Ergodic Optimization for a Sequence of Continuous
Functions
Chen Yangyang Zhao Yun∗
Department of mathematics, Soochow University
Suzhou 215006, Jiangsu, P.R.China
[email protected]
Abstract. Let T : X → X be a continuous map on a compact metric space X, and F = {fn }n≥1 a sequence of continuous functions on
X. If F is asymptotically subadditive, then
1
n→∞ n
sup lim sup n1 fn (x) = lim
x∈X
n→∞
lim n1 fn (x)
n→∞
x∈Reg(F ,T )
sup
=
maxx∈X fn (x) = sup{F ∗ (µ) : µ ∈ MT },
where MT denotes the space of T −invariant Borel probability measures, Reg(F, T ) denotes the set of all regular points for F, and
R
F ∗ (µ) = lim n1 fn dµ. This generalizes some results from [6, 7, 8]
n→∞
for asymptotically subadditive potentials. And it also provides a sufficient condition for the subordination principle of a subadditive potential.
Some applications are given at the end of this paper.
Key words and phrases ergodic measures, subadditive potentials, maximizing measures.
1
Introduction
Let (X, T ) denote a topological dynamical system(TDS for short) in the sense that
T : X → X is a continuous map on a compact metric space X with metric d, and let
C(X) denote the space of continuous real-valued functions on X. The term MT and ET
denote the space of T −invariant Borel probability measures and the set of T −invariant
ergodic Borel probability measures, respectively.
* Corresponding author
2000 Mathematics Subject classification: 37A30, 37L40.
1
R
Given a continuous function f ∈ C(X), let β(f ) := supµ∈MT f dµ denote the
R
maximum integral of f . A T −invariant measure µ is f −maximizing if f dµ = β(f ),
denote by Mmax (f ) the set of all f −maximizing invariant measures. The study of the
functional β and sets Mmax (f ) has been termed ergodic optimization. For example,
the subordination principle, i.e., if ν ∈ MT , µ ∈ Mmax (f ) and suppν ⊆ suppµ, then
ν ∈ Mmax (f ), and the maximizing set, i.e., a closed set K ⊆ X is a maximizing set for
f if it has the property that µ ∈ MT is f −maximizing if and only if suppµ ⊆ K, see
[1, 2, 3, 4, 5] for related discussions. We point out that Morris has provided a sufficient
condition for the subordination principle in [5].
Although the ergodic optimization is extensively studied for a single function f ,
see [6] and the references where there in for an overview, there exist only a few results
of ergodic optimization for a sequence of functions. If Φ = {φn }n≥1 ⊆ C(X) is a
subadditive potential, i.e., φn+m (x) ≤ φn (x) + φm (T n x), ∀x ∈ X, n, m ∈ N, Schreiber
[7, theorem 1] and Sturman and Stark [8, theorem 1.7] independently proved that
1
1
sup lim sup φn (x) = lim max φn (x) = sup{Φ∗ (µ) : µ ∈ MT }
(1.1)
n→∞ n x∈X
x∈X n→∞ n
R
where Φ∗ (µ) = limn→∞ n1 φn dµ. And since Φ∗ (µ) is upper semi-continuous with
respect to µ ∈ MT and MT is weak∗ compact, the supremum in (1.1) can be replaced
by maximum, moreover, using ergodic decomposition theorem, the supremum in (1.1)
is indeed attained by a T −invariant ergodic measure. Cao [9] also generalized this
result to random dynamical systems. See Proposition 7.7 and Theorem 10.2(1) of
Dooley and Zhang’s paper [10] for a more general related discussions of the second
equality of the formula (1.1).
