Ana Cristina Araujo (IMPA, Brazil)
Title: Hölder regularity of the top Lyapunov exponent in random matrix products
Abstract:
This seminar will discuss the regularity of Lyapunov exponents for products of random invertible matrices in arbitrary dimension.
In particular, we show that the top Lyapunov exponent depends in a pointwise Hölder continuous way on the underlying probability measure, with respect to the Wasserstein-Hausdorff distance on the space of compactly supported measures. This holds under the assumptions that the top exponent is simple and that the associated equator has dimension at most one. This is a joint work with El Hadji Yaya Tall, Adriana Sánchez and Marcelo Viana.
Emma Dinowitz (CUNY, USA)
Title: Dimension and entropy via point to set principles
Abstract
TBA
Gonzalo Cousillas (Universidad de la República, Uruguay)
Title: Topology of a global attractor for homeomorphisms with the topological shadowing property in R^m
Abstract:
Iván Rodriguez (USP, Brazil)
Title: Extendable shift maps on generalized countable Markov shifts
Abstract:
It is well known that the topological entropy (or pressure) of the shift map on a Markov shift with a finite alphabet coincides with the noncommutative entropy (or noncommutative pressure) for the corresponding Cuntz–Krieger algebra.
Generalized countable Markov shifts correspond to the spectrum of commutative C*-subalgebras of Cuntz–Krieger algebras with infinitely many symbols (Exel–Laca algebras). We consider matrices for which the generalized space is a compactification of the standard countable Markov shift, where a special set of allowed finite words determines the additional points. In this setting, the shift map yields a partial dynamical system.
We show that the existence of a canonical map on the Exel–Laca algebra — a map for which, in the case of the renewal shift, the noncommutative entropy coincides with the Gurevich entropy — is equivalent to the existence of a continuous extension of the shift operator from the standard countable Markov shift to this compactification. Jointly work with Thiago Raszeja (PUC-Rio) and Rodrigo Bissacot (USP).
Odylo Costa (Sorbonne Université, France)
Title: Emergence of some periodic flows
Abstract:
The emergence of a dynamical system is a quantitative measure of how far the system is from being ergodic. In this seminar, we study emergence for a class of paradigmatic examples: flows whose every orbit is periodic but whose period function is unbounded. We provide a criterion for detecting low emergence and show that this criterion applies to several classical constructions of such flows. Finally, we present a construction of a counterexample to the Periodic Orbit Conjecture with positive order of emergence.
Yingjian Liu (IMPA, Brazil)
Title: Regularity of Lyapunov exponents at one-point spectra
Abstract:
We study the regularity of Lyapunov exponents for random matrix products at probability measures exhibiting a one-point spectrum, that is, when all Lyapunov exponents coincide. While continuity and Hölder continuity of Lyapunov exponents are well understood under irreducibility and simplicity assumptions, the degenerate case of a collapsed spectrum is typically excluded due to the lack of hyperbolicity.
In this talk, we show that despite this degeneracy, the top Lyapunov exponent enjoys enhanced regularity at one-point spectrum measures. Under compactness and semisimplicity conditions, we prove pointwise logarithmic Hölder continuity of the top Lyapunov exponent with respect to perturbations of the underlying probability measure. The mechanism relies on a quantitative control of the random walk on projective space and a careful analysis of mass accumulation governed by invariant structure.