H\"older properties of Weierstrass-like solutions of $\theta$-twisted cohomological equations

TL;DR

This paper demonstrates that solutions to certain twisted cohomological equations for expanding circle maps are exactly $ heta$-Hölder continuous, providing new examples of functions with precisely determined optimal Hölder exponents.

Contribution

It establishes the exact Hölder regularity of solutions to $ heta$-twisted cohomological equations, revealing their precise Hölder exponents at almost every point.

Findings

Solutions are $ heta$-Hölder continuous

Solutions are not $( heta+ ext{any }eta)$-Hölder continuous for $eta>0$

Provides new examples of functions with known optimal Hölder exponents

Abstract

It is proved that bounded solutions of modified ($\theta$-twisted) cohomological equations for expanding circle maps are $\theta$-H\"{o}lder continuous but are not $(\theta+\gamma)$-H\"{o}lder continuous for every $\gamma>0$ at almost every point. This gives new examples of "nonlinear" Weierstrass-like functions for which the optimal \holder exponent at most points is known.