Seminario de Sistemas Dinámicos
El Seminario de Sistemas Dinámicos de Santiago es el encuentro semanal de matemáticas con mayor tradición en el país pues se realiza ininterrumpidamente desde la década del '80. Se realiza alternadamente en alguna de las instituciones de Santiago donde hay miembros del grupo de Sistemas Dinámicos. Participan así las universidades de Chile, de Santiago, Andrés Bello y Católica de Chile.
2019-06-03
14:30-15:30hrs.
14:30-15:30hrs.
Anibal Velozo. Yale University, Usa
Pressure at infinity and applications.
Auditorio Bralic
Abstract:
There are many important dynamical systems which can be coded, via Markov partitions, into a symbolic dynamical system. Whenever this is possible one gets a fairly good understanding of the ergodic theory of the initial system. In this talk I will motivate the study of (non-compact) symbolic dynamics and elaborate on recent works about its entropy theory. I will focus on semi-continuity properties of the entropy and pressure. Notions of entropy and pressure at infinity play an important role in these results and will be discussed. This talk is partially based on joint works with G. Iommi and M. Todd.
Pressure at infinity and applications.
Auditorio Bralic
Abstract:
There are many important dynamical systems which can be coded, via Markov partitions, into a symbolic dynamical system. Whenever this is possible one gets a fairly good understanding of the ergodic theory of the initial system. In this talk I will motivate the study of (non-compact) symbolic dynamics and elaborate on recent works about its entropy theory. I will focus on semi-continuity properties of the entropy and pressure. Notions of entropy and pressure at infinity play an important role in these results and will be discussed. This talk is partially based on joint works with G. Iommi and M. Todd.
2019-06-03
15:45-16:45hrs.
15:45-16:45hrs.
Salomon Rebollo. Universidad del Bío-Bío
Polynomial perturbations of planar vector fields with curves of singularities
Auditorio Bralic
Abstract:
We will consider polynomial perturbations of planar polynomial vector fields that have centers and curves of singularities. For the perturbed vector field we will study its limit cycles that bifurcate from the centers of the unperturbed vector field. The bifurcation phenomena of limit cycles is richer in perturbation of vector fields with curves of singularities than in perturbation of vector fields with only isolated singularities. For example, more limit cycles can bifurcate in the former case than in the latter one. We will give some results about the maximum number of this kind of limit cycles that the perturbed vector field can support.
Polynomial perturbations of planar vector fields with curves of singularities
Auditorio Bralic
Abstract:
We will consider polynomial perturbations of planar polynomial vector fields that have centers and curves of singularities. For the perturbed vector field we will study its limit cycles that bifurcate from the centers of the unperturbed vector field. The bifurcation phenomena of limit cycles is richer in perturbation of vector fields with curves of singularities than in perturbation of vector fields with only isolated singularities. For example, more limit cycles can bifurcate in the former case than in the latter one. We will give some results about the maximum number of this kind of limit cycles that the perturbed vector field can support.
2019-06-03
16:50-17:50hrs.
16:50-17:50hrs.
Carlos Vásquez . Pontificia Universidad Católica de Valparaíso
Invariance of entropy for maps isotopic to Anosov
Auditorio Bralic
Abstract:
We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with ''controlled geometry '') and such that their linear part is hyperbolic.
In absence of the simplicity condition it is possible to construct a robustly transitive counter-example, evidencing the necessity of our assumptions.
Work in progress joint to Pablo Carrasco (UFMG-Brazil), Cristina Lizana (UFBA -Brazil) and E. Pujals (CUNY, USA).
Invariance of entropy for maps isotopic to Anosov
Auditorio Bralic
Abstract:
We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with ''controlled geometry '') and such that their linear part is hyperbolic.
In absence of the simplicity condition it is possible to construct a robustly transitive counter-example, evidencing the necessity of our assumptions.
Work in progress joint to Pablo Carrasco (UFMG-Brazil), Cristina Lizana (UFBA -Brazil) and E. Pujals (CUNY, USA).
2019-05-27
16:30hrs.
16:30hrs.
Ryo Moore. Pontificia Universidad Católica de Chile
Properties of Birkhoff spectra for the generic continuous functions on a shift space
Sala 1, PUC, Facultad de Matemáticas
Properties of Birkhoff spectra for the generic continuous functions on a shift space
Sala 1, PUC, Facultad de Matemáticas
2019-05-27
15:30--16:20hrs.
15:30--16:20hrs.
Sebastián Donoso. Uoh-Uch
Expansiveness and dimension of minimal sets
Sala 1, PUC, Facultad de Matemáticas
Abstract:
A remarkable theorem by Mañé states that a minimal expansive homeomorphism can only occur in a 0 dimensional space (a subshift). I will give the main ingredients and ideas of his proof.
Expansiveness and dimension of minimal sets
Sala 1, PUC, Facultad de Matemáticas
Abstract:
A remarkable theorem by Mañé states that a minimal expansive homeomorphism can only occur in a 0 dimensional space (a subshift). I will give the main ingredients and ideas of his proof.
2019-05-13
15:30--16:20hrs.
15:30--16:20hrs.
Carolina Canales. PUC
Tba
Sala 1, PUC
Tba
Sala 1, PUC
2019-05-13
16:30--17:30hrs.
16:30--17:30hrs.
Antti Käenmäki. University of Jyväskylä, Finland
Assouad dimension of planar self-affine sets
Sala 1, PUC
Abstract:
We consider planar self-affine sets X satisfying the strong separation condition and the projection condition. We show that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets. We also calculate the Assouad dimension of X. Finally, we prove that if X is dominated, then it is minimal for the conformal Assouad dimension. The talk is based on joint work with Balázs Bárány and Eino Rossi.
Assouad dimension of planar self-affine sets
Sala 1, PUC
Abstract:
We consider planar self-affine sets X satisfying the strong separation condition and the projection condition. We show that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets. We also calculate the Assouad dimension of X. Finally, we prove that if X is dominated, then it is minimal for the conformal Assouad dimension. The talk is based on joint work with Balázs Bárány and Eino Rossi.
2019-05-06
17:00-18:00hrs.
17:00-18:00hrs.
María Isabel Cortez. Usach
Algebraic Invariant of Minimal Group Actions on The Cantor Set: Topological Full Group and Group of Automorphism
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Algebraic Invariant of Minimal Group Actions on The Cantor Set: Topological Full Group and Group of Automorphism
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
2019-04-22
15:30-16:20hrs.
15:30-16:20hrs.
Cristobal Rivas. Usach
Sobre el grupo de Higman
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
Les contaré sobre el grupo de Higman. Porqué no tiene cocientes finitos y porqué no admite representaciones lineales. Si aún hay tiempo, diré algunas palabras sobre sus representaciones en grupos de difeomorfismos y homeomorfismos.
Sobre el grupo de Higman
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
Les contaré sobre el grupo de Higman. Porqué no tiene cocientes finitos y porqué no admite representaciones lineales. Si aún hay tiempo, diré algunas palabras sobre sus representaciones en grupos de difeomorfismos y homeomorfismos.
2019-04-22
16:30--17:30hrs.
16:30--17:30hrs.
Mónica Moreno Rocha. Cimat
On the dynamics of elliptic functions of the form P+b
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
The dynamical system obtained by iteration of the Weierstrass P function over real square lattices can be characterized by the behavior of its single free critical orbit. In contrast, as soon as P is “perturbed” by the addition of a complex parameter b, the elliptic function P+b exhibits at least two free critical orbits, which complicates the study of its dynamics and connectedness locus. This talk I will present some of the results and open questions regarding the rich structures found in dynamical and parameter plane of P+b when b is restricted to a complex line and P is defined over real square lattices. This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.
On the dynamics of elliptic functions of the form P+b
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
The dynamical system obtained by iteration of the Weierstrass P function over real square lattices can be characterized by the behavior of its single free critical orbit. In contrast, as soon as P is “perturbed” by the addition of a complex parameter b, the elliptic function P+b exhibits at least two free critical orbits, which complicates the study of its dynamics and connectedness locus. This talk I will present some of the results and open questions regarding the rich structures found in dynamical and parameter plane of P+b when b is restricted to a complex line and P is defined over real square lattices. This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.
2019-04-15
15:30--16:20hrs.
15:30--16:20hrs.
2019-04-15
16:30--17:30hrs.
16:30--17:30hrs.
Thomas Jordan. University of Bristol
Multifractal analysis for self-affine systems
Sala 1, PUC, Facultad de Matemáticas
Abstract:
joint work with Balazs Barany, Antti Kaenmaki and Michal Rams. If you consider a uniformly expanding Markov map on the interval and a continuous function. You can consider level sets of point for which the Birkhoff average is some fixed point. A typical problem I need multifractal analysis is to look at the dimension of these level sets. We will show how this can be done using the topological pressure and then how results can be obtained in the setting of certain self-affine sets in two dimensions using the sub-additive pressure and approximation by dominated subsystems.
