  It is known that all uniformly expanding or hyperbolic dynamics have no phase
transition with respect to H\"older continuous potentials. In \cite{BC21}, is
proved that for all transitive $C^{1+\alpha}-$local diffeomorphism $f$ on the
circle, that is neither a uniformly expanding map nor invertible, has a unique
thermodynamic phase transition with respect to the geometric potential, in
other words, the topological pressure function $\mathbb{R} \ni t \mapsto
P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also
it is proved spectral phase transitions, in other words, 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 does not have the
spectral gap property for all $t\geq t_0$. Our goal is to prove that the
results of thermodynamical and spectral phase transitions imply a multifractal
analysis for the Lyapunov spectrum. In particular, we exhibit a class of
partially hyperbolic endomorphisms that admit thermodynamical and spectral
phase transitions with respect to the geometric potential, and we describe the
multifractal analysis of your central Lyapunov spectrum.
