Transience and multifractal analysis

TL;DR

This paper investigates the multifractal spectra of dissipative dynamical systems, revealing how transience affects the validity of variational principles and leads to discontinuities in Lyapunov spectra.

Contribution

It establishes a conditional variational principle for Birkhoff averages on recurrent parts and demonstrates the impact of transience on multifractal spectra, including the first example of a transitive map with discontinuous Lyapunov spectrum.

Findings

Conditional variational principle holds on recurrent parts

Discontinuous Lyapunov spectrum in a transitive map

Transience causes pathological multifractal features

Abstract

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit the first example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.