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In \cite{BC21} is shown that if $f: \mathbb{S}^{1} \rightarrow\mathbb{S}^{1}$ is a $C^{1+\alpha}$-local diffeomorphism non-invertible andnon-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 phase transition withrespect to potential $\phi := -\log|Df|$. On the other hand, by the works\cite{KQW21,KQ22}, for continuous potentials the topological pressure functioncan be wild; in particular, it can be infinite phase transitions. In thispaper, we study the possibilities of behaviour of the topological pressurefunction and transfer operator for transitive local diffeomorphism with breakpoints on the circle and Holder continuous potentials. In particular, we showedthat: 1-that there is an open and dense subset of continuous potentials suchthat if a Holder continuous potential belongs to subset then it has no phasetransition and the transfer operator has spectral gap property; 2-for Holdercontinuous potentials, not cohomologous to constant, the existence of a phasetransition is equivalent to the topological pressure function not beingstrictly convex; 3-if a Holder continuous potential has phase transition, thenthe topological pressure function and the transfer operator associated havebehaviour similar to \cite{BC21}. Consequently, every Holder continuouspotential has at most two phase transitions and the set of smooth potentialsuch that $\mathcal{L}_{f,\phi}$ has spectral gap, acting on the Holdercontinuous space, is dense in the uniform topology. Furthermore, we obtainapplications for multifractal analysis of Birkhoff's average.

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