LisMath Seminar 

21/04/2022, 14:30 — 15:00 — Online
Carllos Eduardo Holanda, Instituto Superior Técnico, Universidade de Lisboa

Nonadditive thermodynamic formalism and multifractal analysis for flows

In this work we consider the study of the nonadditve thermodynamic formalism and multifractal analysis for continuous time dynamical systems. We introduce a version of the nonadditive topological pressure for flows and we describe some of its main properties. In particular, we discuss how the nonadditive topological pressure varies with the data and we establish a variational principle in terms of the Kolmogorov-Sinai entropy. In the more specific context of almost additive families of continuous functions, we establish an appropriate version of the classical variational principle for the topological pressure, and obtain the existence and uniqueness of equilibrium and Gibbs measures for families with bounded variation.

Building on the construction of equilibrium measures, we establish a conditional variational principle for the multifractal spectra of an almost additive family with respect to a continuous flow $\Phi$ such that the entropy map $\mu \mapsto h_{\mu} (\Phi) $ is upper-semicontinuous. We also show that the spectrum is continuous and that in the case of hyperbolic flows the corresponding irregular sets have full topological entropy.

In the last part, we introduce a version of the nonlinear thermodynamic formalism for flows. In this context, building on the multifractal analysis developed earlier, we discuss the existence, uniqueness, and characterization of equilibrium measures for an almost additive family with tempered variation. Moreover, we consider with some care the special case of an additive family for which it is possible to strengthen some of the results.