Facultad de Matemáticas
Abstract:
We prove a bilateral Mañé Lemma assuming some hyperbolicity on the dynamical system $T : X \rightarrow X$ and some regularity on $f : X \rightarrow \mathbb{R}$, there exist $\theta : x \rightarrow \mathbb{R}$ in the same regularity class and such that $\alpha(f) \leqslant f-\theta+\theta \circ T \leqslant \beta(f)$, where $\alpha(f)$, $\beta(f)$ are the infimum and the supremum of the averages of $f$ along periodic orbits.
2019-06-11
15:30 - 16:30hrs.
Angela Flores. PUC Chile
Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type, Part 2
Sala 1, Facultad de Matemáticas
2019-06-04
15:30 - 16:30hrs.
Angela Flores. PUC Chile
Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type, Part 1
sala 1, Facultad de Matemáticas
2019-05-28
15:30 - 16:30hrs.
Jairo Bochi. PUC Chile
Genericity of Periodic Maximization, Part 2
Sala 1, Facultad de Matemáticas
2019-05-14
15:30 - 16:30hrs.
Jairo Bochi. PUC Chile
Genericity of periodic maximization
Sala 1, Facultad de Matemáticas
Abstract:
A theorem due to Gonzalo Contreras and published in 2016 essentially says that given a expanding map $T \colon X \to X$ and a generic Lispchitz (or Hölder) function $f \colon X \to \mathbb{R}$, there is a unique $T$-invariant probability measures that maximizes the average of $f$, and this measure is supported on a periodic orbit.
In this expository note, we present a proof of Contreras' theorem following the recent preprint of Wen Huang, Zeng Lian, Xiao Ma, Leiye Xu, and Yiwei Zhang.
2019-05-07
15:30 - 16:30hrs.
Erik Contreras. PUC Chile
The Bressaud-Quas Closing Lemma
Sala 1, Facultad de Matemáticas
Abstract:
Let $X$ be a compact metric space. Given a homeomorphism $T:X\to X$ belonging to a specific class called \textit{Hyperbolic homeomorphisms}, and given $Y\subseteq X$ a non empty compact $T$-invariant set, we want to prove the existence of periodic orbits of period at most $n$, supported on the $O(n^{-\tau})$-neighborhood of $Y$, where $n\in\mathbb{N}$ and $\tau>0$ are given. This fact is known as the Bressaud-Quas Closing Lemma, and we follow [1, Appendix A.6] for the proof.
 
References
 
[1] Bochi, J.; Garibaldi, E. Extremal Norms for Fiber Bunched Cocycles. https://arxiv.org/abs/1808.02804
2019-04-30
15:30 - 16:30hrs.
Sebastián Burgos. PUC
A Revelation Theorem in Expanding case.
Sala 1, Facultad de Matemáticas
Abstract:
Let $X$ be a compact metric space and $T:X\to X$ a continuous dynamic. For a continuous function $f:X\to\mathbb{R}$, we are interested in the ergodic maximum
$$\beta(f):=\sup_{\mu\in\mathcal{M}_T}\int fd\mu$$
and in the set $\mathcal{M}_{\max}(f):=\{\mu\in\mathcal{M}_T:\int fd\mu=\beta(f)\}\neq\emptyset.$
 
In the last session we introduced a useful tool called revelations:
 
\textbf{Definition.} We say that a continuous function $f:X\to\mathbb{R}$ is \textit{revealed} if there exists a compact set $K\subset f^{-1}(\max f)$ such that $TK\subset K$.
 
\textbf{Definition.} We say that a continuous coboundary $\psi=\varphi-\varphi\circ T$ is a \textit{revelation} for $f$ if the function $f+\psi$ is revealed.
 
In this talk we are going to define the notions of \textit{topologically transitivity} and \textit{expansiveness} of a dynamical system, and we will prove the following theorem:
 
\textbf{Theorem.} Suppose that $T:X\to X$ is topologically transitive and expansive. Then every Lipschitz function $f:X\to\mathbb{R}$ has a revelation $\psi=\varphi-\varphi\circ T$, where $\varphi$ is Lipschitz.
 
