Phase transitions for transitive local diffeomorphism with break points on the circle and Holder continuous potentials

It is known that if $f: \mathbb{S}^{1} \rightarrow \mathbb{S}^{1}$ is a transitive $C^{1+\alpha}$-local diffeomorphism non-invertible and non-uniformly expanding, then there is a unique parameter $t_{0} \in (0 , 1]$ such that the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f , -t\log|Df|)$ is not analytic, in particular $f$ has a phase transition with respect to potential $\phi := -\log|Df|$. On the other hand, it is known that for continuous potentials, the topological pressure function can exhibit an infinite number of phase transitions. In this paper, we study the possibilities of the behaviour of the topological pressure function and transfer operator for transitive local diffeomorphism with break points on the circle and H\"older continuous potentials. In particular, we showed that: (1) there is an open and dense subset of continuous potentials such that if a H\"older continuous potential belongs to this subset, then it has no phase transition and the transfer operator has the spectral gap property; (2) if a H\"older continuous potential has a phase transition, then the topological pressure function and the associated transfer operator are described. Consequently, every H\"older continuous potential has at most two phase transitions and the set of smooth potentials such that $\mathcal{L}_{f,\phi}$ has the spectral gap property, acting on the H\"older continuous space, is dense in the uniform topology. Furthermore, we obtain applications for multifractal analysis of the Birkhoff average.