Dimension theoretical properties of generalized Baker's transformations

TL;DR

This paper investigates the parameter-dependent properties of generalized Baker's transformations, revealing regions with absolutely continuous ergodic measures and others where classical dimension principles fail.

Contribution

It identifies parameter domains where the variational principle for Hausdorff dimension does not hold, highlighting complex dimension behavior.

Findings

Existence of parameter regions with absolutely continuous ergodic measures

Regions where the variational principle for Hausdorff dimension fails

Demonstration of intricate dimension-theoretic properties in generalized Baker's transformations

Abstract

We show that for generalized Baker's transformations there is a parameter domain where we have an absolutely continuous ergodic measure and in direct neighborhood there is a parameter domain where not even the variational principle for Hausdorff dimension holds.