Generic Points of shift-Invariant Measures in the Countable Symbolic Space

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by  Ai-hua Fan

GENERIC POINTS OF SHIFT-INVARIANT MEASURES IN
THE COUNTABLE SYMBOLIC SPACE
6
1 AIHUAFAN, MINGTIAN LI,AND JIHUAMA
0
2

n
a
J Abstract. Weareconcernedwithsetsofgenericpointsforshift-invariant

9 measures in the countable symbolic space. We measure the sizes of the
2 setsbytheBillingsley-HausdorffdimensionsdefinedbyGibbsmeasures.
It is shown that the dimension of such a set is given by a variational

] principle involvingtheconvergenceexponentof theGibbsmeasure and
S
therelativeentropydimensionoftheGibbsmeasurewithrespecttothe
D
invariant measure. This variational principle is different from that of
.
h the case of finite symbols, where the convergent exponent is zero and
t is not involved. An application is given to a class of expandinginterval
a
dynamical systems.
m
[

1. Introduction
1
v ConsiderthecountablesymbolicspaceX = NN endowedwiththeproduct
9
1 topology and the shift mapping T on X defined by
0
8 T(x1x2x3···) = (x2x3x4···).
0
. For any T-invariant Borel probability measure µ (we write µ ∈ M(X,T)),
1
0 define the set of µ-generic points by
6
1
1 G := x ∈ X : lim S f(x) = fdµ for all f ∈ C (X) ,
µ n b
: n→∞n
v n ZX o
i
X where S f(x) := n−1f(Tix) is the n-th ergodic sum of f and C (X)
n i=0 b
r denotes the space of all bounded real-valued continuous functions on X.
a P
Our aim in this paper is to investigate the size of G by studying its
µ
Hausdorff dimension with respect to different metrics. Let ν be another
probability measure supported on the whole space X. It induces a metric
ρ on NN as follows: if x = y, define ρ (x,y) = 0; otherwise
ν ν
ρ (x,y) = ν([x ···x ]),
ν 1 n
where n = inf{k ≥ 0 : x 6= y } and [x ···x ] (called cylinder) is the
k+1 k+1 1 n
set of all sequences having x ···x as prefix. The Hausdorff dimension of
1 n
a subset of X with respect to the metric ρ is the Billingsley dimension
ν
defined by ν ([2]).
Inthispaper,weonlyconsidermetricsdefinedbyGibbsmeasures(seethe
definition of Gibbs measure in Section 2. See also [25]). Let ϕ : X → R be
1
2 AIHUAFAN,MINGTIANLI,ANDJIHUAMA
a function, called a potential. For any n > 1, we define its n-order variation
by
var ϕ := sup{|ϕ(x)−ϕ(y)| :x = y , for 1 6 i6 n}.
n i i
We say that ϕ has summable variations if
∞
var ϕ < ∞.
n
n=2
X
The Gurevich pressure of a potential ϕ with summable variations is defined
to be the limit
1
P := lim ln eSnϕ(x)1 (x),
ϕ [a]
n→∞n
Tnx=x
X
where a ∈ N ([23]). It is shown that the limit exists and is independent of
a. It is known ([24]) that a potential function ϕ with summable variations
admits a unique Gibbs measure ν iff var ϕ < ∞ and the Gurevich pressure
1
P < ∞.
ϕ
As we shall prove, the Billingsley dimension dim G is tightly related to
ν µ
the convergence exponent of ν, which is defined by
∞
α := inf t > 0 : ν([n])t < +∞ .
ν
n nX=1 o
Itisevidentthatα 6 1. Wewillprovethatifthemeasuretheoreticentropy
ν
h is the infinity we have α = 1 (see Section 2.2). For µ ∈ M(X,T), define
ν ν
the (relative) entropy dimension of ν with respect to µ by
µ([ω])lnµ([ω])
ω∈Σk
β(ν|µ) := limsuplimsup N
µ([ω])lnν([ω])
k→∞ N→∞ Pω∈Σk
N
where Σk = {1,··· ,N}k. Our main rePsult is the following.
N
Theorem 1. Let µ ∈ M(X,T) be an invariant Borel probability measure
and ϕ be a potential function of summable variations admitting a unique
Gibbs measure ν with convergence exponent α . We have
ν
(1) dim G = max{α ,β(ν|µ)}.