Next we first recall the definition of asymptotically subadditive potentials(ASP for
short) which was introduced by Feng and Huang [11]. A sequence F = {fn }n≥1 ⊆ C(X)
is called an ASP, if for each k > 0, there exists a subadditive potential Φk = {ϕkn }n≥1 ⊆
C(X) such that
1
1
lim sup ||fn − ϕkn || ≤
k
n→∞ n
where ||fn − ϕkn || := maxx∈X |fn (x) − ϕkn (x)|. This kind of potential arises naturally
in the study of the dimension theory in dynamical systems, see [11, 12] for related
examples. This paper first generalizes the formula (1.1) for ASP, moreover, it is proved
that all of the items in (1.1) are also equal to supx∈Reg(F ,T ) limn→∞ n1 fn (x), where
Reg(F, T ) is the set of x ∈ X for which the limit lim n1 fn (x) exists.
n→∞
Given an ASP F = {fn }n≥1 , the following result is proved by Feng and Huang [11],
we cite here just for completeness.
Theorem 1.1. Given an ASP F = {fn }n≥1 and µ ∈ MT . Then the following properties hold:
2
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(1)The limit F ∗ (µ) = limn→∞ n1 fn dµ exists(which may take value −∞). FurtherR
more, the limit λF (x) = limn→∞ n1 fn (x) exists for µ−a.e. x ∈ X, and λF (x)dµ =
F ∗ (µ). In particular, when µ ∈ ET , λF (x) = F ∗ (µ) for µ−a.e. x ∈ X.
(2) The map F ∗ : MT → R ∪ {−∞} is upper semi-continuous and there exists C ∈ R
such that for all µ ∈ MT , F ∗ (µ) ≤ C and λF (x) ≤ C for µ−a.e. x ∈ X.
R
(3)Let µ = ET mdτ (m) be the ergodic decomposition of µ ∈ MT , then F ∗ (µ) =
R
F ∗ (m)dτ (m).
ET
Now, we give some definitions and notations. Given a subadditive potential Φ =
{φn }n≥1 , put Λ(Φ) := sup{Φ∗ (µ) : µ ∈ MT }. A T −invariant measure µ is Φ−maximizing
if Λ(Φ) = Φ∗ (µ), such a measure always exists since Φ∗ (µ) is upper semi-continuous
with respect to µ and MT is weak∗ compact, and the set of all Φ−maximizing invariant
measures is denoted by Mmax (Φ).
Definition 1.1. We say that the maximizing measures of Φ satisfy the subordination
principle if the following property holds: if ν ∈ MT , µ ∈ Mmax (Φ) and suppν ⊆
supp µ, then ν ∈ Mmax (Φ).
Definition 1.2. We say that a closed set K ⊆ X is a maximizing set for Φ if it has
the property that µ ∈ MT is Φ−maximizing if and only if suppµ ⊆ K.
It is not difficult to see that the existence of an Φ-maximizing set implies the subordination principle. This paper provides a sufficient condition for the subordination
principle for subadditive potentials, moreover, it is proved that an Φ−maximizing set
exists if and only if Mmax (Φ) satisfies the subordination principle.
The remainder of the paper is organized as follows. Section 2 generalizes formula
(1.1) for ASP, and all the items in (1.1) are also equal to supx∈Reg(F ,T ) limn→∞ n1 fn (x).
Section 3 provides a sufficient condition for the subordination principle of subadditive
potentials. Section 4 gives some applications, particularly, it provides a simple proof
of the main result in [15].
2
Growth of asymptotically subadditive potentials
In this section, we consider the growth of an ASP, and show that the formula (1.1) for
ASP still holds.
Let F = {fn } be an ASP on X, for any positive integer k, let Φk = {ϕkn }n≥1 be
as in the definition of ASP. Put Λ(F) := sup{F ∗ (µ) : µ ∈ MT }, we get the following
result.
Proposition 2.1. The following properties hold:
(a) Let Λ(Φk ) := sup{Φ∗k (µ) : µ ∈ MT }, we have lim Λ(Φk ) = Λ(F);
k→∞
3
(b) Let {µk }k≥1 be a sequence in MT , and µk ∈ Mmax (Φk ) for each k. Then any weak∗
limit point of {µk }k≥1 is an F−maximizing measure.
