Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Thermodynamical and spectral phase transition for local diffeomorphisms
  in the circle   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 $t_0 =
1.$

30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas