  In \cite{BC21} is shown that if $f: \mathbb{S}^{1} \rightarrow
\mathbb{S}^{1}$ is a $C^{1+\alpha}$-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 phase transition with
respect to potential $\phi := -\log|Df|$. On the other hand, by the works
\cite{KQW21,KQ22}, for continuous potentials the topological pressure function
can be wild; in particular, it can be infinite phase transitions. In this
paper, we study the possibilities of behaviour of the topological pressure
function and transfer operator for transitive local diffeomorphism with break
points on the circle and Holder continuous potentials. In particular, we showed
that: 1-that there is an open and dense subset of continuous potentials such
that if a Holder continuous potential belongs to subset then it has no phase
transition and the transfer operator has spectral gap property; 2-for Holder
continuous potentials, not cohomologous to constant, the existence of a phase
transition is equivalent to the topological pressure function not being
strictly convex; 3-if a Holder continuous potential has phase transition, then
the topological pressure function and the transfer operator associated have
behaviour similar to \cite{BC21}. Consequently, every Holder continuous
potential has at most two phase transitions and the set of smooth potential
such that $\mathcal{L}_{f,\phi}$ has spectral gap, acting on the Holder
continuous space, is dense in the uniform topology. Furthermore, we obtain
applications for multifractal analysis of Birkhoff's average.