Proof. (a) As the potentials Φk = {ϕkn }n≥1 were given from the assumption of Proposition 2.1, we have
1 k
1
1
1
1
ϕn (x) − ≤ fn (x) ≤ ϕkn (x) + , ∀x ∈ X
n
k
n
n
k
(2.2)
for all sufficiently large n. By the second part of theorem 1.1, there exists µ0 ∈ MT
such that F ∗ (µ0 ) = Λ(F). Using the second inequality of (2.2), we have
Z
Z
1
1
1
1
1
Λ(F) = lim
fn dµ0 ≤ lim
ϕkn dµ0 + = Φ∗k (µ0 ) + ≤ Λ(Φk ) + .
n→∞ n
n→∞ n
k
k
k
Therefore, we have
1
Λ(F) ≤ lim inf [Λ(Φk ) + ] = lim inf Λ(Φk ).
k→∞
k→∞
k
(2.3)
On the other hand, for each µ ∈ MT , using the first inequality in (2.2), we have
Z
Z
1
1
1
1
∗
k
Φk (µ) − = lim
ϕn dµ − ≤ lim
fn dµ ≤ Λ(F).
k n→∞ n
k n→∞ n
Hence, we have that Λ(Φk ) −
1
k
≤ Λ(F). It immediately follows that
1
lim sup Λ(Φk ) = lim sup[Λ(Φk ) − ] ≤ Λ(F).
k
k→∞
k→∞
(2.4)
Thus, the desired result immediately follows from (2.3) and (2.4).
(b)Without loss of generality, we assume that µk converges weakly to ν. Using
(2.2), we have
Z
Z
1
1
1 k
1
Λ(Φk ) − = lim
ϕn dµk − ≤ lim
fn (x)dµk = F ∗ (µk ).
k n→∞ n
k n→∞ n
Let k → ∞, using (a) and the upper semi-continuity of F ∗ (µ) with respect to µ, we
have Λ(F) ≤ F ∗ (ν), it follows that ν ∈ Mmax (F). This completes the proof.
Using the formula (1.1) for subadditive potentials and the above proposition, we
have the following theorem.
Theorem 2.1. Let (X, T ) be a TDS and F = {fn }n≥1 an ASP. Then we have
1
1
sup lim sup fn (x) = lim max fn (x) = sup{F ∗ (µ) : µ ∈ MT }.
n→∞ n x∈X
x∈X n→∞ n
4
Proof. For each positive integer k, by the definition of ASP, there exists a subadditive
potentials Φk = {ϕkn }n≥1 such that
1 k
1
1
1
1
ϕn (x) − ≤ fn (x) ≤ ϕkn (x) + , ∀x ∈ X
n
k
n
n
k
(2.5)
for all sufficiently large n. Therefore, we have
1
1
1
1
1
lim [ max ϕkn (x) − ] ≤ lim max fn (x) ≤ lim [ max ϕkn (x) + ]
n→∞
n→∞
x∈X
x∈X
x∈X
n
k
n
n
k
n→∞
Using the formula (1.1) for subadditive potentials, we have
Λ(Φk ) −
1
1
1
≤ lim max fn (x) ≤ Λ(Φk ) + .
k n→∞ n x∈X
k
Letting k → ∞, proposition 2.1 implies that
lim
n→∞
1
max fn (x) = sup{F ∗ (µ) : µ ∈ MT }.
n x∈X
For the other equality in the theorem, inequality (2.5) implies that
1
1
1
1
1
sup lim sup[ ϕkn (x) − ] ≤ sup lim sup fn (x) ≤ sup lim sup[ ϕkn (x) + ].
k
k
x∈X n→∞ n
x∈X n→∞ n
x∈X n→∞ n
Using formula (1.1) for subadditive potentials, we have
Λ(Φk ) −
1
1
1
≤ sup lim sup fn (x) ≤ Λ(Φk ) + .
k x∈X n→∞ n
k
Letting k → ∞, proposition 2.1 implies that
1
sup lim sup fn (x) = sup{F ∗ (µ) : µ ∈ MT }.
x∈X n→∞ n
This finishes the proof of the theorem.
Using the above theorem, we have the following corollary.