2019-04-23
15:30- 16:30hrs.
Sebastian Burgos. PUC
Revelaciones: Introducción y ejemplos.
Sala 1, Facultad de Matemáticas
Abstract:
Sea $X$ un espacio métrico compacto y $T : X\to X$ una dinámica continua. Para una función continua $f : X\to\mathbb{R}$, la optimización ergódica trata del estudio de medidas $f$ -maximizantes, es decir, las medidas en donde se alcanza el máximo ergódico
$$\beta(f):=\sup_{\mu\in\mathcal{M}_T}\int fd\mu.$$
En esta charla introduciremos una herramienta técnica, llamadas funciones reveladas y revelaciones, que (cuando existen) nos permitirán calcular estos máximos ergódicos y caracterizar las medidas $f$ - maximizantes.
También revisaremos ejemplos de estas funciones en el doubling map $(\mathbb{R}/\mathbb{Z},T(x)=2x$ mod $1)$ y en el full shift en dos simbolos $(\{0,1\}^{\mathbb{N}},\sigma((x_n)_{n\in\mathbb{N}})=(x_{n+1})_{n\in\mathbb{N}})$.
2019-04-16
15:30-16:30hrs.
Sebastián Pavez. PUC
Teorema de Kucherenko-Wolf
Sala 1, Facultad de Matemáticas PUC
2019-04-09
15:30 - 16:30hrs.
Sebastián Pavez. PUC
Optimización Ergódica: Introducción y ejemplo del Pescado
Sala 1, Facultad de Matemáticas PUC
Abstract:
El objeto de estudio de la optimizaci\'on erg\'odica es describir las \'orbitas de cierto sistema din\'amico que maximizan cierta funci\'on \textit{performance} dada. En el contexto de esta charla, consideraremos el caso de $(X,T)$ un sistema din\'amico, con $X$ un espacio m\'etrico compacto, $f \in \mathcal{C}(X)$, y queremos estudiar qu\'e ocurre con las \'orbitas que maximizan el problema:
 
$$\displaystyle \beta(f)= \sup_{x \in X} \lim_{n \rightarrow \infty}  \frac{1}{n} (f(x)+f(T(x))+...+f(T^{n-1}(x)))$$
 
donde este l\'imite exista. El problema (1) se puede trabajar equivalente como un problema de Teor\'ia Erg\'odica
 $$\beta(f)= \sup_{\mu \in \mathcal{M}_{T}} \int f d\mu$$
donde $\mathcal{M}_{T}$ denota las medidas de probabilidad $T$-invariantes. Luego de enunciar algunos resultados en el contexto del problema (2), vamos a hablar en espec\'ifico del ejemplo del Pescado de Bousch. A lo largo de esta charla, vamos tanto a repasar como motivar los conceptos y resultados que son conocidos en Teor\'ia Erg\'odica que se usar\'an para los prop\'ositos de lo que vamos a introducir.
2018-05-29
10:00hrs.
Renato Velozo. Pontificia Universidad Católica de Chile
Characterization of uniform hyperbolicity for fiber-bunched cocycles
Sala 3
Abstract:
We prove a characterization of uniform hyperbolicity for fiber-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched SL(2,R)-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an alpha-Holder cocycle which has uniform gap between the Lyapunov exponents, but it is not uniformly hyperbolic.
2017-01-09
16:30 Hrs.hrs.
Ian Morris, Surrey.
Matrix thermodinamic formalism
Sala 1 de la Facultad de Matemáticas de la Universidad Católica
Abstract:

Equilibrium states of real-valued potentials over subshifts of finite type have been investigated since the 1970s and their basic ergodic properties have long been well understood: they are exponentially mixing, Bernoulli and have positive entropy. Much more recently a theory has emerged of equilibrium states associated to matrix-valued potentials. In this talk I will describe how the ergodic properties of a matrix equilibrium state depend on the semigroup generated by the underlying matrices. At the end I will discuss some consequences for self-affine fractals in the plane.

2016-11-08
10:00hrs.
Rafael Potrie. Universidad de la Republica, Montevideo
El Lema de Morse en el plano hiperbólico
Sala 2
Abstract:
Voy a contar la relación entre el plano hiperbólico y las matrices 2x2 de determinante uno y utilizando un resultado de J-C. Yoccoz contar una prueba del Lema de Morse para el disco hiperbólico que dice que toda quasi-geodésica (a ser definido) en el plano hiperbólico puede ser aproximada uniformemente por una geodésica. Esto involucra traducir el problema a una pregunta básica de productos de matrices 2x2. 
2016-10-25
10:00hrs.
Italo Cipriano. .
Introduction to telescoping product measures (part 2)
Sala 2
Abstract:
In the last 4 years an important extension of the classic theory of multifractal analysis has been achieved after the introduction of a purely probabilistic machinery called telescoping product measure by Peres and Solomyak. In some sense, that it is still not completely understood, telescoping product measures play the role of the Gibbs measure in classic thermodynamics formalisms. An extension of the spectral theory of the Ruelle-Perron-Frobenius operator has also been initiated and a future extension of the theory of mutiple-ergodic averages in this direction looks now plausible.

