Multifractal analysis for countable shift space with
matrix potentials
Yun Zhao
Department of Mathematics Suzhou University
Suzhou215006, Jiangsu, P.R. China [email protected]
YongLuo Caoy
Department of Mathematics Suzhou University
Suzhou215006, Jiangsu, P.R. China [email protected]
[email protected] JungChao Banzx
Department of Mathmatics National Dong Hwa University
Hualien 970003, Taiwan [email protected]
September 7, 2010
Abstract
Let ( A; ) be a subshift of …nite type with incidence matrix A
is over countably in…nite alphabet, i.e., A 2 MI I with I = N; and
M : A ! L Rd; Rd be a continuous function on A taking values
Zhao is partially supported by NSF for College and Universities in Jiangsu Province (09KJB110007) and by a Pre-research Project of Suzhou University.
yCao is partially supported by NSFC Project (2007CB814800). zAuthor to whom any correspondence should be addressed
xBan is partially supported by the National Science Council, ROC (Contract No NSC
98-2628-M-259-001-), National Center for Theoretical Sciences (NCTS) and CMPT(Center for Mathematics and Theoretical Physics) in National Central University.
in the set of positive matrices. We extend some of the theory of ther-modynamic formalsim on M with I is a …nite set (cf. [3], [5], [6]). More precisely, it is proved that the Gibbs measure exists uniquely; variational principle holds between pressure, entropy and Lyapunov exponents on M and the di¤erentiability of the pressure. We should remark that the main di¢ culty is that Ais not compact if I = N and
the potential is no longer real-valued. Finally, we e¤ect a multifractal structure for the entropy spectrum of the upper Lyapunov exponents.
1
Introduction
In this paper we study the entropy spectrum of the Lyapunov exponents and extend some standard thermodynamical results to countable shift space, this is mainly motivated by the recent works of [3] for the matrix potentials, and [17] for the generalization of underlined symbolic space to countable one.
We …rst give some notations and background before formulating our main results. Let be the shift map on A (In the following we simply write
for A) with A is indexed either …nite or countably many alphabets and
denote by jAj the matrix dimension of A, i.e., kAk = 1TA1; 1T = (1; : : : ; 1).
If jAj < 1; the mathematical foundation for the thermodynamic formalism was developed by Ruelle [20], Bowen [2] and some other works (cf. [25]), which include the variational principle, existence, uniqueness of Gibbs and equilibrium measures, di¤ erentiability of the pressure (this e¤ects the phase transition dynamics) and so on.
Later, Pesin and Weiss (cf. [18]) and some other works consider the multifractal analysis on Gibbs and equilibrium measures for conformal map which explore the …ne structure for a given dynamical systems equipped with some real-valued potential. On the other hand, for the potential function is matrix-valued and the underlined space is of …nite symbols, i.e., jAj < 1; Feng and Lau [6] …rst construct the Gibbs measure for such potential and Feng [5] establish the variational principle according to this setting. Feng [3] also set up the dimension spectrum for upper Lyapunov exponent. We refer [7], [12] and [10] to those readers who are interested in the motivations of the matrix-valued potentials and their applications to the study of mul-tifractal structure of self-similar measures generalized by iterative function systems (IFS brie‡y) with no overlap and general Sierpinski (or McMullen [12]) carpet .
If jAj = 1 ( jAj = 1 means countably many herein), Gurevic (cf. [8], [9]) adopt the ideas of renewal theory to show that the Perron number of the incidence matrix is equal to the supremum of the metric entropy. Later,
Sarig [21] extends Gurevic’s results to pressure function and establish the variational principle. On the other hand, Mauldin and Urbanski [17] gener-alized the classical results of thermodynamics to jAj = 1 with the potential is real-valued. To be precise, they prove the existence and uniqueness of Gibbs measure, exponential decay of correlations and central limit theorem. We note here that they use the weaker smoothness for the potential function and stronger speci…cation property than Sarig [21]. As we know, the reason to extend those works of jAj < 1 to jAj = 1 is that for a hyperbolic system, it always admits the …nite Markov partition [2] and all the investigation of statistic properties are related to study the …nite symbolic dynamical sys-tem which lifts the origin one. However, it is no longer true for parabolic or non-hyperbolic systems and the phase transition dynamics occurs in this cir-cumstance (cf. [22], [23] and [24]). Standard method demonstrates that one could code this system to a countable symbolic system and then developing the …nite approximation theory therein.
In this paper, we study the thermodynamic properties for Awith jAj =
1 and matrix-valued potential under the setting of [3], [5] and [6]: The problem is more subtle because is not compact herein and M is no longer real-valued. Note that the product of matrix deeply e¤ects the admissible words of the underlined space . For instance, let = f0; 1; 2gZ be the full shift of 3 symbols, de…ne the matrix-valued potential depends on one coordinate as M0 = 0 1 0 1 ; M1= 1 1 1 0 and M2 = 0 0 1 1 ;
it can be checked that M0M1 0 and M1M2 0 and this means [01] and
[12] are still admissible words of length 2: However, one can easily checked that M0M1M2 = 02 2 and [012] forms a new forbidden word of length 3
for ; and since the underlined space is mixing, the admissible words which connect to [012] are also forbidden. One can see that these forbidden patterns grow exponentially and make the original dynamical system more complicated.