Corollary 1. Let (Xi , Ti )(i = 1, 2) be two TDS, and F = {fn }n≥1 an ASP on X2 .
If ϕ : X1 → X2 is a surjective continuous map with ϕ ◦ T1 = T2 ◦ ϕ, then we have
ϕ∗ (Mmax (F ◦ ϕ)) ⊆ Mmax (F), where F ◦ ϕ = {fn ◦ ϕ}n≥1 is a sequence of functions
on X1 and ϕ∗ µ(A) = µ(ϕ−1 A) for each µ ∈ MT1 and Borel subset A ⊂ X2 . Moreover,
if ϕ : X1 → X2 is a homeomorphism with ϕ ◦ T1 = T2 ◦ ϕ, we have ϕ∗ (Mmax (F ◦ ϕ)) =
Mmax (F).
5
Proof. It is easy to see that F ◦ ϕ = {fn ◦ ϕ}n≥1 is an ASP on X1 . For each µ ∈
Mmax (F ◦ ϕ), i.e., (F ◦ ϕ)∗ (µ) = Λ(F ◦ ϕ), using theorem 2.1 we have
Z
1
1
1
∗
fn ◦ ϕdµ = lim max fn ◦ ϕ(x) = lim max fn (y) = Λ(F).
(F ◦ ϕ) (µ) = lim
n→∞ n x∈X1
n→∞ n
n→∞ n y∈X2
Note that
1
lim
n→∞ n
Z
1
fn ◦ ϕdµ = lim
n→∞ n
Z
fn dϕ∗ µ = F ∗ (ϕ∗ µ).
It follows that ϕ∗ (Mmax (F ◦ ϕ)) ⊆ Mmax (F). And it is easy to see that ϕ∗ (Mmax (F ◦
ϕ)) = Mmax (F) if ϕ : X1 → X2 is a homeomorphism with ϕ ◦ T1 = T2 ◦ ϕ.
Let F = {fn }n≥1 be an ASP. We define Reg(F, T ) as the set of x ∈ X for which
the limit limn→∞ n1 fn (x) exists, and we call Reg(F, T ) the regular set for F. Then, we
consider the following quantity
Ω(F) :=
1
fn (x)
x∈Reg(F ,T ) n→∞ n
sup
lim
and have the following result.
Proposition 2.2. Let (X, T ) be a TDS and F = {fn }n≥1 an ASP. Then we have
Ω(F) = Λ(F).
Proof. By theorem 2.1, it immediately follows that Ω(F) ≤ Λ(F). To prove the reverse
inequality, using (2) and (3) in theorem 1.1, we may choose a T −invariant ergodic
measure µ ∈ ET such that F ∗ (µ) = Λ(F). Then, using (1) in theorem 1.1, there exists
a point x ∈ X such that
1
fn (x) = F ∗ (µ) = Λ(F).
n→∞ n
lim
This immediately implies that Ω(F) ≥ Λ(F), finishing the proof.
Theorem 2.1 and proposition 2.2 immediately imply the following result.
Corollary 2. Let (X, T ) be a TDS and F = {fn }n≥1 an ASP. Then we have
1
1
1
fn (x) = sup lim sup fn (x) = lim max fn (x)
n→∞ n x∈X
x∈X n→∞ n
x∈Reg(F ,T ) n→∞ n
sup
lim
= sup{F ∗ (µ) : µ ∈ MT }.
6
3
Subordination principle for subadditive potentials
This section provides a sufficient condition for the subordination principle of subadditive potentials. The arguments are similar to that in [5].
Theorem 3.1. Let (X, T ) be a TDS and F = {fn }n≥1 a subadditive potential with
Λ(F) 6= −∞, and suppose that supn≥1 supx∈X [fn (x) − nΛ(F)] < ∞. Then Mmax (F)
satisfies the subordination principle.
Before we give the proof of this theorem, we first give some remarks on it.
Remark 1. (1) In order to remove the triviality, we add the condition Λ(F) 6= −∞,
since Λ(F) = −∞ implies that F ∗ (µ) = −∞ for each µ ∈ MT , and this immediately
implies the subordination principle of F without any assumptions;
(2) It suffices to assume that supn≥L supx∈X [fn (x) − nΛ(F)] < ∞ for some positive
integer L.
