Curriculum Vitae Zhiren Wang Positions
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Centralizers of Hyperbolic and Kinematic-Expansive Flows
CENTRALIZERS OF HYPERBOLIC AND KINEMATIC-EXPANSIVE FLOWS LENNARD BAKKER, TODD FISHER, AND BORIS HASSELBLATT ABSTRACT. We show generic C1 hyperbolic flows (Axiom A and no cycles, but not transitive Anosov) commute with no C1-diffeomorphism other than a time-t map of the flow itself. Kinematic expansivity, a substantial weakening of expansivity, implies that C0 flows have quasi-discrete C0-centralizer, and additional conditions broader than transitivity then give discrete C0-centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set. CONTENTS 1. Introduction 1 1.1. Discrete-time centralizers: commuting diffeomorphisms 2 1.2. Continuous-time centralizers: commuting flows 2 1.3. Diffeomorphisms commuting with flows 3 2. Background 4 3. Centralizers for kinematic-expansive flows 6 4. Centralizers for Axiom-A flows 8 4.1. Fixing the basins 9 4.2. Rigidity: Local coincidence to global coincidence 10 4.3. Linearization theorems for flows and maps 11 4.4. Lie group of commuting matrices 12 4.5. Perturbations near attractors 13 References 17 1. INTRODUCTION It is natural to expect a dynamical system to have no symmetries unless it is quite special. Symmetries correspond to the existence of a diffeomorphism or flow that commutes with the given dynamics, so one expects flows to “typically” have small centralizers, i.e., to commute with few flows or diffeomorphisms. Our topological results imply that expansive continuous flows have essentially discrete centralizer; the natural condition for this is a weakening of expansivity that does not allow reparameterizations and hence requires only a “kinematic” or dynamical separation of orbits rather than a “geometric” one. -
María Alejandra RODRIGUEZ HERTZ FRUGONI
Curriculum Vitae María Alejandra RODRIGUEZ HERTZ FRUGONI Actualizado: 23/03/2012 Publicado: 23/03/2012 Sistema Nacional de Investigadores Ciencias Naturales y Exactas / Matemáticas Categorización actual: Nivel II Ingreso al SNI: Nivel II (01/03/2009) Datos generales Información de contacto E-mail: [email protected] Teléfono: 27114462 int 123 Dirección: IMERL Julio Herrera y Reissig 565 - 11300 Montevideo Uruguay URL: http://www.fing.edu.uy/~jana Institución principal IMERL / Facultad de Ingeniería - UDeLaR / Universidad de la República / Uruguay Dirección institucional Dirección: Facultad de Ingeniería - UDeLaR / IMERL / 11300 / Montevideo / Montevideo / Uruguay Teléfono: (+598) 2711 44 62 Fax: 2711 44 62 E-mail/Web: [email protected] / http://www.fing.edu.uy/~jana Formación Formación concluida Formación académica/Titulación Posgrado 1994 - 1999 Doctorado Doctorado en Matemática (UDELAR-PEDECIBA) Facultad de Ciencias - UDeLaR , Uruguay Título: Propiedades Co genéricas de conjuntos estables e inestables de difeomorfismos Tutor/es: -- Obtención del título: 1999 Becario de: Programa de Desarrollo de las Ciencias Básicas , Uruguay Areas del conocimiento: Ciencias Naturales / Matemáticas / Matemática Pura / Sistemas Dinámicos Grado 1988 - 1993 Grado Licenciatura en Matemática Facultad de Ciencias Exactas, Ingeniería y Agrimensura - UNR , Argentina Tutor/es: M. Castagnino Obtención del título: 1994 Becario de: Fundación Bolsa de Comercio de Buenos Aires , Argentina Areas del conocimiento: Ciencias Naturales / Matemáticas / Matemática Pura / -- -
STABLE ERGODICITY 1. Introduction a Dynamical System Is Ergodic If It
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 1, Pages 1{41 S 0273-0979(03)00998-4 Article electronically published on November 4, 2003 STABLE ERGODICITY CHARLES PUGH, MICHAEL SHUB, AND AN APPENDIX BY ALEXANDER STARKOV 1. Introduction A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen are separated by a wall. When the wall is removed, the gasses mix thoroughly as time tends to infinity. In contrast think of the rotation of a sphere. All points move along latitudes, and ergodicity fails due to existence of invariant equatorial bands. Ergodicity is stable if it persists under perturbation of the dynamical system. In this paper we ask: \How common are ergodicity and stable ergodicity?" and we propose an answer along the lines of the Boltzmann hypothesis { \very." There are two competing forces that govern ergodicity { hyperbolicity and the Kolmogorov-Arnold-Moser (KAM) phenomenon. The former promotes ergodicity and the latter impedes it. One of the striking applications of KAM theory and its more recent variants is the existence of open sets of volume preserving dynamical systems, each of which possesses a positive measure set of invariant tori and hence fails to be ergodic. Stable ergodicity fails dramatically for these systems. But does the lack of ergodicity persist if the system is weakly coupled to another? That is, what happens if you have a KAM system or one of its perturbations that refuses to be ergodic, due to these positive measure sets of invariant tori, but somewhere in the universe there is a hyperbolic or partially hyperbolic system weakly coupled to it? Does the lack of egrodicity persist? The answer is \no," at least under reasonable conditions on the hyperbolic factor. -
Instituto Nacional De Matemática Pura E Aplicada
Instituto Nacional de Matemática Pura e Aplicada Doctoral Thesis KNEADING SEQUENCES FOR TOY MODELS OF HÉNON MAPS Ermerson Rocha Araujo Rio de Janeiro July 30, 2018 Instituto Nacional de Matemática Pura e Aplicada Ermerson Rocha Araujo KNEADING SEQUENCES FOR TOY MODELS OF HÉNON MAPS Thesis presented to the Post-graduate Program in Math- ematics at Instituto Nacional de Matemática Pura e Aplicada as partial fulfillment of the requirements for the degree of Doctor in Philosophy in Mathematics. Advisor: Enrique Ramiro Pujals Rio de Janeiro 2018 Amar e mudar as coisas me interessa mais. Alucinação - Belchior To Sheila Cristina and José Ribamar Acknowledgments I would like to especially express my gratitude to my advisor Enrique Pujals. I thank him for constant encouragement and for many insightful conversa- tions during the preparation of the thesis. Our meetings were almost always held in the IMPA’s restaurant. Which was very good for me, because this took away the pressure that the elaboration of a thesis puts on the students. At each meeting with him many new ideas arose for the problems we were working on. He made extremely complicated arguments seem simple. Besides being an extraordinary mathematician, he is a great person, generous and very attentive. So I can say that it was an honor and an immense adventure to have been a Pujals’ student. I thank IMPA for these four years in which I had the opportunity to grow as a mathematician. In addition, I would like to express my gratitude to the Graduate Center-CUNY in New York City, USA, where I have been twice.