  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 $t_0 =
1.$
