Weighted topological pressure revisited

TL;DR

This paper generalizes the concept of weighted topological pressure to higher dimensions, proving a variational principle that facilitates calculating the Hausdorff dimension of complex affine-invariant sets.

Contribution

It extends Tsukamoto's approach to higher dimensions, redefining weighted topological entropy and pressure, and establishes a variational principle for these invariants.

Findings

Provides a method to compute Hausdorff dimensions of self-affine sponges

Generalizes weighted topological pressure to arbitrary dimensions

Proves a variational principle for the new definitions

Abstract

Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto (2022) redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto's approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.