FACTORS OF GIBBS MEASURES FOR FULL SHIFTS

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1 FACTORS OF GIBBS MEASURES FOR FULL SHIFTS M. POLLICOTT AND T. KEMPTON Abstract. We study the images of Gibbs measures under one block factor maps on full shifts, and the changes in the variations of the corresponding potential functions. 1. Introduction In this note we consider the images of invariant measures under simple factor maps. Let σ 1 : Σ 1 Σ 1 and σ 2 : Σ 2 Σ 2 denote two full shifts, on finite alphabets A and B, respectively. Assume that Π : Σ 1 Σ 2 is a one block factor map (i.e., a semi-conjugacy satisfying Πσ 1 = σ 2 Π where (Π(x)) 0 = π(x 0 ) for a surjective map π : A B). Furthermore, let us consider a σ 1 -invariant probability measure µ ψ1 which is a Gibbs measure for a potential function ψ 1 : Σ 1 R. In particular, the image ν = π µ ψ1 will be a σ 2 -invariant probability measure. A very natural question would be to ask under which hypotheses on ψ 1 would ν be a Gibbs measure for a potential ψ 2 : Σ 2 R (i.e., ν = µ ψ2 ), and how the regularity of ψ 1 is reflected in that of ψ 2. In the particular case that ψ 1 is locally constant, and thus µ ψ1 is a generalized Markov measure, it was shown by Chazottes and Ugalde in (2) that ν is a Gibbs measure for a Hölder continuous function ψ 2 : Σ 2 R (i.e., ν = µ ψ2 ), although not necessarily still a generalized Markov measure. One of our main results is the following. Theorem 1.1. Assume that n=0 nvar n(ψ 1 ) < + then n=0 var n(ψ 2 ) < +. The method we describe can be extended to the case of suitable subshifts of finite type and factor maps. These results will appear in a separate paper by the first author. In section 2 we recall some basic properties of Gibbs measures. In section 3, we discuss a construction of the potential function ψ 2. In section 4 we present the proof of a key step in this construction (Proposition 3.2). Finally, in section 5 we present and prove the main results. After completing this work, Ugalde informed the authors of his contemporaneous work with Chazottes. In (3) they have proved related results 1

2 2 M. POLLICOTT AND T. KEMPTON using a somewhat different method. In particular, in (3) it is shown that if n=0 n2+ɛ var n (ψ 1 ) < + then n=0 var n(ψ 2 ) < +. However, the methods there appear to give sharper bounds on the actual rates of decay of the terms var n (ψ 2 ) as n + than we can obtain. 2. Gibbs measures and Equilibrium states We begin with some general definitions and results. Given k 2, let Σ = {1,, k} Z+ denote a full shift space, with the shift map σ : Σ Σ defined by (σw) n = w n+1. For each n 0, we define the nth level variation var n (ψ) := sup{ ψ(x) ψ(y) : x i = y i, 0 i n 1}. of ψ : Σ R. Continuity of the function ψ corresponds to var n (ψ) 0 as n +. We say that ψ has summable variation if n=0 var n(ψ) < +. Stronger still, we say that ψ is Hölder continuous if there exists 0 < θ < 1 such that { } varn (ψ) ψ θ := sup < +. n 1 θ n Given any continuous function ψ : Σ R on a subshift of finite type Σ, we can define the pressure by { } P (ψ) = sup µ M h(µ) + ψdµ where the supremum is over the space M of all σ-invariant probability measures. This is equivalent to other standard definitions using the variational principle (5). A measure which realizes this supremum is called an equilibrium state. Any continuous function has at least one equilibrium state and every invariant probability measure is the equilibrium state for some continuous potential (cf. (4)). If ψ has summable variation, then there is a unique equilibrium state µ ψ. Given x 0,, x n 1 {1,, k} we can denote [x 0,, x n 1 ] = {w Σ : w i = x i for 0 i n 1}. We have the following alternative characterization (cf. (1)). Lemma 2.1. If ψ has summable variation then the following are equivalent: (1) A σ-invariant probability measure µ is the unique equilibrium state for ψ; and