Proof. We follow the arguments in [5] by Morris. It suffices to prove the following statement: if ν, µ ∈ MT satisfy supp ν ⊆ supp µ and F ∗ (ν) < Λ(F), then F ∗ (µ) < Λ(F).
We assume without loss of generality that Λ(F) = 0. Let B ≥ supn≥1 supx∈X fn (x)
with B > 0.
Let ν, µ ∈ MT with supp ν ⊆ supp µ and F ∗ (ν) < 0. By Kingman’s sub-additive
ergodic theorem(see [13] or [14, theorem 10.1]) applied to ν, there is a function λF ∈
R
L1 (ν) such that λF (x) = limn→∞ n1 fn (x) for ν−a.e. x, where λF (x)dν = F ∗ (ν) < 0.
Hence the set of points x such that λF (x) < 0 is a set of positive ν−measure. Since
every set of positive ν−measure intersects supp ν, there exists a point x0 ∈ supp ν
such that limn→∞ n1 fn (x0 ) < 0. Therefore, we can take a sufficiently large N > 0 such
that fN (x0 ) ≤ −3B. Since fN is continuous, there exists an open neighborhood U of
x0 such that fN (x) < −2B for all x ∈ U .
R
Let µ = ET mdτ (m) be the ergodic decomposition of µ ∈ MT , then µ(Z) =
R
m(Z)dτ (m) for each Borel set Z ⊆ X. Suppose that m is an ergodic component of
ET
µ with m(U ) > 0. For x ∈ U , define the sequence of return times rn (x) in the following
way: let r1 (x) = 0, and for n > 1 define rn (x) = inf{k > rn−1 (x) : T k x ∈ U }. Applying
Kac’s lemma and Kingman’s sub-additive ergodic theorem to the ergodic measure m,
we can find a point xm ∈ U such that rn (xn m ) → m(U ) and n1 fn (xm ) → F ∗ (m) as
n → ∞.
Let n > 0. The number of integers k satisfying 0 ≤ k ≤ rn (xm ) such that T k xm ∈ U
is equal precisely to n. We may therefore choose an increasing sequence of Mn =
n
[n/N ]([n/N ] denotes the integral part of n/N ) integers {ni }M
i=1 with the following
properties: n1 = 0, nMn = rn (xm ), and for each i, T ni xm ∈ U and ni+1 ≥ ni + N .
7
Using our hypothesis that B ≥ supn≥1 supx∈X fn (x) together with the choice of U , we
have
frn (xm ) (xm ) ≤
≤
M
n −1
X
i=1
M
n −1
X
[fN (T ni xm ) + fni+1 −ni −N (T ni +N xm )]
[−2B + B]
i=1
≤ −B(Mn − 1)
here by convention we set f0 to be the constant zero function. Hence
F ∗ (m) =
frn (xm ) (xm )
−B(Mn − 1)
B
≤ lim
= − m(U )
n→∞
n→∞
rn (xm )
rn (xm )
N
lim
whenever m(U ) > 0. Since U ∩ supp µ 6= ∅ and U is open, it follows that µ(U ) > 0.
Therefore, we have
Z
∗
F ∗ (m)dτ (m)
F (µ) =
ET
Z
Z
∗
=
F (m)dτ (m) +
F ∗ (m)dτ (m)
{m∈ET : m(U )>0}
Z
≤
{m∈ET : m(U )=0}
B
− m(U ) dτ (m) +
N
{m∈ET : m(U )>0}
Z
Λ(F)dτ (m)
{m∈ET : m(U )=0}
B
µ(U ).
N
This completes the proof of the theorem.
= −
Theorem 3.2. Let (X, T ) be a TDS and F = {fn }n≥1 a subadditive potential. Then an
F−maximizing set exists if and only if Mmax (F) satisfies the subordination principle.
