Thermodynamical and spectral phase transition for local diffeomorphisms in the circle

Abstract: It is known that all uniformly expanding dynamics have no phase transition with respect to H\"older continuous potentials. In this paper we show that given a local diffeomorphism $f$ on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , -t\log |Df|)$ is not analytical. In other words, $f$ has a thermodynamic phase transition with respect to geometric potential. Assuming that $f$ is transitive and that $Df$ is H\"older continuous, we show that there exists $ t_{0} \in (0 , 1]$ such that the transfer operator $\mathcal{L}{f, -t\log|Df|}$, acting on the space of H\"older continuous functions, has the spectral gap property for all $t < t{0}$ and has not the spectral gap property for all $t \geq t_{0}$. Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case $f$ has a unique thermodynamic phase transition and it occurs in $t_{0}$. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then $t_0 = 1.$