The variation of invariant graphs in forced systems

TL;DR

This paper investigates how topological features of forcing systems influence the regularity of invariant graphs, providing sharp conditions for bounded variation based on Markov map parameters and illustrating with examples.

Contribution

It introduces a sharp condition for bounded variation of sync functions in forced systems, linking topological entropy to regularity properties.

Findings

Derived a sharp condition for bounded variation of sync functions

Linked topological entropy of Markov maps to invariant graph regularity

Provided illustrative examples demonstrating the theoretical results

Abstract

In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, that involves the Markov map topological entropy. The results are illustrated with examples.