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.
2019-07-01
15:30-16:30hrs.
15:30-16:30hrs.
Eduardo Oregon Reyes . Berkeley
The Avalanche Principle and Negative Curvature
Auditorio Jorge Krause ubicado en el segundo piso de la Facultad de Física, Av. Vicuña Mackenna 4860, Macul, La Florida
Abstract:
In 2001, M. Goldstein and W. Schlag introduced the Avalanche Principle, a quantitative sufficient condition for the operator norm $\|A_N\cdots A_1\|$ of a product of matrices in $\mathrm{SL}_2(\mathbb{R})$, to being similar to the product $\|A_N\|\cdots \|A_1\|$. Since then several refinements and generalizations have appeared in the literature. In this talk I will present a reformulation of this principle in terms of the geometry of the hyperbolic plane, and show how to extend it to metric spaces of negative curvature.
The Avalanche Principle and Negative Curvature
Auditorio Jorge Krause ubicado en el segundo piso de la Facultad de Física, Av. Vicuña Mackenna 4860, Macul, La Florida
Abstract:
In 2001, M. Goldstein and W. Schlag introduced the Avalanche Principle, a quantitative sufficient condition for the operator norm $\|A_N\cdots A_1\|$ of a product of matrices in $\mathrm{SL}_2(\mathbb{R})$, to being similar to the product $\|A_N\|\cdots \|A_1\|$. Since then several refinements and generalizations have appeared in the literature. In this talk I will present a reformulation of this principle in terms of the geometry of the hyperbolic plane, and show how to extend it to metric spaces of negative curvature.
2019-06-26
09:00hrs.
09:00hrs.
Paulina Cecchi. Usach
Invariant measures in Symbolic Dynamics: a topological, combinatorial and geometrical approach
Aula Magna de la Facultad de Derecho, USACH
Abstract:
Defensa de tesis de la estudiante Paulina Cecchi, para optar a los grados de Doctora en Ciencia mención Matemática de la Universidad de Santiago de Chile, y de Docteur, spécialité Informatique de la Universidad Paris-Diderot (Paris 7).
Directoras de tesis: Valérie Berthé y M. Isabel Cortez
Jurado de tesis: Emmanuel Jeandel, Alejandro Maass, Kostya Medynets, Ana Rechtman
Invariant measures in Symbolic Dynamics: a topological, combinatorial and geometrical approach
Aula Magna de la Facultad de Derecho, USACH
Abstract:
Defensa de tesis de la estudiante Paulina Cecchi, para optar a los grados de Doctora en Ciencia mención Matemática de la Universidad de Santiago de Chile, y de Docteur, spécialité Informatique de la Universidad Paris-Diderot (Paris 7).
Directoras de tesis: Valérie Berthé y M. Isabel Cortez
Jurado de tesis: Emmanuel Jeandel, Alejandro Maass, Kostya Medynets, Ana Rechtman
2019-06-24
16:30-17:30hrs.
16:30-17:30hrs.
Ana Rechtman Bulajich. Unam
El conjunto minimal de los flujos de Kuperberg
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
En 1994, K. Kuperberg construyó ejemplos de flujos suaves sin órbitas periódicas en cualquier variedad cerrada y sin frontera de dimensión 3, demostrando así que la conjetura de Seifert es falsa. En la plática presentaré una descripción topológica del conjunto minimal de estos ejemplos, que es un conjunto minimal excepcional de dimensión topológica 2. Explicaré además algunos resultados relacionados con la forma (shape theory) y la entropía de dichos ejemplos.
Los resultados fueron obtenidos en colaboración con Steve Hurder.
El conjunto minimal de los flujos de Kuperberg
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
En 1994, K. Kuperberg construyó ejemplos de flujos suaves sin órbitas periódicas en cualquier variedad cerrada y sin frontera de dimensión 3, demostrando así que la conjetura de Seifert es falsa. En la plática presentaré una descripción topológica del conjunto minimal de estos ejemplos, que es un conjunto minimal excepcional de dimensión topológica 2. Explicaré además algunos resultados relacionados con la forma (shape theory) y la entropía de dichos ejemplos.
Los resultados fueron obtenidos en colaboración con Steve Hurder.
2019-06-24
15:30-16:20hrs.
15:30-16:20hrs.
Italo Cipriano. PUC
Non-dense $\alpha\beta$-sets
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
An $\alpha\beta$-set E is a subset of [0,1] with the property if x belongs to E then $x+\alpha$ (mod 1) or $x+\beta$ (mod 1) too, or both. In this talk, I will provide the original proof by Katznelson of the existence of non-dense $\alpha\beta$-sets, answering a question by Engelking. For some time $\alpha\beta$-sets were left unattended, indeed, until recently, see for example [Affine embeddings of Cantor sets and dimension of $\alpha\beta$-sets, Feng, De-Jun; Xiong, Ying, 2016] and [Multi-rotations on the unit circle, Han Yu, 2019].
Non-dense $\alpha\beta$-sets
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
An $\alpha\beta$-set E is a subset of [0,1] with the property if x belongs to E then $x+\alpha$ (mod 1) or $x+\beta$ (mod 1) too, or both. In this talk, I will provide the original proof by Katznelson of the existence of non-dense $\alpha\beta$-sets, answering a question by Engelking. For some time $\alpha\beta$-sets were left unattended, indeed, until recently, see for example [Affine embeddings of Cantor sets and dimension of $\alpha\beta$-sets, Feng, De-Jun; Xiong, Ying, 2016] and [Multi-rotations on the unit circle, Han Yu, 2019].