For jAj = 1, if M is positive or non-negative under some additional assumptions, we prove the existence and uniqueness for the Gibbs measure. It can also be proved that the pressure function is di¤erentiable, i.e., such system is lack of phase transition in the sense of Ruelle [20]. Finally, we develop the multifractal analysis for the entropy spectrum of the Lyapunov exponents. Precisely, let
EM( ) = 2 : lim n!1
log k nM ( )k
where nM (x) is the product of matrices which will be de…ned in (3).
Then, for any = PI0(q) with respect to q 1; the entropy function forms a Legendre-Fenchel transformation with respect to pressure PI(q), i.e.,
htop(EM( )) = inf
q2Rf q + PI(q)g : (2)
where PI(q) denotes the pressure function with respect to q 2 R and M
which will be de…ned in (5).
The paper is organized as follows. In Section 2, we review some of the standard facts for the Gibbs states and pressure, then we develop the …-nite approximation theorem and variational principle for countable Markov shifts. In Section 3, we set up the di¤erentiability result for the pressure function and establish the entropy spectrum for Lyapunov exponents by means of …nite approximation theorem. In Section 4 we consider non-negative matrix potentials, a new …nite approximation theorem to this type of potential can be proved and we establish the entropy spectrum by us-ing this approximation result. Finally, we complete the proof of the main theorem in Section 5.
2
Preliminaries
Some of the materials in this section are known. We recall them here both for our convenience and that of readers. Let A be an incident matrix indexed by I 2 N, say is irreducible if for all i; j 2 I; there exists a path ! 2 = S
n2N n such that i!j 2 ; where n denotes the set of all n-cylinders in
. We also call is …nitely irreducible if there exists a …nite set
such that for all i; j 2 I; there exists a path ! 2 such that i!j 2 : Finally we call is …nitely primitive if it is …nitely irreducible and j!j = p for all ! 2 ; here j j denotes the length of 2 : In the following we always assume is …nitely primitive. Let G I; we de…ne the restriction space of to G by
G= GZ\ =n! 2 GZ : A
!i!i+1 = 1 for all i 1
o :
For ! 2 , we also de…ne [! \G] = 2 G : jj!j= ! ; where jndenotes
the restriction of n-cylinder of and Gn be the collection of n-cylinders on
G: Suppose M :
! L+ Rd; Rd is continuous function taking values in the set of positive d d matrices (Here we say K = (Kij) is positive if
Kij > 0 for all 1 i; j d). For 2 and n 2 N; we simply write
where kBk denotes the matrix norm, i.e., kBk 1tB1, 1t = (1; 1; : : : ; 1): For all q 2 R; the topological pressure on G for M with respect to G I is de…ned: PG(q) = PG(M; q) = lim n!1 1 nlog X !2 G n sup 2[!\G]k nM jG( )kq; (4)
where M jG is the restriction of M to G I: And the topological pressure
on I is de…ned similarly, i.e.,
PI(q) = PI(M; q) = lim n!1 1 nlog X !2 n sup 2[!]k nM ( )kq; (5)
whenever the limit exists. The existence of above limits (4) are the immedi-ate consequences of the …nitely primitivity of G and the sub- (or super-)
additive property of positive matrix M (see Lemma 2.1 [6] and Lemma 2.1 [3]). Next we recall some results for real-valued potential of [17] and [2]. We call f : ! R is acceptable (cf. De…nition 1.1 [17])if
osc(f ) sup i2I ( sup x2[i] f (x) inf x2[i]f (x) ) < 1: (6)
Set F = ff 2 ! R : f is F-continuousg ; where F stands for H; A and C for Holder, acceptable and continuous respectively. It can be easily checked that
H A C:
For M : ! L Rd; Rd we also call M is acceptable if (6) holds for f (x) = log kM(x)k : Let F+ be de…ned similarly as Ffor the positive matrix-valued
potentials, i.e.,
F + =
n
M : ! L Rd; Rd j M is F-continuous positive matrixo;
and we also have
H
+ A+ C+:
For M 2 C+; q 2 R; a measure 2 M ( ) (the collection of -invariant
Borel measure on ) is Gibbs for M if there exists Q > 0 such that for all ! 2 n; and 2 [!]; we have
Q 1 ([!])
exp ( nPI(q)) k nM ( )kq
The classical results shows that Gibbs measure exists for f 2 H and is mixing with jAj < 1 (cf. [2]). Feng and Lau [6] prove that the Gibbs measure exists for positive (or non-negative with some additional conditions) matrix potential M 2 H+, Mauldin and Urbanski [17] show such measure
exists for jAj = 1 with is …nitely primitive and f 2 A: Finally, Sarig [21] shows Gibbs measure exists with jAj = 1, is mixing and f 2 H: In this paper, we …rst demonstrate the existence and uniqueness for Gibbs measure with jAj = 1 and M 2 H+. In the following we present the result
of Feng and Lau and formulate our main result later.
Theorem 1 (Theorem 1.1. of [6]) Let ( A; ) be a …nite Markov system
with A is mixing and M 2 H+: Then for all q 2 R; there exists a unique
-invariant , ergodic probability measure q on A satisfying (7).
