Banca de DEFESA: VICTOR BORGES CARNEIRO

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : VICTOR BORGES CARNEIRO
DATE: 23/03/2022
TIME: 14:00
LOCAL: Defesa remota
TITLE:

Thermodynamical and spectral phase transitions for local diffeomorphisms on the circle

KEY WORDS:

Thermodynamic formalism, Transfer Operator; Linear Response Formula; Large Deviation Principles; Multifractal Analysis.

PAGES: 79
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUBÁREA: Geometria e Topologia
SPECIALTY: Sistemas Dinâmicos
SUMMARY:

It is known that all uniformly expanding dynamics have no phase transition with respect to H\"older continuous potentials. In this work we show that given a local diffeomorphism $f$ on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function $\R \ni t \mapsto P_{top}(f,-t\log|Df|)$ is not analytical. In other words, $f$ has a thermodynamic phase transition with respect to geometric potential. Assuming that $f$ is transitive and that $Df$ is Hölder continuous, we show that there exists $t_0 \in (0,1]$ such that 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 has not the spectral gap property for all $t\geq t_0$. Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case $f$ has a unique thermodynamic phase transition and it occurs in $t_0$. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then $t_0=1$. As a consequence of thermodynamical and spectral phase transition, we obtain applications on multifractal analysis for the Lyapunov spectrum.

BANKING MEMBERS:
Presidente - 2006794 - THIAGO BOMFIM SAO LUIZ NUNES
Interno - 1318879 - AUGUSTO ARMANDO DE CASTRO JUNIOR
Interno - 1654597 - PAULO CESAR RODRIGUES PINTO VARANDAS
Externo à Instituição - DANIEL SMANIA BRANDAO - USP
Externo à Instituição - LEANDRO MARTINS CIOLETTI - UnB