On orbit complexity of dynamical systems: intermediate value property and level set related to a Furstenberg problem

TL;DR

This paper investigates the orbit complexity of certain dynamical systems, providing solutions for entropy and Hausdorff dimension problems, and applies these results to systems like beta-transformations and Furstenberg's problem.

Contribution

It introduces new methods to determine the Hausdorff dimension of level sets and orbit closures in symbolic dynamics, addressing the Furstenberg problem.

Findings

Solved the lowering topological entropy problem for subsystems.

Determined the Hausdorff dimension of level sets with given complexity.

Calculated the dimension of Furstenberg level sets.

Abstract

For symbolic dynamics with some mild conditions, we solve the lowering topological entropy problem for subsystems and determine the Hausdorff dimension of the level set with given complexity, where the complexity is represented by Hausdorff dimension of orbit closure. These results can be applied to some dynamical systems such as $\beta$-transformations, conformal expanding repeller, etc. We also determine the dimension of the Furstenberg level set, which is related to a problem of Furstenberg on the orbits under two multiplicatively independent maps.