Theorem 2 Let M 2 H+ and be …nitely primitive, if P
i2I
sup
2[i]kM ( )k < 1;
then there exists a unique Gibbs measure 2 M ( ) for q 1: Remark 3 1. The reason of why we impose the condition
P
i2I
sup
2[i]kM ( )k < 1
is that in [17], if the potential function f : ! R is real-valued, then the condition P
i2I
exp sup 2[i]f (e) < 1 allows us to de…ne the Perron-Frobenius operator Lf : Cb( ) ! Cb( ) acting on the space of
bounded continuous functions Cb( ) and the existence of Gibbs
mea-sure is equivalent to …nd the eigenmeamea-sure for the conjugate operator Lf : Cb( ) ! Cb( ): Although we do not adopt this approach, the
assumption remains.
2. We note here in Theorem 1 the Gibbs measure exists for all q 2 R: Nevertheless, for countable system it only exists for q 1 and we suspect that this measure may not exist for q < 1 and the problem is still not clear.
We left the proof of Theorem 2 in Section 5 and develop the …nite ap-proximation theorem which is the analog of Theorem1.2 and Theorem 1.3 of [17] and also Theorem 1 and Corollary 2 of [21].
Proposition 4 Under the same assumptions of Theorem 2 then for all q 0
PI(q) = sup PG(q) j G I with G is mixing : (8)
For the proof of Proposition 4, we need the following lemmas which come from Lemma 2.1. and Lemma 2.2. of [6] and the proofs are omitted herein. Lemma 5 Let G I and G is mixing and M 2 C+: There exists a constant C > 0 such that for any 2 G and n; m 2 N
C k nM ( )k k mM ( n( ))k k n+mM ( )k k nM ( )k k mM ( n( ))k :
Lemma 6 Let M 2 H
+; there exists a constant D > 0 such that for all
n 2 N; ! 2 n and 1; 22 [!]; we have n Q j=1 j ! 1 k nM ( 1)k k nM ( 2)k n Q j=1 j; where n= sup Mij(x) Mij(y) : I 2 n; x; y 2 I; 1 i; j d and lim n!1 n Q j=1 j < D:
Proof of Proposition 4. First we prove for q 0 and 8G I
PG(q) PI(q): (9)
The partition function Zn(G; M; q) and Zn(M; q) are de…ned as
Zn(G; M; q) = X !2 G n sup 2[!\G]k nM jG( )kq; (10) Zn(M; q) = X !2 n sup 2[!]k nM ( )kq: Then Zn(G; M; q) X !2 G n sup 2[!]k nM ( )kq X !2 n sup 2[!]k nM ( )kq:
Therefore (9) holds for q 0: For the opposite inequality. Fix 0 q 2 R; since P
i2I
sup
2[i]kM( )k < 1, it can be easily checked that P
combining Lemma 5 and standard argument we can assert that (log Zn(M; q))n 1
is also sub-additive. Therefore, for any k 2 N; we have
PI(q)
1
klog Zk(M; q): (11) For " > 0; choose G I and N 1 large enough such that G (thus
G is mixing) and if n N 1 nlog X !2 G n sup 2[!]k nM ( )kq 1 nlog Zn(M; q) " 2: (12) De…ne ^ Zn(M; q) = X !2 G n sup 2[!]k nM ( )kq:
We claim that for all r 2 N is …xed there exists C = C(r) > 0 such that for all m 2 N we have
^
Zmr+(r 1)p(M; q) CZm(G; M; q)r: (13)
Indeed, for !(1); ; !(r) 2 Gm; since G and 1; ; r 1 can be pick
such that !(i)n i!1(i+1)2 G for i = 1; ; r 1; where !k denotes the
k-th coordinate of !: Let
! = !(1) 1!(2): : : r 1!(r)
and i 2 !(i) for i = 1; ; r 1; and if 2 [!], q 0; mr+(r 1)pM ( ) q is less than k mM ( )kqk pM ( m( ))kq mM (r 1)m+(r 1)p( ) q D1q(r 1) m Q j=1 j !qr r Y j=1 k mM ( j)kq: (14)
(14) and the de…nition of ^Zn(M; q) and Zn(M; q); we have ^ Zmr+(r 1)p(M; q) = X !2 G mr+(r 1)p sup 2[!]k nM ( )kq D1q(r 1) m Q j=1 j !qr X !(1); ;!(r)2 G m r Y j=1 k mM ( j)kq D1q(r 1) m Q j=1 j !qr Zm(G; M; q)r D1q(r 1) lim m!1 m Q j=1 j !qr Zm(G; M; q)r = D1q(r 1)DqrZm(G; M; q)r
Taking C = D1q(r 1)Dqr and (13) follows: Combine (12) with (13) yields for mr + (r 1)p N: 1 mr + (r 1)plog Zmr+(r 1)p(M; q) 1 mr + (r 1)plog Zmr+(r 1)p(M; q) + " 2 1 mr + (r 1)plog[CZm(G; M; q) r] + " 2 1 mlog Zm(G; M; q) + ": (15) Therefore, it follows from (11) and (15) that
PI(q) " +
1
mlog Zm(G; M; q) : Taking m ! 1 and the arbitrariness of " leads to
PI(q) sup fPG(q) : G Ig : (16)
This completes the proof.