ν µ ν
Let us apply (1) to two examples. First, whenµ = ν, we have β(ν|ν) = 1.
Then dim G = 1. This can be obtained by the Birkhoff ergodic theo-
ν ν
rem, because ν is ergodic and of dimension 1. Second, consider the invari-
ant measure δ where x = 1∞. It is easily seen that β(ν|δ ) = 0. Thus
x x
dim G = α . This is not trivial. It reflects the difference between the
ν δx ν
case of finite symbols and that of countable symbols.
A potential function ϕ admitting a Gibbs measure is upper bounded,
so that the integral ϕdµ is well defined as a number in the interval
X
[−∞,+∞). For µ ∈ M(X,T), define the relative entropy of ν with respect
R
to µ by
1
h(ν|µ) := limsup− µ([ω])lnν([ω]).
k
k→∞ ωX∈Nk
3
As we shall see (see Proposition 8),
h(ν|µ) = P − ϕdµ.
ϕ
ZX
The dimension formula (1) can be expressed by entropies. Suppose µ 6= ν.
If h(ν|µ) is finite, then β(ν|µ) = hµ (see Proposition 11). Thus we have
h(ν|µ)
h
µ
dim G = max α , .
ν µ ν
h(ν|µ)
(cid:26) (cid:27)
If h(ν|µ) is the infinity, we have α > β(ν|µ) (see Proposition 11). It follows
ν
that
dim G = α .
ν µ ν
Let us present the idea of the proof. In the case of µ = ν, we have
dim G = 1 because the Gibbs measure ν is ergodic and of dimension
ν ν
1. In the case of µ 6= ν, we first prove that for any µ ∈ M(X,T) there
exists a sequence of ergodic Markov measures {µj}j>1 which converge in
w∗-topology to µ, and h tends to h whenever h is finite. Second, we
µj µ µ
show that α is a universal lower bound by constructing a Cantor subset
ν
of G . For the other part of lower bound we distinguish two cases: in the
µ
case of h(ν|µ) < +∞, we construct a subset of G by using the sequence of
µ
ergodic Markov measures {µj}j>1 and show dimνGµ > h(hνµ|µ); in the case
of h(ν|µ) = +∞, we show α > β(ν|µ). For the upper bound, we adapt
ν
a standard argument by using an estimation on the entropy of subword
distribution which has combinatoric feather (see [12]).
In 1973, Bowen considered the set of generic points G in the setting of
µ
topological dynamical system T : X → X over compact metric space X.
Bowen ([3]) proved that for any T-invariant Borel probability measure µ
the topological entropy of the set of generic points G is bounded by the
µ
measure theoretic entropy h . Fan, Liao and Peyri`ere ([10]) showed that
µ
an equality holds if T satisfies the specification condition. In the case of
finite symbolic space, a study of the Billingsley dimension of G with re-
µ
spect to a shift-invariant Markov measure ν was performed by Cajar ([4]).
He proved that dim G is equal to the entropy dimension of ν with respect
ν µ
to µ. Olivier ([17]) extended this result to Billingsley dimension with re-
spect to a shift-invariant g-measure. Furthermore, Ma and Wen ([15]) even
showed that theHausdorff and Packing measureof G satisfy a zero-infinity
µ
law. On the other hand, Gurevich and Tempelman ([13]) consider G on
µ
high-dimensional finite symbolic systems. They evaluated the Hausdorff
dimension of G with respect to a wide class of metrics including Billlings-
µ
ley metrics generated by Gibbs measures. Actually, there have been many
works done on the generic points set ([4, 6, 13, 19, 20], see also the refer-
ences therein). In the case of infinite symbolic space, the situation changes.
Liao, Ma and Wang ([14]) considered the set of continued fractions with
4 AIHUAFAN,MINGTIANLI,ANDJIHUAMA
maximal frequency oscillation. They proved that the set possessed Haus-
dorff dimension 1. This constant 1 was first observed there. Fan, Liao and
2 2
Ma ([8]) considered sets of real numbers in [0,1) with prescribedfrequencies
of partial quotients in their regular continued fraction expansions. They
showed that 1 is a universal lower bound of the Hausdorff dimensions of
2
these frequency sets. Furthermore, Fan, Liao, Ma and Wang ([9]) consid-
ered the Hausdorff dimension of Besicovitch-Eggleston subsets in countable
symbolic space. They found that the dimensions possess a universal lower
bound depending only on the underlying metric.