3 FACTORS OF GIBBS MEASURES 3 (2) There exists C 1, C 2 > 0 such that C 1 for all w Σ. µ[w 0,, w n 1 ] exp(ψ n (w)) np (ψ)) C 2 (1) We shall refer to the unique invariant measure satisfying the Bowen- Gibbs inequality (1) as a Gibbs measure (in the sense of Bowen (1)). Since we are primarily interested in measures, rather than functions, we can replace ψ by ψ P (ψ) (and still have µ ψ1 = µ ψ1 P (ψ 1 )) and so in the sequel we shall assume, without loss of generality, that P (ψ) = 0. Remark 2.2. For the two sided shift shift σ : Σ Σ on Σ = {1,, k} Z and a function ψ : Σ R we can define var n (ψ) := sup{ ψ(x) ψ(y) : x i = y i, i n 1}. If n=0 n2 var n (ψ) < + we can add a coboundary to obtain a function ψ (x) which depends only on (x n ) n=0 and for which n=0 nvar n(ψ ) < +. Then Theorem 1.1, for example, applies. 3. Constructing the potential ψ 2 If µ ψ1 is a Gibbs measure for a continuous function ψ 1 (satisfying P (ψ 1 ) = 0) then we can write ν[z 0,, z n ] = n+1 µ ψ1 [x 0,, x n ] 1 (x) x 0,,x n x 0,,x n e ψ with each summation being over finite strings x 0,, x n from Σ 1 for which π(x i ) = z i, for i = 0,..., n, and any x [x 0,, x n 1 ]. 1 This motivates the construction of the potential function via the limit of a sequence of functions in C(Σ 2, R). We begin by fixing, for the moment, a sequence w Σ 2. Definition 3.1. We would like to consider a sequence of functions (u n (z)) n=1 in C(Σ 2, R) defined by: x=x u n (z) = 0 x n exp(ψ1 n+1 (xw)) x =x 1 x n exp(ψ1 n (x w)) where z Σ 2 for n 2, and u 1 (z) = x 0 exp(ψ 1 (x 0 w)), where: (1) ψ n 1 = n 1 i=0 ψ 1 σ i and ψ n+1 1 = n i=0 ψ 1 σ i ; (2) each summation is over finite strings from Σ 1 for which x i projects to z i, for i = 0,..., n; and 1 Here means that the ratio of the two sides are bound above and below (away from zero) independently of n.

4 4 M. POLLICOTT AND T. KEMPTON (3) xw Σ 1 denotes the concaternation of words to give the sequence (x 0,, x n, w 0, w 1 ). It is clear that u n (z) depends only on z 0,, z n, i.e., u n : Σ 2 R is a locally constant function. The sequence (u n (z)) n=1 has an explicit dependence on the sequence w. When appropriate, we will change the notation to reflect the w dependence by writing u w,n (z). We need the following result. Proposition 3.2. The sequence {u w,n (z)} converges uniformly to a continuous function u(z) > 0 which is independent of w. We will return to the proof of this proposition in the next section. Before stating a corollary we present a lemma we need in its proof. Lemma 3.3. If µ ψ1 is a Gibbs measure with continuous potential ψ 1 then there exists C > 0 such that for any x = x 0,, x n, w, w Σ 1, n N, exp(ψn+1 1 (xw)) exp(ψ1 n+1 (xw )) C. Proof. By definition, there exists C 1, C 2 > 0 such that C 1 exp(ψ n+1 1 (xw)) µ ψ1 [x 0,, x n ] C 2 exp(ψ n+1 1 (xw )) (2) for all w Σ, and thus exp(ψ1 n+1 (xw)) exp(ψ1 n+1 (xw )) C 2 C 1 The result follows with C = C 2 /C 1. Corollary 3.4. The measure ν = Π µ ψ1 is Gibbs measure for ψ 2 (z) := log u(z) (i.e., ν = µ ψ2 ). Proof. Fix n 1. We can write ψ2 n (z) = lim log u w,m(z) + + lim log u w,m(σ n z) m + m + ( ) = lim log x=x 0 x m exp(ψ1 m+1 (xw)) m + x=x n+1 x m exp(ψ1 m n. (xw)) Moreover, for m > n we can factor the summation exp(ψ1 m+1 (xw)) x=x 0 x m = exp(ψ1 n+1 (x xw)) exp(ψ1 m n (xw)) x=x n+1 x m x =x 0 x n

