Nonexistence of spectral gaps in Hölder spaces for continuous time dynamical systems

Israel. J. Math. 247 (2022) 987-991.

Ian Melbourne, Nicolò Paviato and Dalia Terhesiu

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 Hölder space with Hölder exponent greater than 1/2 unless it consists entirely of coboundaries.