# 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
2019-10-22
11:30hrs.
Nicolás Alvarado. PUC
Variational Principle
Sala 1
2019-10-15
11:30hrs.
Nicolás Alvarado. PUC
Construction of Gibbs Measures and Variational Principle
Sala 1
2019-10-08
11:30hrs.
Nicolás Alvarado. PUC
Construction of Gibbs Measures
Sala 1
2019-10-01
11:30hrs.
Nicolas Pinto. PUC
Ruelle?s Perron Frobenius Theorem
Sala 1
2019-09-24
11:30hrs.
Nicolas Pinto. PUC
Gibbs Distribution and Ruelle's Perron-Frobenius Theorem III
Sala 1
2019-09-10
11:30hrs.
Nicolas Pinto. PUC
Gibbs Distribution and Ruelle's Perron-Frobenius Theorem II
Sala 1
2019-09-03
11:30hrs.
Nicolas Pinto. PUC
Gibbs Distribution and Ruelle's Perron-Frobenius Theorem
Sala 1
2019-08-27
11:30hrs.
Sebastian Pavez. PUC
Genericity of Strictly Convex Rotation Sets II
Sala 1, Facultad de Matemáticas
2019-08-20
11:30hrs.
Sebastian Pavez. PUC
Genericity of Strictly Convex Rotation Sets
Sala 1, Facultad de Matemáticas
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.