5 and then bound 1 C FACTORS OF GIBBS MEASURES 5 x =x 0 x n exp(ψ n+1 1 (x w)) x=x 0 x m exp(ψ1 m+1 (xw)) x=x n+1 x m exp(ψ1 m n (xw)) }{{} C =exp( P n i=0 log uw,m(σi z)) exp(ψ1 n+1 (x w)) x =x 0 x n where C is as in the previous lemma. Since µ ψ1 is a Gibbs measure for ψ 1, we can apply (2) and then summing over strings with π(x i ) = z i, for i = 0,, n 1, and letting m + gives C 1 C ν[z 0 z n 1 ] exp(ψ2 n (z)) C 2C, It then follows from the definitions that ν is a Gibbs measure for ψ Proof of Proposition 3.2 We return to the proof postponed from the previous section. Definition 4.1. Let s > n, x = x n+1 x s be a given finite string and w Σ. We define P (s,n) x=x (x, w) = 1 x n exp(ψ1 n (xxw)) x =x 1 x s exp(ψ1 n (x w)) (where π(x i ) = z i for i = 1,, s). We require the following lemma. Lemma 4.2. For s > n we have the identity u w,s (z) = u xw,n (z)p (s,n) (x, w) where z Σ 2 (and π(x i ) = z i for i = n + 1,, s). Proof. By definition, the numerator of u w,s (z) is exp(ψ1 s+1 (xw)) x=x 0 x s = exp(ψ1 n+1 (xxw)) exp(ψ1 s n (xw)). x=x 0 x n

6 6 M. POLLICOTT AND T. KEMPTON (where π(x i ) = z i for i = 0,..., s) and we have used ψ1 s+1 (xxw) = ψ1 n+1 (xxw) + ψ1 s n (xw). We can further rewrite this as ( ) x=x 0 x n exp(ψ1 n+1 (xxw)) exp(ψ x =x 1 x n exp(ψ1 n (x 1 n (x xw)) exp(ψ1 s n (xw)) xw)) x }{{} =x 1 x n }{{} u xw,n (z) Px =x exp(ψs+1 1 xn 1 (x xw)) We can now divide by the denominator of u s (z) to get the result. Corollary 4.3. We can write u w,s (z) u w,s(z) = The following special case is illustrative. u xw,n (z)p (s,n) (x, w) u xw,n(z)p (s,n) (x, w ). Example 4.4 (Markov case). Assume ψ 1 depends on only the first two coordinates. Let s = n + 2 and then observe from the definitions that u xnxn+1 w,n(z) is independent of w. In particular, u w1 w 2,n+2(z) u w 1 w 2,n+2 (z) = x n,x n+1 u xnx n+1,n(z)p (n+2,n) (x n x n+1, w 1 w 2 ) x n,x n+1 u xnx n+1,n(z)p (n+2,n) (x n x n+1, w 1w 2) In this case Lemma 4.2 corresponds to a linear action by a strictly positive matrix, which contracts the simplex and so converges at an exponential rate to a fixed line. It remains to use the corollary to complete the proof of Proposition 3.2. We require the following. Lemma 4.5. There exists c > 0 such that P (s,n) (x,w) P (s,n) (x,w ) x n x s and w, w Σ. Proof. Since xxw and xxw agree in s places we see that ψ1 n (xxw) ψ1 n (xxw C2 ) log and the result follows with c = ( C 1 C 2 ) 2. For each n 0 and z Σ 2 we denote { } uw,n (z) λ n = λ n (z) := sup u w,n(z) : w, w Σ 2 1. C 1 c for all x = Lemma 4.6. The sequence λ 0 λ 1 λ 2 λ n 1 is a monotone decreasing sequence. Furthermore, the intervals [inf w u w,n (z), sup w u w,n (z)] form a nested sequence in n.

