Multifractal Analysis of the Asympyotically Additive Potentials ()
1. Introduction
The theory of multifractal analysis is a subfield of the dimension theory in dynamical systems. A general framework for multifractal analysis of dynamical systems was laid out in [1] [2] . It studies a global dimensional quantity that assigns to each level set a “size” or “complexity”, such as its topological entropy or Hausdorff dimension. Broadly speaking, let
be a continuous transformation of a compact metric space; let
,
be potential functions defined on
with value in
. Given
, we consider the level set:
![](//html.scirp.org/file/1-1720208x13.png)
The dimension spectrum
(of potential
) is defined by
which has been extensively studied for Hólder continuous potentials for
conformal repellers in [3] - [5] .
In [6] , Barreira, Saussol, and Schmeling extended their work to higher-dimensional multifractal spectra, moreover, for which they consider the more general
-dimension in place of the topological entropy. Precisely, they consider functions
,
with
and examine the level sets
![]()
for
. We denote by
the family of
-invariant Borel probability measures on
, and define a continuous function
:
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Given a positive function
we denote by
the
-dimension of the set
(see Section 2 for the definition). Let
be the family of continuous functions with a unique equilibrium measure, they obtain the following result:
Theorem 1. Assume that the metric entropy of
is upper semi-continuous, and that
.
If
,
. Otherwise, if
,
, and the following properties hold:
(I)
satisfies the variational principle:
![]()
(II)
, where
is the unique real number satisfying:
![]()
(III) There exists ergodic measure
with
and
such that
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In [7] , Barreira and Doutor study the spectrum of the
-dimension for the class of almost additive sequences with a unique equilibrium measure and establish a conditional variational principle for the dimension spectra in the context of the nonadditive thermodynamic formalism. We recall that a sequence of functions
is said to be almost additive (with respect to a transformation
) if there is a constant
such that for every
, we have:
![]()
In [8] Climenhaga proved a generalisation of Theorem 1 provided that there is a dense subspace of
comprising potentials with unique equilibrium states, i.e., the result applies to all continuous functions, not just those whose span lies inside the collection of potentials with unique equilibrium states.
This paper is devoted to the study of higher-dimensional multifractal analysis for the class of asymptotically additive potentials. We consider the multifractal behavior of
-dimension spectrum of level sets and establish the conditional variational principle under the assumption proposed by Climenhaga.
Section 2 gives definitions and notions, and Section 3 gives precise formulations of the result and proofs.
2. Preliminaries
We recall in this section some notions and results from the thermodynamic formalism.
2.1. Nonadditive Topological Pressure
We first introduce the notion of nonadditive topological pressure. We also refer the reader to [2] and [7] for further references.
Let
be a continuous transformation of a compact metric space. We denote by
the space of continuous functions on
and
the set of all
-invariant measures. Given a finite open cover
of
, we denote by
the collection of vectors
with
. For each
, we write
, and we consider the open set
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Now let
be a sequence of continuous functions
. For each
we define:
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We always assume that
(1)
For each
we write:
(2)
Given a set
and
, we define the function:
![]()
where the infimum is taken over all finite or countable collections
, such that
. We also define
![]()
It was shown in [9] that the limit
![]()
exists. The number
is called the nonadditive topological pressure of
in the set
(with respect to
). In particular, if
, we get the topological entropy
. We also write
.
The following proposition was established in [2] .
Proposition 1. For any
, we have
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2.2.
-Dimension
We recall here a notion introduced by Barreira and Schmeling in [10] . Let
be a strictly positive continuous function. Likewise, we define
![]()
where
is defined as in (2) and where the infimum is taken over all finite or countable collections
such that
. We also define
![]()
Theorem 2. ( [10] ) The following limits exist:
![]()
We call
the
-dimension of
. If
, then the number
coincides with the topological entropy of
on
. The following result is an easy consequence of the definitions.
Proposition 2. The number
is the unique root of the equation
, where
with
for each
.
Furthermore, given a probability measure
in
, we set:
![]()
We can show that the limit
exists, and we call it the
-dimension of
. When
is ergodic, one can show that (see [10] )
(3)
2.3. Asymptotically Additive Sequences
This kind of potential was introduced by Feng and Huang ( [11] ).
Definition 1. A sequence
of functions on
is said to be asymptotically additive if for any
, there exists
such that
![]()
We denote by
the family of asymptotically additive sequences of continuous functions (satisfying (1)). Now we give two propositions whose proof can be found in [11] .
Proposition 3. If
is a continuous transformation of a compact metric space,
is an asymptotically additive sequence, and
, then
(I) The limit
exists for
;
(II) The limit
exists;
(III) If
is ergodic, then for
,
(4)
(IV) The function
is continuous with the weak* topology in
.