Then we can use Proposition 4 to prove the following variational principle for A 2 MI I:
Corollary 7 Under the same assumptions of Proposition 4, we have for all q 0 PI(q) = sup 2M ( )fh ( ) + qM ( )g ; (17) where M ( ) lim n!1 1 n Z log k nM ( )k d ( ) : (18)
Proof. Let G I is …nite and M jGbe the restriction of M to G: Since M 2 H
+ C+then MGis also continuous on G: It follows from Theorem 1 that
there exists a measure 2 M G such that PG(q) = h ( ) + qM ( ):
Proposition 4 is thus applied to show that PI(q) = sup fPG(q)j G Ig
= sup h ( ) + qM ( )j 2 M ( G) and G I with G is mixing sup fh ( ) + qM ( )j 2 M ( )g :
To deal with the other inequality, we let be the natural partition on I; i.e., = f[1]; [2]; ; [m]; [> m]g and n 0 = n 1W i=0 i( ) : For 2 M ( ), we have for q 0; 1 n H ( n 0) + Z
log k nM (y)kqd (y)
1 n X !2 n 0 ([!]) log ([!]) + sup y2[!]log k nM (y)kq ! = 1 n X !2 n 0
([!]) logsupy2[!]log k nM (y)k
q ([!]) ! 1 nlog X !2 n 0 sup y2[!]log k nM (y)kq= 1 nlog Zn(M; q) : (19)
It follows from Lemma 5 that R log k nM (y)kqd (y) n 1is sub-additive
and consequently the limit exists lim n!1 1 n Z log k nM (y)kqd = qM ( ) : Letting n ! 1 yields h ( ; ) + qM ( ) PI(q) :
The proof is completed.
To end this section, we give some remarks about Proposition 2 and Corollary 3.
Remark 8 1. The existence of the limit (18) follows from the sub addi-tive property ofR log k nM ( )k d ( ) (cf. [25] ).
2. (8) and (17) hold for f 2 H with is mixing (cf. [21]) or f 2 A with is …nitely primitive [17].
3
Di¤erentiability of the pressure and multifractal
analysis
Let us turn to study the entropy spectrum for upper Lyapunov exponent. We …rst present the following results concerning the di¤erentiability of the pressure function and the entropy spectrum for upper Lyapunov exponent in …nite Markov systems.
Theorem 9 (Theorem 1.2. and Theorem 1.3. of [6]) Under the same assumption of Theorem 1 and if P (q) denotes the topological pressure for positive matrix-valued potential M : A! L(Rd; Rd): Then
1. P (q) is di¤ erentiable for q 6= 0: 2. For any = P0(q); q 6= 0; we have
htop(EM( )) = q + P (q): (20)
Remark 10 In Theorem 1.3. of [6] the authors develop the dimension spec-trum of (20). To be precise, they have
dimH(EM( )) =
1
log m( q + P (q))
with the underlined space is full shift of m symbols. And it is not di¢ cult to extend this formula to SFT and entropy spectrum (20).
Theorem 11 Let M 2 C+ with PI(q) 6= 1 and is …nitely primitive.
Then for all q 0 PI0 q+ lim "!0+ PI(q + ") PI(q) " = sup fM ( ) : 2 I ( )g ; PI0 q lim "!0 PI(q + ") PI(q) " = inf fM ( ) : 2 I ( )g ; where I ( ) denotes the collection of equilibrium measures associated with (17).
Proof. Up to minor modi…cations the proof is identical with the proof of Theorem 1.2. [5] and we omit here.
Here we present the existence and uniqueness of equilibrium measure for M . We follow the idea of Theorem 3.5. of [17] to show the unique-ness of equilibrium measure and refer to [17] for the proof of uniqueunique-ness of equilibrium measures under settings of is in…nite and the potential is real-valued.
Theorem 12 Under the same assumptions of Theorem 2 and for q 1; let
q be the corresponding unique Gibbs measure, then q is also the unique
equilibrium measure for M with respect to (17):
Proof. Since the proof of that Gibbs measure is equilibrium is straight-forward, we only give the proof for the uniqueness of equilibrium measures. Indeed, suppose q6= q is another equilibrium measure for M , without loss
of generality we assume q is ergodic. Let be the nature partition as in
Corollary 7, since H q( n 0) n 1and nR logk nM (y)kq n d q o are sub-additive, then 8n 1 0 = n h q( ) + qM ( q) PI(q) H q( n 0) + Z
(log k nM (y)kq nPI(q)) d q(y)
= X
I2 n
q([I]) log q([I])
1 q([I]) Z I(log k nM (y)kq nPI(q)) d q(y) X I2 n
q([I]) log q([I]) sup y2[I](log k
nM (y)kq nPI(q))
!
= X
I2 n
q([I]) (log q([I]) (log k nM (yI)kq nPI(q)))
= X
I2 n
q([I]) (log q([I]) log exp ( nPI(q)) k nM (yI)kq) (21)
Since q satis…es (7), it follows form (21) we obtain that
0 X
I2 n
q([I]) (log q([I]) log Q q([I]))
= log Q X I2 n q([I]) log q ([I]) q([I]) (22)
where Q comes from (7). Since q and q are both ergodic and they must
be singular to each other and for all K > 0
lim n!1 q I 2 n: q([I]) q([I]) K = 0: De…ne Fn;j = I 2 n: e j q ([I]) q([I]) e j+1 ;
then q(Fn;j) = Z Fn;j q([I]) q([I]) d q([I]) e j+1:
For each integer k 1 we have X I2 n q([I]) log q ([I]) q([I]) = Z log q([I]) q([I]) d q X j2Z q(Fn;j) k X j k q(Fn;j) + X j 1 je j+1 k q( I 2 n: e k q ([I]) q([I]) ) +X j 1 je j+1! k +X j 1 je j+1 as n ! 1: Therefore lim n!1 X I2 n q([I]) log q ([I]) q([I]) = 1;
a contradiction. This completes the proof.
Combining Theorem 11 and Theorem 12, we obtain the di¤erentiability for PI(q) :
Theorem 13 Under the same assumptions of Theorem 2; q 7! PI(q) is
di¤ erentiable for q 1.
Finally, we give the analysis for the entropy function on EM( ) which
is de…ned in (1). More precisely, we have the following transformation for htop(EM( )) and PI(q).
Theorem 14 Under the same assumptions of Theorem 2, if = PI0(t) with t 1; we have
htop(EM( )) = inf
q2Rf q + PI(q)g
= t + PI(t) :
Proof. We …rst prove that 8q 1 and 2 R;
q + PI(q) htop(EM( )) : (23)
Indeed, for " > 0 and n 2 N; let
F (n; ; ") = x 2 nj
1
and f (n; ; ") be its cardinality. It can be easily checked that EM( ) [ k2N 1 \ n=k F (n; ; ") and S !2F (n; ;") [!] is a cover for G (k; ; ") = 1T n=k F (n; ; ") for n k: Thus it su¢ ces to show that
htop(G (k; ; ")) q + PI(q) for all k 2 N: (25)
Since M 2 H+, it can be easily checked that
Zn(M; q) f ( ; n; ") exp (qn ( "))
Hence (25) follows and so does (23). Next we prove that htop(EM( )) inf
q2Rf q + PI(q)g : (26)
From Lemma 21 below we take subsequence if necessary, we choose
(k)
k 1 = Gk
k 1
is a sequence of irreducible subshifts of …nite type with S1
k21
(k) = I and
if k is large enough we have S
n2N (k)
n ( recall that is the …nite set
in the de…nition of …nite irreducibility of ) and (k) is thus mixing SFT. Therefore EM( ) = 1 [ k21 EMk( ) : (27)
According to Theorem 9, for each k 2 N; since q 7! P (k)(q) is di¤erentiable
thus there exists (qk)k 1 such that
htop(EMk( )) = qk+ P (k)(qk): (28)
Combining Theorem 13, Proposition 4 and PI(q) < 1 we obtain (qk)k 1
is bounded. Therefore, there exists a q such that lim
Proposition 4 and 1S
k21
(k) = I we conclude that there exists (
k)k 1 # 0
such that
jPI(q ) P (k)(qk)j jPI(q ) PI(qk)j + jPI(qk) P (k)(qk)j
jPI(q ) PI(qk)j + k: (29)
It also follows from Proposition 4 and continuity of q 7! P (k)(q) 8k 1
we conclude that q 7! PI(q) is continuous. Then (29) leads to that for all
" > 0; there exists N1 > 0 such that if k N1; we have
jPI(q ) P (k)(qk)j
"
2: (30)
On the other hand, since lim
k!1qk= q ; then for all 2 R there exists N2> 0
such that for k N2
qk q
"
2: (31)
Therefore, combining (27), (28), (30) and (31) we have if k max fN1; N2g
htop(EM( )) htop(EMk( )) = qk+ P (k)(qk)
q + PI(q ) " inf
q2Rf q + PI(q)g ":
This establishes the desire formula (26). Finally we prove that if = PI0(t) then
htop(EM( )) = t + PI(t) : (32)
For 2 R is …xed; Theorem 9 is applies to show that for all k 2 N there exists qk2 R such that = P0(k)(qk) with
htop(EMk( )) = qk+ P (k)(qk): (33)
It follows from Proposition 4, Theorem 13 and PI(q) < 1 for all q 1 we
obtain
lim
k!1qk= t: (34)
Therefore, combining (27), (34) and (33) htop(EM( )) = sup k2N htop(EMk( )) = lim k!1htop(EMk( )) = lim k!1( qk+ P (k)(qk)) = t + PI(t) :
Thus the desired formula (32) follows and the proof is completed. If the smoothness of M is weak i.e., M 2 C+; we also have.
Theorem 15 For all q 1; if M 2 C+ and is …nitely primitive, we have. htop(EM( )) = inf
q2Rf q + PI(q)g : (35)
Proof. This proof is identical to the proof of Proposition 2.7. [5]. The idea is to approximate the continuous M by the Holder M(k) which is the restriction of M to the …rst k-coordinates, thus M(k)is Holder, thus Theorem 14 is applied to establish the formula (35) and the proof is omitted.