Proof. The forward implication is simple: if an F−maximizing set K exists, then
µ ∈ Mmax (F) and supp ν ⊆ supp µ imply that supp ν ⊆ K so that ν is F−maximizing.
Suppose that F satisfies the subordination principle. Since F ∗ (µ) is upper semicontinuous with respect to µ and the space X is compact, Mmax (F) is closed and
separable in the weak∗ topology. Let {µn }n≥1 be a sequence of measures (not necessarily
P
−n
distinct) which is dense in Mmax (F), and take µ = ∞
µn . Define K = supp µ, it
n=1 2
is easy to see that µ ∈ Mmax (F) and K is T −invariant. We claim that ν ∈ Mmax (F)
if and only if supp ν ⊆ K.
If ν ∈ MT satisfies supp ν ⊆ K, then it is maximizing by the subordination
principle. Conversely, suppose that ν ∈ Mmax (F). Since supp µn ⊆ K, ∀n, we have
µn (K) = 1, ∀n. Note that ν is a weak∗ limit point of {µn }n≥1 (since {µn }n≥1 is dense
in Mmax (F)) and K is closed, it follows that ν(K) = 1. This finishes the proof.
8
We point out that theorem 3.2 also holds for ASP without changing the above
proof.
4
Applications
This section provides some applications.
We first use formula (1.1) to give a simple proof of the results in [15]. Let M be
a smooth s−dimensional Riemannian manifold. Let U be an open subset of M and
f : U → M a C 1 local diffeomorphism. Suppose J ⊂ U is a compact f −invariant
subset. The map f is said to be uniformly expanding on J if there exist constants
C > 0 and λ > 1 such that ||Dx f n (v)|| ≥ Cλn ||v|| for all x ∈ J, v ∈ Tx M and n ≥ 1.
Let m(Dx f ) denote the minimum norm of the operator Dx f : Tx M → Tf x M , i.e.,
x f (v)||
m(Dx f ) = inf 06=v∈Tx M ||D||v||
, and M(f |J ), E(f |J ) denote the set of all f −invariant
measures and the set of all f −invariant ergodic measures supported on J respectively.
In [15], the author proved the following interesting result:
f is uniformly expanding on J if and only if inf{F ∗ (µ) : µ ∈ M(f |J )} > 0
R
where F = {log m(Dx f n )}n≥1 and F ∗ (µ) = limn→∞ n1 log m(Dx f n )dµ is the minimum Lyapunov exponent with respect to µ. Now, we give a simple proof of the above
result by using formula (1.1).
Proposition 4.1. Let f be a C 1 local diffeomorphism on a smooth s−dimensional
Riemannian manifold M , and J is a compact f −invariant subset. Then
f is uniformly expanding on J if and only if inf{F ∗ (µ) : µ ∈ M(f |J )} > 0
R
where F = {log m(Dx f n )}n≥1 and F ∗ (µ) = limn→∞ n1 log m(Dx f n )dµ is the minimum Lyapunov exponent with respect to µ.
Proof. Suppose f is uniformly expanding on J. It is easy to see that m(Dx f n ) ≥ Cλn
for some constants C > 0 and λ > 1. It immediately follows that
Z
1
∗
F (µ) = lim
log m(Dx f n )dµ ≥ log λ > 0, ∀µ ∈ M(f |J ).
n→∞ n
Conversely, suppose that inf{F ∗ (µ) : µ ∈ M(f |J )} > 0. Let κ := inf{F ∗ (µ) :
µ ∈ M(f |J )} > 0. Since −F = {− log m(Dx f n )}n≥1 is a sequence of subadditive
continuous functions on J, using formula (1.1), we have
1
lim max − log m(Dx f n ) = sup{−F ∗ (µ) : µ ∈ M(f |J )}.
n→∞ x∈J
n
And since
sup{−F ∗ (µ) : µ ∈ M(f |J )} = − inf{F ∗ (µ) : µ ∈ M(f |J )} = −κ < 0,
9
we have that
1
lim max − log m(Dx f n ) = −κ < 0.
n→∞ x∈J
n
Thus, there exists a positive integer N such that if n ≥ N then
−κ
1
− log m(Dx f n ) ≤
n
2
for all x ∈ J. It follows that
κ
m(Dx f n ) ≥ (e 2 )n , ∀n ≥ N, x ∈ J.