Proposition 4. If
is a continuous transformation of a compact metric space,
is an asymptotically additive sequence, then the topological pressure
satisfies the following variational principle:
![]()
We call
an equilibrium measure for the potential
if
![]()
Note that if the function
is upper semicontinuous, then every sequence in
has an equilibrium measure.
3. Main Result
Let
and take
. We write
and
, and also
,
.
We assume that
(1) There exists constant
such that
for any
and any
.
(2) For every
,
for
and every
, where the limit exists by proposition 3.
Given
, we define:
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and function
by
.
We also consider the function
defined by:
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Given vectors
and
we use the notations:
![]()
![]()
and
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We also consider the positive sequence of functions
with
.
Our main result is the following theorem.
Theorem 3. Let
be a continuous transformation of a compact metric space
such that the entropy map
is upper semicontinuous, and assume that there exists a dense subset
such that every
has a unique equilibrium measure.
If
, then
. Otherwise, if
, then
, and the following properties hold:
(I)
satisfies the variational principle:
![]()
(II)
, where
is the unique real number satisfying:
![]()
(III) There exists ergodic measure
with
,
, and
![]()
which is arbitrarily close to
.
Proof. We first establish several auxiliary results.
Lemma 1. For
there exists constant
such that for every
we have
(5)
where
denotes the supremum norm.
Proof. For any
, since the sequence
is asymptotically additive, there exists
such that
![]()
Therefore, there exists
, such that for every
and
, we have
![]()
and thus
□
Lemma 2. If
, then
.
Proof. Using (5), a slight modification of the proof of Lemma 2 in [7] yields this statement, and thus we omit it. □
Lemma 3. If
, then
. Otherwise, if
, then
.
Proof. Take
with
and let
. Then
for
. We consider the sequence
of probability measures in
defined by
. Let
be a limitpoint of
, clearly
. We always assume
is ergodic, or else taking an ergodic decomposition of
. The desired statements are thus immediate consequences of (4). □
Now proceed with the proof of (1) in theorem 3. We use analogous arguments to those in the proof of lemma 3 in [7] . First show that
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Let
be the distance of
to
. Take
and define:
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Given
with
for each
, we have:
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and hence
. Therefore, there exists
such that
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where
,
. Moreover,
![]()
Since
![]()
we obtain:
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Since
, it follows that
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It implies that
takes arbitrarily large values for
sufficiently large, and hence there exists
such that
for every
with
. The continuity of
implies that it attains a minimum at some point
with
.
Note that
is a dense subset such that every
has a unique equilibrium measure, then for every
and
there exists
,
and
with the following properties:
(1)
has a unique equilibrium measure
which depends continuously on
(for fixed
);
(2)
;
(3)
.
Therefore,
![]()
and thus
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Denote
a limit point of
as
, then
(6)
For each vector
with
, let
and let
be taken as in (6). We have
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If
, then
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Now assume that
when
for some measure
. The upper semicontinuity of the entropy implies that
![]()
This shows that
is an equilibrium measure of
![]()
satisfying
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Similarly, one can consider
and find an invariant measure
that is an equilibrium measure of
satisfying
![]()
For each
, let
. Then the function
![]()
is continuous. Moreover,
and
. Hence, there exists
such that
. Since
and
are equilibrium measures of
, this implies that
is also an equilibrium measure of
. Therefore, for each unit vector
there exists an equilibrium measure
of
such that
(7)
We claim that there exists an equilibrium measure
of
such that
(8)
Let us assume that such a measure does not exist. We denote by
the set of all equilibrium measures of
. Then
![]()
is a compact convex subset of
. Hence, there exist a unit vector
and
such that
for
.
For every
, we have
![]()
which contradicts (7). This completes the proof of claim. Observe that this claim implies
.
By lemma 2, for the measure
satisfying (8), we have
![]()
and hence
![]()
We now to prove the reverse inequality. We need the following lemma.
Lemma 4. ([8] ) Under the assumptions of theorem 3, for
and
, we have:
![]()
In fact, this is a particular case of Theorem C in [8] .
For any
, there exists
with
such that
. Therefore
![]()
and hence by proposition 2 we have
. The arbitrariness of
implies that
![]()
and thus
(9)
Furthermore, since the map
is upper semicontinuous on the compact set
, then the supremum of (9) can be obtained, i.e.
![]()
This completes (I) of theorem 3.
We now proceed with the proof of (II) and (III). By lemma 2 we have
![]()
for every
. Therefore,
![]()
On the other hand, for any
we have
![]()
and hence
. So we conclude that
![]()
By ergodic decomposition we obtain
![]()
For any
, there exists ergodic
with
such that
![]()
Note that
, then by (3)
![]()
It follows that statement (III) in theorem 3 holds. □
Acknowledgements
The author wishes to thank Professor Cao Yongluo for his invaluable suggestions and encouragement.