4
Non-negative matrices
In this section we extend the result of Theorem 2, Theorem 14 to non-negative matrix potentials. For the convenience of the readers we repeat the relevant ingredients from [4] and [6] without proofs, thus making our exposition self-contained. Here we follow the conditions which Feng and Lau [6] used in non-negative matrices M (x) and recall M jG is the restriction of
M to subset G I as above:
(H1) M (x) = Mi if x 2 [i]; i = 1; ; m;
(H2) For all G I and G ; there exists r = r(G) > 0 such that for any i; j 2 f1; ; jGjg ; r X k=1 X K2 G K;i;j (MG)K > 0
where GK;i;j denotes the set of all K 2 Gk such that iKj 2 Gk+2: For clarity we make the assumption for is the full shift from now on, i.e., A 2 MI I with all entries are 10s. Since M satis…es (H1) i.e., M only
depends only 1-coordinate, thus it is reasonable to write nM ( ) = M! for
! 2 n and 2 [!] : The following Lemma comes from Lemma 2.1. [4].
Lemma 16 For all G I and G is mixing, there exists C = C(G) > 0 such that
1. For any ! 2 Gn; v 2 Gl there exists K 2
Sr
k=1 Gk such that
2. For any ! 2 Gn and v 2 N, there exists K 2 Gl such that
k(MjG)!Kk Clk(MjG)!k :
3. For any ! 2 Gn and v 2 Gl there exists 1 k r; K1 2 Gk and
K22 G2r k such that
(M jG)!K1vK2 C k(MjG)!k k(MjG)vk :
4. for any !1; !2; : : : ; !n 2 Gl ; there exists K1; K2; : : : ; K2n 2S2r 1j=1 Gj
such thatP2nj=1jKjj = 2nr and
Cn n Y j=1 (M jG)!j (M jG)!1(K1!2K2)(K3!3K4):::(K2n 3!nK2n 2)K2n 1K2n C n n Y j=1 (M jG)!j :
For all G I and G ; de…ne the fat set as in [3] and [4].
EM jG( ) = x 2 G: lim n!1 1 nlog k n(M jG) (x)k = ; (36) EM j(l) G( ) = x 2 G: lim n!1 1 nllog Qn j=1 l(M jG) lj(x) = : (37) FG( ; n; ") = ! 2 Gn : 1 nlog k(MjG)!k " ; (38) FG;l( ; nl; ") = ! 2 Gnl: 1 nllog Qn j=1 (M jG)!(j 1)l+1 !jl " : (39) And …nally fG;l( ; nl; ") = #FG;l( ; nl; ") ; (40) fG( ; n; ") = #FG( ; n; ") (41)
Lemma 17 For all G I, there exists D = D(G) > 0 such that for any " > 0, there exists N > 0 such that for all 2 LMG,
fG( ; p(l + 2r); ; 2") DpfG;l( ; pl; ") Dp(fG( ; l; "))p; 8p 2 N; 8l > N
where LM jG = 2 R : = M jG(x) for some x 2 G :
In the following we present a …nite approximation theorem for non-negative M:
Theorem 18 Let M 2 C satisfy the condition (H1) and (H2) and A is
…nitely primitive. Then for q 0 we have
PI(q) = sup fPG(l; q)j G I and l 2 Ng ; where PG(l; q) = 1 l log X !2NG l k(MjG)!k q and NG l = ! 2 Gl j k(MjG)!k 6= 0 :
Proof. First we prove that 8G I and 8" > 0 there exists l 2 N such that for all q 0 1 l X !2Nl k(MjG)!kq PI(q) + ": (42)
Indeed, let l 2 N and G I be large enough such that S
n2N G n: If
!(1); ; !(n)2 NG
l ; Lemma 16 is applied to show that there exists Ki 2 G
for i = 1; ; n 1 with
! !(1)K1!(2) !(n 1)Kn 1!(n)2 Nnl+kG ;
for some k 2 [n 1; (n 1)r]: Up to a minor modi…cation of the proof of Lemma 2.3. of [4] we obtain that
bq X !(1); ;!(n)2N l k(MjG)!(1)kq k(MjG)!(n)kq X !2Nnl+k k(MjG)!kq;
for some b > 0: This means for q 0
bq 0 @X !2Nl k(MjG)!k q 1 A n X !2Nnl+k k(MjG)!k q: (43)
On the other hand, since M depends on 1-coordinate, for all k 2 N X !2NG k k(MjG)!k q = X !2NG k (M jG)!1(M jG)!2 (M jG)!k q = X !2NG k sup x2[!]k kM (x)kq= X !2NG k sup x2[!\G]k kM (x)kq = Zk(G; M; q) : (44)
Combining (43) and (44) we have
bqZl(G; M; q)n Znl+k(G; M; q) : (45)
Since PI(q) < 1, there exists a B < 1 such that for all m 2 N
1
mZm(G; M; q) < B: (46) Therefore we can choose n and l 2 N large enough such that (n 1)rBnl
" 2
