Abstract
Abstract
It is known that all uniformly expanding dynamics have no phase transition with respect to Hölder 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
ℝ
∋
t
↦
P
t
o
p
(
f
,
−
t
log
|
D
f
|
)
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 Hölder continuous, we show that there exists
t
0
∈
(
0
,
1
]
such that the transfer operator
L
f
,
−
t
log
|
D
f
|
, acting on the space of Hölder continuous functions, has the spectral gap property for all
t
<
t
0
and has not the spectral gap property for all
t
⩾
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.
Funder
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics