Abstract
Abstract
Pointwise Hölder exponents describe the degree of regularity of a function near a point. For a function
f
:
R
→
R
, a number
α
>
0
and a point
t
0
∈
R
, write
f
∈
C
α
(
t
0
)
if there exist a constant
C
>
0
, a number
h
>
0
and a polynomial
P
of degree less than
α
such that
|
f
(
t
)
−
P
(
t
−
t
0
)
|
⩽
C
|
t
−
t
0
|
α
for all
t
∈
(
t
0
−
h
,
t
0
+
h
)
.
The pointwise Hölder exponent of
f
at
t
0
is the number
α
f
(
t
0
)
:=
sup
{
α
>
0
:
f
∈
C
α
(
t
0
)
}
.
A simpler quantity, also frequently called pointwise Hölder exponent in the mathematical literature, is the number
α
~
f
(
t
0
)
:=
sup
{
α
>
0
:
f
∈
C
~
α
(
t
0
)
}
,
where
f
∈
C
~
α
(
t
0
)
means that there exist
C
>
0
and
h
>
0
such that
|
f
(
t
)
−
f
(
t
0
)
|
⩽
C
|
t
−
t
0
|
α
for all
t
∈
(
t
0
−
h
,
t
0
+
h
)
. Clearly
α
f
(
t
)
⩾
α
~
f
(
t
)
, but strict inequality is possible and in fact common. In this paper we consider the case when
f
=
F
μ
is the distribution function of a Gibbs measure
μ
associated with an arbitrary Hölder continuous potential
ψ
on a self-conformal set, and show that, under a very mild condition on
ψ
,
α
f
(
t
)
=
α
~
f
(
t
)
for all
t
. As a consequence, we deduce that the pointwise Hölder spectrum of
f
satisfies the multifractal formalism. As an application, we derive the pointwise Hölder spectrum of conjugacy maps between expanding piecewise
C
1
+
ϵ
maps of an interval.