Multifractal analysis for self-affine systems
Sala 1, PUC, Facultad de Matemáticas
Abstract:
joint work with Balazs Barany, Antti Kaenmaki and Michal Rams. If you consider a uniformly expanding Markov map on the interval and a continuous function. You can consider level sets of point for which the Birkhoff average is some fixed point. A typical problem I need multifractal analysis is to look at the dimension of these level sets. We will show how this can be done using the topological pressure and then how results can be obtained in the setting of certain self-affine sets in two dimensions using the sub-additive pressure and approximation by dominated subsystems.
2019-04-08
15:30--16:20hrs.
15:30--16:20hrs.
Italo Cipriano. PUC
Typical determinant of random matrices
Sala 1, PUC, Facultad de Matemáticas
Abstract:
Let $M_n$ be an $n\times n$ matrix with random entries in $\{+1,-1\}.$ In this talk I will discuss properties of the determinant of these matrices when $n$ tends to infinity. In particular, what is the probability that the determinant is zero? What is the typical value of the determinant? The talk is based on a result in [Tao and Vu, On random +-1 matrices, singularity and determinant. Random Structures and Algorithms, 2005 ].
Typical determinant of random matrices
Sala 1, PUC, Facultad de Matemáticas
Abstract:
Let $M_n$ be an $n\times n$ matrix with random entries in $\{+1,-1\}.$ In this talk I will discuss properties of the determinant of these matrices when $n$ tends to infinity. In particular, what is the probability that the determinant is zero? What is the typical value of the determinant? The talk is based on a result in [Tao and Vu, On random +-1 matrices, singularity and determinant. Random Structures and Algorithms, 2005 ].
2019-04-08
16:30--17:30hrs.
16:30--17:30hrs.
Gabriela Alexandra Estevez Jacinto. Universidade de São Paulo
Renormalization of multicritical circle maps
Sala 1, PUC, Facultad de Matemáticas
Abstract:
We study $C^3$ orientation preserving circle homeomorphisms with irrational rotation number and non-flat critical points. By Yoccoz, two of these maps with same irrational rotation are topologically conjugate. In this talk, we define the Renormalization operator of this kind of maps and assuming some properties of this operator we prove that the conjugacy is a $C^{1+\alpha}$ diffeomorphism. This result is valid for a total Lebesgue measure set of irrational rotation numbers. This is a joint work with Pablo Guarino (Universidade Federal Fluminense, Brazil).
Renormalization of multicritical circle maps
Sala 1, PUC, Facultad de Matemáticas
Abstract:
We study $C^3$ orientation preserving circle homeomorphisms with irrational rotation number and non-flat critical points. By Yoccoz, two of these maps with same irrational rotation are topologically conjugate. In this talk, we define the Renormalization operator of this kind of maps and assuming some properties of this operator we prove that the conjugacy is a $C^{1+\alpha}$ diffeomorphism. This result is valid for a total Lebesgue measure set of irrational rotation numbers. This is a joint work with Pablo Guarino (Universidade Federal Fluminense, Brazil).
2019-04-02
16:00-17:00hrs.
16:00-17:00hrs.
Roberto Markarian . Universidad de la República
Billares Caóticos. Resultados y métodos
PUC, Sala 1
Abstract:
La teoría matemática de billares estudia modelos simples de dinámicas en que hay choques de partículas y con las fronteras de un recipiente. Luego de indicar algunas motivaciones se expondrán propiedades ergódicas y estadísticas sobre las que se han hecho avances sustantivos en los últimos decenios, siguiendo muchos de los lineamientos abiertos, en particular, por la escuela de Ya. G. Sinai
Billares Caóticos. Resultados y métodos
PUC, Sala 1
Abstract:
La teoría matemática de billares estudia modelos simples de dinámicas en que hay choques de partículas y con las fronteras de un recipiente. Luego de indicar algunas motivaciones se expondrán propiedades ergódicas y estadísticas sobre las que se han hecho avances sustantivos en los últimos decenios, siguiendo muchos de los lineamientos abiertos, en particular, por la escuela de Ya. G. Sinai
2019-04-01
15:30-16:20hrs.
15:30-16:20hrs.