7 Proof. Fix n 1. By definition u w,n+1 (z) = FACTORS OF GIBBS MEASURES 7 x=x 0 x n x n+1 exp(ψ1 n+1 (xx n+1 w) exp(ψ 1 (x n+1 w)) x =x 1 x n x n+1 exp(ψ1 n (x x n+1 w) exp(ψ 1 (x n+1 w)) max x n+1 x=x 0 x n exp(ψ n+1 1 (xx n+1 w) x =x 1 x n exp(ψ n 1 (x x n+1 w) }{{} u xn+1 w,n(z) The inequality here comes from the fact that x n+1 a(x n+1 ) x n+1 b(x n+1 ) max x n+1 a(x n+1 ) b(x n+1 ). Similarly, min x n+1 u x n+1 w,n(z) u w,n+1 (z). Thus if w, w Σ 2 then u w,n+1 (z) u w,n+1(z) u x nw,n(z) u x n w,n(z) λ n. Taking the supremum over w, w Σ 2 gives that λ n+1 λ n. The following inequality is central to our analysis. Lemma 4.7. Let s > n. Then λ s c exp ( s k=s n var k (ψ 1 ) Proof. If λ n exp ( s k=s n var k(ψ 1 ) ) then λ s = c.λ s + (1 c)λ s c exp ( s k=s n ) + (1 c)λ n (3) var k (ψ 1 ) ) + (1 c)λ n as required. Therefore, we can assume that λ n exp ( s k=s n var k(ψ 1 ) ). For ease of notation, we denote b s,n := exp ( s k=s n var k(ψ 1 ) ). We want to combine Corollary 4.3 and Lemma 4.6. For any w, w Σ we

8 8 M. POLLICOTT AND T. KEMPTON can bound u w,s (z) u w,s(z) = = cu xw,n (z)p (s,n) (x, w) + (1 c)u xw,n (z)p (s,n) (x, w) cu xw,n(z)p (s,n) (x, w) + (P (s,n) (x, w ) cp (s,n) (x, w))u xw,n(z) c ( b s,n min x {u x w,n(z)} ) P (s,n) (x, w) + (1 c) max x {u xw,n (z)}p (s,n) (x, w) c min x {u x w,n(z)}p (s,n) (x, w) + (P (s,n) (x, w ) cp (s,n) (x, w)) min x {u x w,n(z)} cb s,n P (s,n) (x, w) + (1 c) max x{u xw,n (z)} min x {u x w,n (z)} P (s,n) (x, w) cp (s,n) (x, w) + (P (s,n) (x, w ) cp (s,n) (x, w)) cb s,n + (1 c) max x{u xw,n (z)} min x {u x w,n(z)} cb s,n + (1 c)λ n as required. For the second line here we decreased the denominator by replacing u xw,n(z) with min x {u x w,n(z)} and increased the numerator by replacing u xw,n (z) with max x {u xw,n (z)} and b s,n min x {u x w,n(z)}. Choosing s = 2n gives the following. Corollary 4.8. For any n 1 we can bound ) λ 2n c exp ( 2n k=n var k (ψ 1 ) } {{ } b 2n,n +(1 c)λ n In particular, since b 2n,n 1 as n + it is clear that lim n + λ n = 1. Moreover, the regularity of u, and thus ψ 2, is determined by the speed of convergence. To complete the proof of Proposition 3.2 we need the following simple lemma. Lemma 4.9. We have that (1) for any w Σ 2 the limit u(z) = lim n + u w,n (z) exists; and (2) ψ 2 := log(u(z)) is continuous in z Σ 2. Proof. For the first part, it suffices to use the previous observation that lim n + λ n = 1.