κ
Thus, it is easy to see that f is uniformly expanding on J with λ = e 2 > 1. This
completes the proof.
By the Oseledec multiplicative ergodic theorem, for each µ ∈ E(f |J ), we can define
Lyapunov exponents λ1 (µ) ≤ λ2 (µ) ≤ · · · ≤ λs (µ). If λ1 (µ) = λ2 (µ) = · · · = λs (µ) for
each µ ∈ E(f |J ), then J is a set with one Lyapunov exponent.
Proposition 4.2. Suppose J is a set with one Lyapunov exponent, then
1
(log ||Dx f n || − log m(Dx f n )) = 0
n→∞ n
lim
uniformly on J, where ||Dx f || denotes the maximum norm of the operator Dx f .
Proof. Let
φn (x) = log ||Dx f n || − log m(Dx f n ), n ∈ N, x ∈ J
and F = {φn }n≥1 . Then it is easy to see that F is a non-negative subadditive potential.
Using formula (1.1) and ergodic decomposition theorem, we have
1
max φn (x) = sup{F ∗ (µ) : µ ∈ E(f |J )} = 0,
n→∞ n x∈J
lim
the last equality holds since J is a set with one Lyapunov exponent. This immediately
implies the desired result.
Remark 2. In [16], a set J with one positive Lyapunov exponent is called an average
conformal repeller. In that paper, the authors proved the above result for an average
conformal repeller(see theorem 4.2 in [16]). The above proposition gives a more general
version of theorem 4.2 in [16]. Moreover, we provide a simpler proof by using formula
(1.1).
In the end of the paper, we provide an application of theorem 3.1. Let T be the
shift map on Σ = {1, 2, · · · , m}N , m ≥ 2. As usual Σ is endowed with the metric
d(x, y) = m−n where x = (xi ), y = (yi ) and n = inf{i : xi 6= yi }. Given an m × m
matrix A with entries 0 or 1, we consider the sub-shift of finite type (ΣA , T )(see [17]).
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We assume that A is primitive, i.e., (ΣA , T ) is topologically mixing. Finally, for a
subadditive potential F = {fn }n≥1 on ΣA and j ≥ 1 we define
varfj =
sup
|fj (x) − fj (y)|.
{x,y∈ΣA : xi =yi , ∀1≤i≤j}
We have the following result.
Corollary 3. Let T : ΣA → ΣA be a topologically mixing sub-shift of finite type.
Suppose that F = {fn }n≥1 is a subadditive potential on ΣA satisfies that supj≥1 varfj <
∞ and supj≥1 supp∈Sl≥1 Fixl (T ) [fj (p) − jΛ(F)] < ∞, where Fixl (T ) denotes the set of all
periodic points of T with period l. Then Mmax (F) satisfies the subordination principle.
Proof. By remark 1, without loss of generality, we assume that Λ(F) is finite and
Λ(F) = 0. Let A = supj≥1 varfj and B = supj≥1 supp∈Sl≥1 Fixl (T ) fj (p). Since T : ΣA →
ΣA is a topologically mixing sub-shift of finite type, there exists an integer N > 0 such
that for every x ∈ ΣA and n > 0, we may find a periodic point p ∈ Fixn+N (T ) such
that x and p agree in their first n symbols. Let x ∈ ΣA and n > 0, and choose such a
periodic point p. We have
fn (x) ≤ fn (p) + varfn ≤ A + B < ∞.
The desired result immediately follows by theorem 3.1.
Acknowledgements. This work was initiated when the second author visited CTS,
whose finial support and research condition were greatly appreciated. Zhao is partially
supported by NSFC(11001191), NSF from Jiangsu Province(09KJB110007) and Ph.D.
Programs Foundation of Ministry of Education of China (20103201120001). Chen is
partially supported by National University Student Innovation Program(111028508).
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