Nonexistence of spectral gaps in H\"older spaces for continuous time dynamical systems

TL;DR

This paper demonstrates that for continuous dynamical systems, the transfer operator cannot have a spectral gap in H"older spaces with exponent greater than 1/2 unless the space is trivial, highlighting a fundamental limitation on smoothness.

Contribution

It establishes a fundamental restriction on the smoothness of spaces supporting spectral gaps for transfer operators in continuous dynamical systems.

Findings

Spectral gaps cannot exist in H"older spaces with exponent > 1/2 unless the space is trivial.

The result applies to the transfer operator in continuous dynamical systems.

Spaces with higher regularity do not support spectral gaps unless they consist solely of coboundaries.

Abstract

We show that there is a natural restriction on the smoothness of spaces where the transfer operator for a continuous dynamical system has a spectral gap. Such a space cannot be embedded in a H\"older space with H\"older exponent greater than 1/2 unless it consists entirely of coboundaries.