and q log bnl "2( Note r = r(G) as in (H2), thus l 2 N can be chosen and depends on G I) and combining (45) with (46) that we have for all q 0
1 l log Zl(G; M; q) q log b nl + 1 nllog Znl+k(G; M; q) q log b nl + nl + k nl 1 nl + klog Znl+k(G; M; q) q log b nl + 1 nl + klog Znl+k(G; M; q) + (n 1)rB nl PI(q) + ":
Since " > 0 is arbitrary we conclude that
sup fPG(l; q)j G I and l 2 Ng PI(q):
Next we prove
sup fPG(l; q)j G I and l 2 Ng PI(q); (47)
Since M only depends on 1-coordinate we choose G I and N1 2 N be
such that if n N1 1 nZn(G; M; q) = 1 nlog P !2 G n sup 2[!\G]k nM ( )kq = 1 nlog P !2 G n sup 2[!]k nM ( )kq 1 nlog Zn(M; q) " 2: (48)
Then we choose !(1); ; !(l)2 NmG; Lemma 16 is applied to show that there exists K1; ; Kl 12 N such that
!(1)K1!(2) Kl 1!(l)2 Nml+k; where k 2 [l 1; 2(l 1)r] and (MG)!(1)K 1!(2) Kl 1!(l) q (l 1)q max 1 j 2rK2maxG j k(MG)Kkq ! l Y k=1 k(MG)!(k)kq:
It follows from (44) we have
Zml+k(G; M; q) (l 1)qCZm(G; M; q)l;
for some C = max1 j 2rmaxK2 G
j k(MG)Kk q> 0: Therefore log C ml + q log(l 1) ml + 1 mlog Zm(G; M; q) ml + k ml 1 ml + klog Zml+k(G; M; q) k ml log Zml+k(G; M; q) + 1 ml + klog Zml+k(G; M; q) 1 ml + k log Zml+k(G; M; q) :
Thus we conclude that for all " > 0 there exists N2 2 N such that if m N2
1 mlog Zm(G; M; q) 1 ml + k log Zml+k(G; q) " 2: (49) Hence, in view of (48) and (49) if we take l 2 N large enough then
1
mlog Zm(G; M; q) PI(q) ": This completes the proof.
For G I is …nite, the following Theorem is taken from Proposition 2.6. [4] for the analysis on the entropy spectrum of EM(l)
G( ).
Theorem 19 (Proposition 2.6. of [4]) Given G I with G is mixing, if = P0
G(l; q) for some 0 < q 2 R; then E (l)
MG( ) 6= ; and
We extend Theorem 19 to countable systems.
Theorem 20 Under the same assumptions of Theorem 18. Then htop(EM( )) = inf
q2Rf q + PI(q)g :
Proof. The proof for htop(EM( )) t + PI(t) is identical to the proof
in Theorem 14. Thus we only prove htop(EM( )) inf
q2Rf q + PI(q)g :
The proof is alone the line of Theorem 1.1. of [4]. 8G I let fG;l( ; nl; ")
and fG( ; n; ") be de…ned as in (40) and (41). By Lemma 17 with G I is
…xed, 2 LMG and " > 0 for all p 2 N we have
fG( ; p(l + 2r); 2") DpfG;l( ; pl; ") DpfG( ; l; ")p: Thus lim p!1 log fG;l( ; pl; ") pl (l + 2r) l p!1lim log fG( ; p(l + 2r); 2") p(l + 2r) 1 l log D: And this means that for " > 0 there exists N = N (G) 2 N such that if l N
htop(EM(l)G( )) " htop(EMG( )): (50)
From Theorem 18 and Lemma 19, for " > 0; choose (G; ljGj) 2 N N with ljGj N (G) be such that (50) is satis…ed, and also there exists qjGj2 R (Here we note that G I can also be chosen large enough such that qjGj for some > 1) and G I large enough with
PI(qjGj) PMG(l; qjGj) + ": (51) and htop E (ljGj) MG ( ) = PG(ljGj; qjGj) qjGj: (52) Since qjGj= PG(ljGj; qjGj) htop E (ljGj) MG ( ) PG(ljGj; qjGj) PI(qjGj) < 1:
Thus qjGj is bounded, and there exists q such that lim
It follows from Lemma 16 and Theorem 18, we also have
PG(ljGj; q ) PI(q ) PG(ljGj; q ) PG(ljGj; qjGj) + PG(ljGj; qjGj) PI qjGj
+ PI qjGj PI(q ) 3" (54)
Combining (50), (51) (53) and (54) we obtain
htop(EMG( )) htop(E
(ljGj)
MG ( )) " = PG(ljGj; qjGj) qjGj "
PI(q ) q 5" inf
q2RfPI(q) qg 5": (55)
Since EMG( ) EM( ) then (55) implies that
htop(EM( )) htop(EMG( )) inf
q2RfPI(q) qg 5":
Since " > 0 is arbitrary; the proof is completed.
5
Proof of Theorem 1
We need some technique lemmas for the proof of Theorem 1. The following lemma comes form Lemma 2.7 of [17].