2019-06-17
16:30-17:30hrs.
16:30-17:30hrs.
Joel Moreira. Northwestern University
The Erdos sumset conjecture
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
The Erdos sumset conjecture predicts that any set of natural numbers with positive density must contain the arithmetic sum A+B of two infinite sets A and B. I will present a recent solution to this conjecture, obtained jointly with F. Richter and D. Robertson. The proof involves a modified version of the correspondence principle devised by Furstenberg in 1977 to convert certain problems from combinatorics into the realm of ergodic theory, and two variations of the decomposition of an arbitrary function on a measure preserving system into an almost periodic and a weak mixing components.
The Erdos sumset conjecture
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
The Erdos sumset conjecture predicts that any set of natural numbers with positive density must contain the arithmetic sum A+B of two infinite sets A and B. I will present a recent solution to this conjecture, obtained jointly with F. Richter and D. Robertson. The proof involves a modified version of the correspondence principle devised by Furstenberg in 1977 to convert certain problems from combinatorics into the realm of ergodic theory, and two variations of the decomposition of an arbitrary function on a measure preserving system into an almost periodic and a weak mixing components.
2019-06-17
15:30-16:20hrs.
15:30-16:20hrs.
Anh Le. Northwestern University
Multiple correlations and nilsequences
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
Multiple correlation sequences first appeared implicitly in Furstenberg's proof of Szemeredi's theorem. Bergelson, Host and Kra later proved they can be decomposed into the sum of a nilsequence and a sequence tending to zero in density. Motivated by this, Frantzikinakis asks whether we have a similar decomposition along the sequence of primes p_n, or Hardy sequence [n^c], or 2^n. In this talk, I'll answer this question affirmatively. Even though the positive answers to the prime and Hardy sequences are expected, the positive answer to 2^n is somewhat surprising and has an interesting connection with harmonic analysis.
Multiple correlations and nilsequences
CMM (Beauchef 851, Torre Norte, 7mo piso, Sala de Seminarios John Von Neumann)
Abstract:
Multiple correlation sequences first appeared implicitly in Furstenberg's proof of Szemeredi's theorem. Bergelson, Host and Kra later proved they can be decomposed into the sum of a nilsequence and a sequence tending to zero in density. Motivated by this, Frantzikinakis asks whether we have a similar decomposition along the sequence of primes p_n, or Hardy sequence [n^c], or 2^n. In this talk, I'll answer this question affirmatively. Even though the positive answers to the prime and Hardy sequences are expected, the positive answer to 2^n is somewhat surprising and has an interesting connection with harmonic analysis.
2019-06-10
16:30-17:30hrs.
16:30-17:30hrs.
Ignacio Huerta. Usach
Linearization of a nonautonomous unbounded system with nonuniform contraction: A Spectral Approach
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
In this session we will address the topic of topological equivalence between a linear system and a nonlinear perturbation of it. We will review, as an introduction, the works carried out in the context of nonautonomous differential equations, in order to give way to my research work which involves the linearization of a nonautonomous unbounded system from a point of view of "nonautonomous spectral theory" .
Linearization of a nonautonomous unbounded system with nonuniform contraction: A Spectral Approach
USACH, Sala de seminarios del 4to piso del Departamento de Matemáticas y Ciencia de la computación ( Las Sophoras nº 173, Santiago, Estación Central).
Abstract:
In this session we will address the topic of topological equivalence between a linear system and a nonlinear perturbation of it. We will review, as an introduction, the works carried out in the context of nonautonomous differential equations, in order to give way to my research work which involves the linearization of a nonautonomous unbounded system from a point of view of "nonautonomous spectral theory" .
2019-06-10
15:30-16:30hrs.
15:30-16:30hrs.
Andreas Koutsogiannis. The Ohio State University, Usa
Multiple averages along polynomials and applications to joint ergodicity
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 limiting behavior of multiple ergodic averages is a central problem in ergodic theory for the past forty years. The main reason for this is that, from such results, one can get deep applications in many areas of mathematics as dynamical systems, combinatorics, number theory etc. In this talk we will provide a method on how someone can start with a multiple ergodic average and find, by using the concatenation theorem of Tao and Ziegler, characteristic factors for the aforementioned, along polynomials, averages. With this approach, which exploits a strong polynomial exhaustion technique induction (known as PET induction), we also answer a question due to Bergelson. This is a joint recent work with S. Donoso and W. Sun.
Multiple averages along polynomials and applications to joint ergodicity
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 limiting behavior of multiple ergodic averages is a central problem in ergodic theory for the past forty years. The main reason for this is that, from such results, one can get deep applications in many areas of mathematics as dynamical systems, combinatorics, number theory etc. In this talk we will provide a method on how someone can start with a multiple ergodic average and find, by using the concatenation theorem of Tao and Ziegler, characteristic factors for the aforementioned, along polynomials, averages. With this approach, which exploits a strong polynomial exhaustion technique induction (known as PET induction), we also answer a question due to Bergelson. This is a joint recent work with S. Donoso and W. Sun.
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
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
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
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.
2019-05-27
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
2019-05-13
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
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.
2019-04-22
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.
2019-04-15
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.
Contáctenos
Edificio Rolando Chuaqui, Campus San Joaquín. Avda. Vicuña Mackenna 4860, Macul, Chile.Teléfono (+56) 95504 4511.
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