Later, Fan, Jordan, Liao and Rams ([11]) considered expanding interval
maps with infinitely many branches. They obtained multifractal decompo-
sitions based on Birkhoff averages for a class of continuous functions with
respect to the Eulidean metric.
Theorem 1 will be applied to study the generic points of invariant mea-
sures in the Gauss dynamics which is related to the continued fractions.
Actually, our result can be applied to a class of expanding interval mapping
system. Recall that the Gauss transformation S : [0,1) → [0,1) is defined
by
1 1
S(0) := 0, S(x) := − , ∀x ∈(0,1).
x x
(cid:22) (cid:23)
Let ℓ be an S-invariant Borel probability measure on [0,1) and let G be the
ℓ
set of ℓ-generic points. Consider the potential function
φ (x) = −sln|S′(x)| = 2slnx
s
for s > 1. The Gauss system is naturally coded by NN. It is known that
2
φ has summable variations and admits a unique Gibbs measure η whose
s s
convergence exponent is denoted by α . By a standard technique of trans-
s
ferring dimension results from the symbolic space to the interval [0,1), we
obtain the following result.
Theorem 2. Let ℓ ∈ M([0,1),S) be an S-invariant Borel probability mea-
sure and s > 1. If − 1lnxdℓ(x) < ∞, then
2 0
R h
ℓ
dim G =max α , ;
ηs ℓ s 1
( Pφs −2s 0 lnxdℓ(x))
otherwise, we have R
dim G = α ;
ηs ℓ s
Remark that the Gurevich pressure P equals the infinity if s 6 1 and
φs 2
the case s= 1 of Theorem 2 corresponds to Theorem 1.2 in [11].
The paper is organized as follows. In Section 2, we give some pre-
liminaries. Section 3 is devoted to the construction of a µ-generic point
x = (xn)n>1 ∈ Gµ satisfying xn 6 an, where {an}n>1 is a sequence of posi-
tive integers tending to the infinity. Using this point as seed we construct a
Cantor subset of G and we obtain the lower bound for dim G in Section
µ ν µ
5
4. Section 5 is concerned with the upper bound for dim G . In Section 6,
ν µ
Theorem 1 is applied to a class of expanding interval dynamics including
the Gauss dynamics and Theorem 2 is proved there.
2. Preliminaries
In this section, we will make some preparations: introducing a metric
to describe the w∗-convergence in M(X), discussing Gibbs measures and
defining the convergence exponent of a given measure, approximating a T-
invariant measure by ergodic Markov measures and discussing the relative
entropy of Gibbs measure with respect to a given T-invariant measure, ap-
proximating the above mentioned Markov measure by orbit measures.
First of all, let us begin with some notation. We denote by X the count-
able symbolic space NN endowed with the product topology and define the
shift map T : X → X by
(Tx) = x .
n n+1
An element (x ···x ) ∈ Nn is called an n-length word. Let A∗ = ∞ Nn
1 n n=0
stand for the set of all finitewords, whereN0 denotes the set of empty word.
S
Given x = (x x ···)∈ X and m > n> 1,
1 2
x|m = (x ···x )
n n m
denotes a subword of x. For ω = (ω ···ω ) ∈ Nn, the n-cylinder [ω] is
1 n
defined by
[ω]= {x ∈ X :x|n = ω}.
1
We will denote by Cn the set of all n-cylinders for n > 0. There is a one-to-
one correspondence between Nn and Cn. Let C∗ = ∞ Cn denote the set
n=0
of all cylinders. For j,N > 1 we will write
S
Σj = {1,··· ,N}j, Cj = {[ω] :ω ∈ Σj }.
N N N
2.1. Metrization of the w∗-topology. Recall that M(X) denotes the set
ofBorel probability measures onX. We endow M(X) withthew∗-topology
induced by C (X). Let us introduce a metric to describe the w∗-topology
b
of M(X).