Lemma 21 (Lemma 2.7. of [17]) Let be irreducible with A 2 MI I
then there exists a sequence (Gn)n 1 with Gn I and Gn is irreducible for
each n 2 N and S
n2N
Gn= I:
Let G I and G is mixing. For q 2 R and ! 2 G de…ne
n;q([!]) = sup 2[!\G]k
nM ( )kq=Zn(G; M; q) ;
where Zn(G; M; q) is de…ned in (10). The following lemma is taken from
Lemma 2.2., Lemma 2.4. and Lemma 2.5. of [6] and Theorem 1.1..
Lemma 22 Let G I with G is topologically mixing and n;q is de…ned
as above. Then there exists a sequence ( nk;q)k 1 convergent to q in the
weak topology such that the followings are satis…ed.
2. There exists C1 = C1(G) and C2= C2(G) > 0 such that for all n 2 N and ! 2 Gn: C1 q ([!]) n;q([!]) C2;
3. There exists C3 = C3(G) and C4= C4(G) > 0 such that
C3exp(nPG(q)) Zn(G; M; q) C4exp(nPG(q));
4. Let q be the limiting measure of m1 m 1P
i=0
q i
m 1
then q also
satis…es (1), (2) and (3) as above. Moreover, it is ergodic and -invariant.
Proof of Theorem 2. Let ( (k))k 1 Gk k 1 be the sequence of
ir-reducible SFTs extracted from by Lemma 21 and denote by (k)n the
collection of n-cylinders of (k) for all k 2 N. 8n 2 N and q > 1 let
(k) n;q k 1; (k) q k 1 and (k) q
k 1 be the measures as in Lemma 22 with
respect to (k) k 1 and we also denote by Q(k); Ci(k) for i = 1; ; 4 as
de…ned in Lemma 22, we note here Q(k); Ci(k) are bounded for i = 1; ; 4
by Proposition 4. For 2 , l 2 N, set l( ) = l; then for …xed k; l 2 N
and e 2 I: (k) q l 1(e) = X !2 (k)l ; !l=e (k) q ([!]) C2(k) X !2 (k)l ; !l=e (k) l;q ([!]) = C2(k) X !2 (k)l , !l=e sup 2[!\ (k)]k lM ( )kq Zl (k); Mk; q (where Mk = M j (k)) C2(k) Zl 1 (k); Mk; q Zl 1 (k); Mk; q sup 2[e]k 1M ( )kq C2(k) C31(k) C4(k) exp ( P (k)(q)) sup 2[e]k 1M ( )kq C2(k) C31(k) C4(k) exp ( P (k)(q)) X j e sup 2[j]k 1M ( )kq (56)
Since H+ A+ and P
i2Ik
1M (i)kq< 1 for q > 1; it follows from (56) that
8" > 0 there exists km 2 N such that
(k) q 0 @ TY m 1 [1; km] 1 A 1 "; Therefore q(k)
k 1 is tight and there exists q such that limk!1 (k)
q = q in
the weak topology. De…ne
r;q= 1 r r 1 X i=0 q i:
By a similar computation as above it can be easily checked that ( r;q)r 1 is
tight and consequently that there exists q such that lim
r!1 r;q = q in the
weak topology, thus q is -invariant. Next we claim q satis…es property (7), and it su¢ ces to show that q satis…es the same property (7). Indeed, Proposition 4 applies to shows that there exists ( k)k 0 # 0 such that 8n 2 N:
(1 k)n exp n (PI(q) PGk(q)) (1 + k)
n
(57) On the other hand, Lemma 22 applies to show that 8k 2 N there exists Q (k) > 0 such that 8n 2 N, ! 2 (k)n and 2 [!]
Q 1(k) (k) q ([!]) exp ( nP (k)(q)) k nMkkq Q (k) : (58) Since lim
r!1 r;q= q then there exists ("k) # 0 such that 8! 2
(1 + "k) 1 q
([!])
(k) q ([!])
(1 + "k) (59)
Combining (59), (57) and (58) yields 8n 2 N; ! 2 n and 8 2 [!]
q([!]) = exp ( nPI(q)) k nM ( )kq Q (k) (k)q([!]) q ([!]) exp n (PI(q) P (k)(q)) k nMk( )k k nM ( )k q Q (k) (1 + "k) (1 + k)n Qnj=1 j q Q1;
for some constant Q1 > 0 and opposite inequality is in the same fashion.
Hence q satis…es (7) and so does q: Next we alone the line in the proof of Theorem 1.1. [6] to show q is ergodic and we only sketch its proof. From the Gibbs property of q; we can …nd a constant C5> 0 such that for v 2 n
and ! 2 l with l n + 2p (p is de…ned in the …nitely primitive)
q [v]
T i[!] C
5 q([v]) q([!]) :
Therefore, q is ergodic and so is q: Finally, if is another -invariant, ergodic, probability measure satis…es (7), one has qfor all q 2 R, where stands for that is absolutely continuous to : Combining the ergodic and probability properties for both q and 2 M ( ). Proposition 1.2.7 of [19] should be applied to show that = q: This proves the uniqueness of Gibbs measure and complete the proof of Theorem 2.
Acknowledgement 23 The second and third author thank Prof. Feng for his valuable suggestions on this topic and hospitable host when Ban visited the CUHK in 2008.
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