Banca de DEFESA: AFONSO FERNANDES DA SILVA

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : AFONSO FERNANDES DA SILVA
DATE: 07/03/2023
TIME: 13:30
LOCAL: Videoconferência
TITLE:

Contributions To Phase Transition Of Intermittent Skew-Product And Piecewise Monotone Dynamics On The Circle

KEY WORDS:

Phase Transition; Thermodynamic formalism; Transfer Operator; Large Deviation Principles; Multifractal Analysis.

PAGES: 76
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUBÁREA: Geometria e Topologia
SPECIALTY: Sistemas Dinâmicos
SUMMARY:

It is well known that all transitive uniformly expanding or hyperbolic dynamics have no phase transition with respect to H\"older continuous potentials. For more general dynamics, It is still an open question to classify all the dynamics having phase transition with respect to a certain class of regular potential. In dimension one, due to the work of Bomfim-Victor \cite{BC21}, it was proved that for all transitive $C^{1+\alpha}-$local diffeomorphism $f$ on the circle that is neither a uniformly expanding map nor invertible, has an unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\R \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except in a point $t_{0} \in (0 , 1]$. Furthermore, they proved spectral phase transitions, more specific, the transfer operator $\Lo_{f,-t\log|Df|}$ acting on the space of H\"older continuous functions, has the spectral gap property for all $t<t_0$ and does not have the spectral gap property for all $t\geq t_0$. We aim to prove similar results for two special cases of dynamics: a co-dimension $1$ partially hyperbolic endomorphism and transitive piecewise monotone on the circle. For the higher dimension, endomorphisms, we prove that thermodynamic and spectral phase transition lead to multifractal analysis of the Lyapunov spectrum, in particular we exhibit a class of partially hyperbolic endomorphism having phase transition with respect to the geometric potential in the central direction and describe the multifractal analysis of the central Lyapunov spectrum. For transitive piecewise monotone maps, we prove that the set of Hölder continuous potentials which doesn't have spectral and thermodynamic phase transition is dense in the uniform topology and the set of Hölder continuous potentials that has phase transition are not dense. Furthermore, we provide a description of phase transition based on the properties of the transfer operator and the type of convexity of the topological pressure function. In particular, we describe the behavior of the topological pressure function and the transfer operator associated.

COMMITTEE MEMBERS:
Presidente - 2006794 - THIAGO BOMFIM SAO LUIZ NUNES
Interno - 1518921 - VITOR DOMINGOS MARTINS DE ARAUJO
Interno - 1654597 - PAULO CESAR RODRIGUES PINTO VARANDAS
Interno - 1318879 - AUGUSTO ARMANDO DE CASTRO JUNIOR
Externa à Instituição - KATRIN GRIT GELFERT - UFRJ
Externo à Instituição - RICARDO TUROLLA BORTOLOTTI - UFPE
Externo à Instituição - ANDERSON REIS DA CRUZ - UFRB