For every cylinder [ω]∈ C∗, we choose a positive number a so that
[ω]
a = 1,
[ω]
[ω]∈C∗
X
where the sum is taken over all cylinders. For µ,ν ∈ M(X), define
d∗(µ,ν) = a |µ([ω])−ν([ω])|.
[ω]
[ω]∈C∗
X
The following proposition shows that the metric d∗ is compatible with the
w∗-topology of M(X).
Proposition 3. Let {µn}n>1 ⊂ M(X) and µ ∈ M(X). Then µn converges
in w∗-topology to µ if and only if lim d∗(µ ,µ)= 0.
n→∞ n
6 AIHUAFAN,MINGTIANLI,ANDJIHUAMA
Proof. Supposeµ converges in w∗-topology to µ, so that lim µ ([ω]) =
n n→∞ n
µ([ω]) for any ω ∈ A∗. Let ǫ > 0. Since (a[ω])[ω]∈C∗ is a probability on the
set of all cylinders, there exists a large integer K > 1 such that
ǫ
(2) a 6 ,
[ω] 4
[ωX]∈/DK
where D = {[ω] ∈ Cm : 1 6 m 6 K}. Since D is a finite set, we can find
K K K
a positive integer N > 1 such that for any n > N we have
ǫ
(3) |µ ([ω])−µ([ω])| 6 .
n
2
[ωX]∈DK
Then, by (2), (3) and the fact that µ ([ω]) 6 1, we have
n
d∗(µ ,µ) = ( + ) a |µ ([ω])−µ([ω])| 6 ǫ
n [ω] n
[ωX]∈/DK [ωX]∈DK
for n > N. Thus we have proved lim d∗(µ ,µ) = 0.
n→∞ n
Conversely, supposelim d∗(µ ,µ) = 0.Thisimplies immediately that
n→∞ n
forany [ω]∈ C∗, µ ([ω]) −→ µ([ω]) as n → ∞ . We finishthe proofby using
n
the following lemma which can be found in ([2], p.17).
Lemma 4. Let Y be a metric space and A be a family of subsets of Y. Then
µ converges in w∗-topology to µ if the following conditions are satisfied:
n
(1) A is closed under the finite intersection,
(ii) any open set U can be written as U = ∞ A with A ∈ A,
n=1 n n
(iii) for any A∈ A, lim µ (A) = µ(A).
n→∞ n
S
The set of cylinders has the above properties of A. (cid:3)
For any x ∈ X and n> 1, define the orbit measure
n−1
1
∆ := δ .
x,n n Tix
i=0
X
By Proposition 3, we can rewrite G as
µ
d∗
G = {x ∈ X :∆ −→ µ as n → ∞}.
µ x,n
The metric d∗ can be extended to finite symbolic measure space over X
and has the sub-linearity described in the following proposition which will
be useful in the sequel.
Proposition 5. For any µ ,µ ,ν ,ν ∈ M(X) and any α,β ∈ R, we have
1 2 1 2
d∗(αµ +βµ ,αν +βν )6 |α|d∗(µ ,ν )+|β|d∗(µ ,ν ).
1 2 1 2 1 1 2 2
Proof. It follows immediately from the definition of the metric d∗. (cid:3)
The following proposition shows that two orbit measures approach each
other, even uniformly, when the two orbit approach each other (under the
Bowen metric).
7
Proposition 6. The following equality holds:
(4) lim sup d∗(∆ ,∆ ) = 0.
x,n y,n
n→∞x|n=y|n
1 1
Proof. First, observe that
∞
A =1, where A := a .
n n [ω]
n=0 [ω]∈Cn
X X
By the sub-additivity of d∗ stated in Proposition 5, we have
n−1
1
d∗(∆ ,∆ ) 6 d∗(δ ,δ ).
x,n y,n n Tix Tiy
i=0
X
For any integer N such that n > N > 1, we break the above sum into two
parts: n−N−1+ n−1 .Assumethatx|n = y|n. By the definition of the
i=0 i=n−N 1 1
metric d∗, we have
P P
n−N−1 n−N−1 ∞ ∞
d∗(δ ,δ ) 6 A 6 (n−N) A .
