Preimage pressure on subsets and multifractal analysis

Abstract

In this paper, pointwise preimage pressures for (non-invertible) continuous maps on any subset (not necessarily compact or invariant) are introduced via Carathéodory-Pesin construction. A variational inequality for preimage pressure of saturated sets is then obtained. In particular, we prove that the preimage pressure of the set of generic points for an ergodic measure equals the metric preimage pressure of the measure, which extends Bowen’s theorem to preimage pressure. We also use the thermodynamic formalism of preimage pressure to obtain some estimates on Birkhoff level sets.