Coordenadas:
Quinta-feira 17 de setembro das 16h00 às 17h45
Sala L856
Palestrante: Tanja Isabelle Schindler
(Department of Mathematics and Statistics, University of Exeter, UK)
Título:
Diffraction, spectral and fractal analysis of (generalized) Thue--Morse measures and applications to large deviations
Resumo:
The classic Thue–Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. It arises by using a diffraction algorithm of the aperiodic Thus--Morse sequence resampling diffraction one gets from physical experiments. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism (a method originating from statistical mechanics, but widely used in ergodic theory). In the generalized case, we consider Riesz products that can be regarded as diffraction measures of generalized Thue–Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising from the doubling map together with an observable exhibiting a logarithmic singularity. For this system, we develop a generalized thermodynamic formalism beyond the standard setting, which yields explicit formulas for Birkhoff and dimension spectra. Moreover, this new formulation of the thermodynamic formalism allows to give precise large deviation results for unbounded observables with a qualitatively different result depending if the logarithmic singularity appears at a periodic or pre-periodic non periodic point.
The talk will first give an intrduction into diffraction measures and the classical thermodynamic formalism and explains then the novelty of the results.
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.