Tix Tiy j j
i=0 i=0 j=n−i+1 j=N+2
X X X X
Then by noting the trivial fact that d∗(δ ,δ ) 6 1, we have
Tix Tiy
∞
n−N N
d∗(∆ ,∆ ) 6 A + .
x,n y,n j
n n
j=N+2
X
Letting n then N tend to the infinity, we finish the proof. (cid:3)
2.2. Gibbs measure. WeuseGibbsmeasurestoinducemetricsonX. The
following facts about Gibbs measures can be found in [25].
Recall that for a function ϕ : X → R, called potential function, the n-
order variation of ϕ is defined by
var ϕ := sup{|ϕ(x)−ϕ(y)| : x,y ∈ X, x|n = y|n}.
n 1 1
We say that a potential ϕ has summable variations if
∞
var ϕ< +∞.
n
n=2
X
It is easy to see that a potential ϕ with summable variations is uniformly
continuous on X. The Gurevich pressure of ϕ with summable variations is
defined to be the limit
1
P := lim ln eSnϕ(x)1 (x),
ϕ [a]
n→∞n
Tnx=x
X
where a ∈ N and it can be shown that the limit exists and is independent of
a (see [23]).
8 AIHUAFAN,MINGTIANLI,ANDJIHUAMA
Aninvariantprobabilitymeasureν iscalledaGibbs measureassociated
to a potential function ϕ if it satisfies the Gibbsian property: there exist
constants C > 1 and P ∈R such that
1 ν([x x ···x ])
(5) 6 1 2 n 6 C
C exp(S ϕ(x)−nP)
n
holds for any n > 1 and any x ∈ X. It is known ([24]) that a potential
function ϕ with summable variations admits a unique Gibbs measure ν iff
var ϕ < +∞ and the Gurevich pressure P < +∞. Assume that ϕ admits
1 ϕ
a unique Gibbs measure ν . Then the constant P in (5) is equal to the
ϕ
Gurevich pressure P . Let ϕ∗ = ϕ−P , we have
ϕ ϕ
Pϕ∗ = 0 and νϕ∗ = νϕ.
Hence, without loss of generality, we always suppose P = 0 in the rest of
ϕ
this paper. A trivial fact is that the Gibbsian property (5) implies:
(6) ∀x ∈X, ϕ(x) 6 lnC.
It follows that the integral ϕdµ is defined as a number in [−∞,+∞) for
X
any probability measure µ. Also, the Gibbsian property implies the quasi
R
Bernoulli property which will beexploited many times in the present paper.
Lemma 7. Let ν be a Gibbs measure associated to potential ϕ. For any k
words ω ,··· ,ω , we have
1 k
C−(k+1)ν([ω ···ω ]) 6 ν([ω ])···ν([ω ]) 6 Ck+1ν([ω ···ω ]).
1 k 1 k 1 k
For any T-invariant Borel probability measure µ, define the relative en-
tropy of ν with respect to µ by
1
h(ν|µ) = limsup− µ([ω])lnν([ω]).
k
k→∞ ωX∈Nk
It is trivially true that h(µ|µ) = h .
µ
When ν is the Gibbs measure associated to ϕ, the relative entropy h(ν|µ)
is equal to the integral − ϕdµ.
X
Proposition 8. AssumeRthat ϕ has summable variations and admits a
unique Gibbs measure ν. Then for any invariant measure µ ∈ M(X,T),
we have
1
h(ν|µ) = lim − µ([ω])lnν([ω]) = − ϕdµ.
k→∞ k ωX∈Nk ZX
Proof. For each cylinder [ω], we arbitrarily choose a point x′ in [ω]. Then
ω
for any λ ∈ M(X), let
I (λ) = λ([ω])ϕ(x′ ).
k ω
ωX∈Nk
9
Invirtueof (6),theaboveinfiniteseriesisdefinedasanumberin[−∞,+∞).
Furthermore, theconvergence of theseries implies theabsolute convergence.