Sebastián Donoso. Cmm, Universidad de Chile
On subsets with no arithmetic progressions
CMM
Abstract:
For $N\in \mathbb{N}$, let $\nu(N)$ be the maximal cardinality of a subset of \{1,\ldots,N\} that contains no arithmetic progression of length 3. Finding upper and lower bounds for $\nu(N)$ has been a challenging problem for decades. In this talk I will survey this problem and present a proof of a theorem by Behrend in the 40's, that gave a surprising lower bound to $\nu(N)$.
On subsets with no arithmetic progressions
CMM
Abstract:
For $N\in \mathbb{N}$, let $\nu(N)$ be the maximal cardinality of a subset of \{1,\ldots,N\} that contains no arithmetic progression of length 3. Finding upper and lower bounds for $\nu(N)$ has been a challenging problem for decades. In this talk I will survey this problem and present a proof of a theorem by Behrend in the 40's, that gave a surprising lower bound to $\nu(N)$.
2019-04-01
16:30-17:30hrs.
16:30-17:30hrs.
Angel Pardo. Cmm, Universidad de Chile
Counting problem on infinite periodic billiards and translation surfaces
CMM
Abstract:
The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.
Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.
In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces. The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.
I will first present some quite precise results that are only valid for the wind-tree model (and some natural generalizations) and then, a general result which is valid for a.e. infinite periodic translation surfaces that uses completely different techniques: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups.
Counting problem on infinite periodic billiards and translation surfaces
CMM
Abstract:
The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus or, periodic trajectories, in a square billiard table.
Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.
In these talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces. The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.
I will first present some quite precise results that are only valid for the wind-tree model (and some natural generalizations) and then, a general result which is valid for a.e. infinite periodic translation surfaces that uses completely different techniques: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups.
2019-03-25
16:30 - 17:30hrs.
16:30 - 17:30hrs.
Michele Triestino. U. Bourgogne
Cantor dynamics and simple left-orderable groups
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central)
Abstract:
I will present a construction of simple groups of homeomorphisms of the real line.
Given a homeomorphism of a Cantor set $\sigma: X \to X$, consider the suspension $Y=X \times [0,1] / (x,1) \sim (\sigma(x),0)$, and look at the group $H_0(Y)$ of homeomorphisms of $Y$, isotopic to the identity. If $\sigma$ is minimal, then $H_0(Y)$ is simple [Aliste-Prieto - Petite], and I will describe countable subgroups $T(Y)$ which are also simple. These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, $T(Y)$ is finitely generated.
Joint work with Nicolás Matte Bon.
Cantor dynamics and simple left-orderable groups
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central)
Abstract:
I will present a construction of simple groups of homeomorphisms of the real line.
Given a homeomorphism of a Cantor set $\sigma: X \to X$, consider the suspension $Y=X \times [0,1] / (x,1) \sim (\sigma(x),0)$, and look at the group $H_0(Y)$ of homeomorphisms of $Y$, isotopic to the identity. If $\sigma$ is minimal, then $H_0(Y)$ is simple [Aliste-Prieto - Petite], and I will describe countable subgroups $T(Y)$ which are also simple. These are reminiscent of the classical Thompson groups, and feature several nice properties. For instance, when \sigma is a minimal subshift, $T(Y)$ is finitely generated.
Joint work with Nicolás Matte Bon.
2019-03-18
16:30-17:30hrs.
16:30-17:30hrs.
Yiwei Zhang. Center for Mathematical Sciences, Huazhong University of Science and Technology, China
Understanding physical mixing processes via transfer operator approach
Sala 1, PUC, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, Macul, La Florida
Abstract:
Industrial and chemical mixing processes of various kinds occur throughout nature and are vital in many technological applications.In the context of discrete dynamical systems, the transfer operator approach has been shown as a powerful tools from both theoretic and numerical viewpoint.
In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introductionon the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has "cutting and shuffling'' behaviour (i.e., composing with a permutation).
If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.
Understanding physical mixing processes via transfer operator approach
Sala 1, PUC, Facultad de Matemáticas, Av. Vicuña Mackenna 4860, Macul, La Florida
Abstract:
Industrial and chemical mixing processes of various kinds occur throughout nature and are vital in many technological applications.In the context of discrete dynamical systems, the transfer operator approach has been shown as a powerful tools from both theoretic and numerical viewpoint.
In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introductionon the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has "cutting and shuffling'' behaviour (i.e., composing with a permutation).
If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.
2019-03-14
16:30 -- 17:30hrs.
16:30 -- 17:30hrs.
Neil Dobbs. University of Geneva
Tba
Sala 5
Tba
Sala 5
Contáctenos
Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile.Teléfono (+56) 95504 4511.
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