In this talk I will focus on the purely probabilistic aspects of telescoping product measures. I will start with a motivation and I will end by exhibiting an original and more general proof of the essential probabilistic result used by Peres and Solomyak. No background will be needed.
2016-10-18
10:00hrs.
Jana Rodriguez Hertz. Universidad de la Republica, Montevideo
Ergodicity and partial hyperbolicity
Sala 2, Fac. de Matematicas.
Abstract:
In this talk we will recall different mixing properties and we'll see why they are relevant in some real-life applications. In particular, we will focus on ergodicity. We will then show some relations of partial hyperbolicity and ergodicity and describe some open problems. 
2016-10-11
10:00hrs.
Italo Cipriano. PUC
Introduction to telescoping product measures
sala 2
Abstract:
In the last 4 years an important extension of the classic theory of multifractal analysis has been achieved after the introduction of a purely probabilistic machinery called telescoping product measure by Peres and Solomyak. In some sense, that it is still not completely understood, telescoping product measures play the role of the Gibbs measure in classic thermodynamics formalisms. An extension of the spectral theory of the Ruelle-Perron-Frobenius operator has also been initiated and a future extension of the theory of mutiple-ergodic averages in this direction looks now plausible.

In this talk I will focus on the purely probabilistic aspects of telescoping product measures. I will start with a motivation and I will end by exhibiting an original and more general proof of the essential probabilistic result used by Peres and Solomyak. No background will be needed.
2016-10-04
10:00hrs.
Ryo Moore. Puc-Chile
A summary of Wiener-Wintner, multiple recurrence, and return times averages, and their recent developments, Part IV
Sala 2, Fac. de Matemáticas.
Abstract:
In this series of talks, we will discuss a variety of extensions of Birkhoff's pointwise ergodic theorem, namely, Wiener-Wintner, multiple recurrence, and the return times averages. We will survey the backgrounds and machinery associated to showing the convergences of these averages while discussing recent developments in these fields, particularly the ones obtained in the speaker's joint work with I. Assani, and partly with D. Duncan. The fourth (and the final) talk will focus on weighted ergodic averages and the return times theorem.
2016-09-27
10:00hrs.
Ryo Moore. PUC
A summary of Wiener-Wintner, multiple recurrence, and return times averages, and their recent developments, Part III
Sala 2, Fac. de Matemáticas
Abstract:
In this series of talks, we will discuss a variety of extensions of Birkhoff's pointwise ergodic theorem, namely, Wiener-Wintner, multiple recurrence, and the return times averages. We will survey the backgrounds and machinery associated to showing the convergences of these averages while discussing recent developments in these fields, particularly the ones obtained in the speaker's joint work with I. Assani, and partly with D. Duncan. The third talk will focus on the the double recurrence Wiener-Wintner theorem and its extensions, as well as the survey of the return times theorem.
2016-09-13
10:00hrs.
Ryo Moore. PUC
A summary of Wiener-Wintner, multiple recurrence, and return times averages, and their recent developments. Part 2.
Sala 2, Facultad de Matemáticas
Abstract:
In this series of talks, we will discuss a variety of extensions of Birkhoff's pointwise ergodic theorem, namely, Wiener-Wintner, multiple recurrence, and the return times averages. We will survey the backgrounds and machinery associated to showing the convergences of these averages while discussing recent developments in these fields, particularly the ones obtained in the speaker's joint work with I. Assani, and partly with D. Duncan. The second talk will focus on the history of the multiple recurrence theorem, and Wiener-Wintner extensions of Bourgain's double recurrence averages.
2016-09-06
10:00hrs.
Ryo Moore. Pontificia Universidad Católica de Chile
A summary of Wiener-Wintner, multiple recurrence, and return times averages, and their recent developments
Sala 2 (Víctor Ochsenius) Facultad de Matemáticas UC.
Abstract:
In this series of talks, we will discuss a variety of extensions of Birkhoff's pointwise ergodic theorem, namely, Wiener-Wintner, multiple recurrence, and the return times averages. We will survey the backgrounds and machinery associated to showing the convergences of these averages while discussing recent developments in these fields, particularly the ones obtained in the speaker's joint work with I. Assani, and partly with D. Duncan. The first talk will focus on the Wiener-Wintner ergodic theorem and its uniform counterpart.