Since ϕ is uniformly continuous on X, for any λ ∈ M(X) we have
k
1
(7) ϕdλ = lim I (λ) = lim I (λ).
k i
ZX k→∞ k→∞k i=1
X
First,weassumethat ϕdµ > −∞.Fork > 1,bytheGibbsianproperty
X
of ν, we have
R
µ([ω])lnν([ω]) 6 lnC + µ([ω])S ϕ(x ).
k ω
ωX∈Nk ωX∈Nk
Notice that
S ϕ(x )≤ [ϕ(x′ )+var ϕ]+···+[ϕ(x′ )+var ϕ].
k ω ω1···ωk k ωk 1
It follows that
k
µ([ω])S ϕ(x ) ≤ var ϕ+[ µ([ω ···ω ])ϕ(x′ )
k ω j 1 k ω1···ωk
ωX∈Nk Xj=1 ω1··X·ωk∈Nk
+ ···+ µ([ω ])ϕ(x′ )]
k ωk
ωXk∈N
k k
= var ϕ+ I (µ).
j i
j=1 i=1
X X
Finally we get
∞ k
µ([ω])lnν([ω]) ≤ lnC + var ϕ+ I (µ).
j i
ωX∈Nk Xj=1 Xi=1
In the same way, we can also get the opposite inequality
∞ k
µ([ω])lnν([ω]) > −lnC − var ϕ+ I (µ).
j i
ωX∈Nk Xj=1 Xi=1
Thus
1 µ([ω])lnν([ω])− 1 k I (µ) 6 lnC + ∞j=1varjϕ.
i
(cid:12)k k (cid:12) k
(cid:12)(cid:12) ωX∈Nk Xi=1 (cid:12)(cid:12) P
(cid:12) (cid:12)
By (7) w(cid:12)e obtain (cid:12)
(cid:12) (cid:12)
h(ν|µ) =− ϕdµ.
ZX
Now assume that ϕdµ = −∞. The similar argument works in com-
X
bination with the following fact: there exists a sequence of points {x′ }
R ω1···ωk
such that
lim µ([ω])ϕ(x′ ) = −∞.
ω
k→∞
ωX∈Nk
10 AIHUAFAN,MINGTIANLI,ANDJIHUAMA
(cid:3)
By the concavity of the logarithm function ln, it is easy to show
(8) h(ν|µ) > h .
µ
By Proposition 8, we can rewrite the variational principle ([25], p. 86) in
the following form
(9) P = sup {h −h(ν|µ) : h(ν|µ) < +∞}.
ϕ µ
µ∈M(X,T)
Recall that we assume that P = 0. It is known that the supremum in the
ϕ
variational principle (9) is attained only by a Gibbs measure ν with h <∞
ν
if such Gibbs measure exits ([25], p. 89). It follows that when ν 6= µ, we
have h(ν|µ) > h , which implies h(ν|µ) >0.
µ
RecallthataGibbsmeasureν inducesametricρ onX: foranyx,y ∈ X,
ν
if x = y, we define ρ (x,y) = 0; otherwise
ν
ρ (x,y) =ν([x|n]),
ν 1
wheren = min{k > 0 :x 6= y }. Onecan show thatρ isaultrametric
k+1 k+1 ν
and induces the product topology on X since ν is non-atomic and has X as
its support. Let n > 1 be an integer. Define
δ = sup{ν([u]) :u∈ Nn}.
n
Thefollowingpropositionmeansthattheρ -distanceoftwopointsuniformly
ν
tends to zero when they approach each other in the sense of Bowen. This
property will be used in the proof for the upper bound of dim G .
ν µ
Proposition 9. For {δn}n>1 defined on the above, one has
lim δ =0.
n
n→∞
Proof. Suppose that lim δ = a > 0. Note that δ is non-increasing.
n→∞ n n
Then there exists a sequence of cylinders [u ]∈ Cn so that
n
ν([u ]) > a/2.
n
Observethattwocylinderseitheraredisjointoroneiscontainedintheother.
Sinceν isaprobabilitymeasure,thereexistsacylinder[u ]whichintersects
n1
infinitely many cylinders {[u ] : k > 1} ⊂ {[u ] : n > 1}. Actually, the
nk n
cylinder [u ] contains every element of {[u ] : k > 1}. By the same
n1 nk
argument one can choose a cylinder [u ] with n > n which contains
n2 2 1
infinite elements of {[u ] : k > 1}. In this way, one choose a sequence of
nk
decreasing cylinders {[unj]}j>1 so that
∀j > 1, ν([u ]) >a/2.
nj
This contradicts the fact that the Gibbs measure ν has no atom. (cid:3)