Palestrante: Sergey Tikhomirov (PUC-Rio)
Coordenadas:
quarta-feira, 11 de março, 17h na sala L856
Título:
Generalized $(C, \lambda)$-structure in Banach spaces: shadowing, robustness and semi-structural stability.
Resumo:
Hyperbolicity is a central concept in the theory of structural stability on manifolds.
This talk is devoted to the notion of hyperbolicity in Banach spaces.
Relatively recently [1, 2], the concept of \emph{generalized hyperbolicity} for linear mappings in Banach spaces was introduced, and it was shown in particular that it implies both the shadowing property and an analogue of the Grobman--Hartman theorem.
In this talk, combining the ideas of the $(C, \lambda)$-structure [3] for finite-dimensional manifolds with the notion of generalized hyperbolicity, we introduce a \emph{generalized $(C, \lambda)$ -structure} for nonlinear diffeomorphisms of Banach spaces [4].
The key novelty is the possibility of discontinuous dependence of "hyperbolic splitting" on a point and "dimension variability" along trajectory for both stable and unstable foliations.
We prove that the generalized $(C, \lambda)$ -structure implies:
-- finite Lipschitz shadowing in arbitrary Banach spaces,
-- infinite Lipschitz shadowing and periodic shadowing in reflexive Banach spaces.
-- robustness of generalized $(C, \lambda)$ -structure under $C^1$ small perturbations.
Situation with structural stability is more involved, we managed to prove only its weak version (semi-conjugacy from both sides with $C^1$-small perturbations) under extra assumption of continuity of the splitting.
[1] N. Bernardes, P. Cirilo, U. Darji, A. Messaoudi, E. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018) 796–816.
[2] P. Cirilo, B. Gollobit, and Enrique Pujals. Dynamics of generalized hyperbolic linear operators. Advances in Mathematics 387 (2021): 107830.
[3] S. Yu Pilyugin. Generalizations of the notion of hyperbolicity. J. Difference Equ. Appl. 12 (2006), 271–282.
[4] S. Tikhomirov. Generalized $(C, \lambda)$-structure for nonlinear diffeomorphisms of Banach spaces. https://arxiv.org/abs/2510.05499
Palestrante:
Jérôme Rousseau (UFBA)
Coordenadas:
Segunda-feira, 14 de abril, 17h na sala L863
Título:
Furacão, ZIP e DNA: uma introdução matemática aos eventos raros
Resumo:
Na primeira parte desta palestra, apresentarei uma introdução probabilística e dinâmica aos eventos raros, e explicarei como essa teoria pode, em certos casos, ter aplicações práticas — por exemplo, no estudo de eventos climáticos extremos, algoritmos de compressão de dados e comparação de sequências de DNA. Serão discutidas as noções de entropia, recorrência de Poincaré e dimensões fractais.
Na segunda parte da palestra (de caráter mais técnico), focarei na distância mínima entre órbitas e no problema da maior subcadeia comum, explicando suas conexões com a dimensão de correlação e a entropia de Rényi.
Palestrante:
Thiago Costa Raszeja (PUC-Rio)
Coordenadas:
4ª-feira, 2 de abril, 17h, sala de seminários (L856), DMat, PUC-Rio
Título:
Generalized thermodynamic formalism: Exel-Laca algebras, generalized Markov shift spaces, eigenmeasures and phase transitions
Resumo:
In 1999, R. Exel and M. Laca solved the problem of extending the Cuntz-Krieger algebras to infinite alphabets. From this construction, the notion of generalized countable Markov shift (GCMS) arises, a completion of the usual Markov shift spaces (CMS) and depends only on the transition matrix, by including families of finite words that are invariant under the shift action. This new space is always locally compact, and even for a large class of non-locally compact CMS', their corresponding GCMS' are compact. We developed the thermodynamic formalism for this generalized context by extending notions of eigenmeasures of the Ruelle's transformation, conformal measures, Gurevich pressure, etc. We proved that the pressure of a point, a notion for pressure that considers the finite words constructed by M. Denker and M. Yuri, coincides with the Gurevich pressure for a wide class of potentials and GCMS. New conformal and eigenmeasures were discovered, as well as new phase transition phenomena. In particular we emphasize the length-type phase transition, where the eigenmeasure passes from living on the CMS to its complement when we cool down the system. A complete topological description of the GCMS was developed and allowed us to connect the new eigenmeasures and conformal measures via weak*-limits on the inverse of temperature parameter.
This is a joint work with R. Bissacot, R. Exel and R. Frausino.