Chaos and frequent hypercyclicity for composition operators

The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Rusza in 2015 that for backward weighted shifts on $\ell_p(\mathbb{Z})$, the notions chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^p$-spaces the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible frequently hypercyclic operator on $\ell_1$ whose inverse is not frequently hypercyclic is constructed.