9 FACTORS OF GIBBS MEASURES 9 For the second part we observe that if z 1, z 2 Σ 2 agree in n terms then for any w, we have u n (z 1 ) = u n (z 2 ). Thus { } u(z 1 ) u(z 2 ) sup uw,n (z 1 ) u w n(z 2 ) : w, w Σ 2 = λ n and again the result follows from lim n + λ n = 1. We now have that ψ 2 = log u is a well defined continuous function which is a potential for ν, which completes the proof of Proposition 3.2. Remark There are more general hypotheses on the hypotheses that one might consider including, for example, the Walters condition. However, it is not immediately clear how the methods in this paper can be adapted to that case. However, observe that any potential ψ 1 satisfying the Bowen-Gibbs inequality must necessarily be uniformly continuous. Since the only condition on ψ 1 that we have used is that it is uniformly continuous, Proposition 3.2 shows that if µ ψ1 is a Gibbs measure. Then ν := Π(µ ψ1 ) is a Gibbs measure. 5. Regularity of the potential ψ 2 In this section we will consider the regularity properties of ψ 2 := log u. The following is our main result. Theorem 5.1. Let κ 0. If n=0 nκ+1 var n (ψ 1 ) < then n=0 nκ var n (ψ 2 ) <. Proof. Let 0 < c < 1 be as in Lemma 4.5. Choose 1 < β < 1/(1 c) and an integer M > 1 sufficiently large that α := β(1 c) M 1 1 κ M < 1. Let us denote a n = log λ n and recall the trivial inequality 1 + x exp(x) 1 + βx, for x > 0 sufficiently small. Thus providing N 0 is sufficiently large we can deduce from (3) that for any n > N 0 n 1 + a n exp(a n ) c. exp var m (ψ 1 ) + (1 c) exp(a n [n/m] ) m=[n/m] c + (1 c) + βc n m=[n/m] and hence that for any N > N 0, N N n n κ a n βc n κ var m (ψ 1 ) + β(1 c) n=n 0 n=n 0 m=[n/m] (where [ ] denotes the integer part ). var m (ψ 1 ) + β(1 c)a n [n/m] N n=n 0 n κ a n [n/m]

10 10 M. POLLICOTT AND T. KEMPTON We can bound N n=n 0 n κ and N n=n 0 n κ a n [n/m] n m=[n/m] var m (ψ 1 ) M κ M κ+1 N 1 N 1 1 κ (n [n/m]) κ a n [n/m] M n=n κ M N M 1 1 κ M [N N M ]+1 m=[n 0 N 0 M ] m κ a m + m=n 0 m κ a m + O(1) n n=n 0 m=[n/m] N N 0 n N M n+1 m κ var m (ψ 1 ) n=n 0 n κ+1 var n (ψ 1 ) (n [n/m]) κ a n [n/m] where we have used that N 0 n N M n+1 and (n [n/m]) κ a n [n/m] 1 M N 0 1 m=n 0 [N 0 /M] m k a m = O(1). N m=n 0 [N 0 /M] m k a m Comparing the above inequalities we can bound ( ) N N M 1 β(1 c) 1 1 κ n κ a n βcm κ+1 n κ+1 var n (ψ 1 )+O(1) M n=n }{{} 0 n=n 0 >0 Letting N + we see that n=n 0 n κ a n <, which completes the proof. When κ = 0 we have the following corollary, which was stated in the Introduction.

11 FACTORS OF GIBBS MEASURES 11 Corollary 5.2. If n=0 nvar n(ψ 1 ) < then n=0 var n(ψ 2 ) <. Another application of (3) is the following. Theorem 5.3. Assume there exists c 1 > 0 and 0 < θ 1 < 1 such that var n (ψ 1 ) c 1 θ n 1, for all n 0 then there exists c 2 > 0 and 0 < θ 2 < 1 such that var n (ψ 2 ) c 2 θ n 2, for all n 0 Proof. By inequality (3) we can write n λ n c exp c 1 k=n [ n] c exp Cθ [ n] θ k 1 + (1 c)λ n [ n] + (1 c)λ n [ n] for any θ 1 < θ < 1 and some C > 0. Using this inequality inductively [ n] times, we can write ( ) λ n c exp Cθ [ n] + (1 c) c exp Cθ [ n] + (1 c)λ n 2[ n] ( c exp Cθ [ n] ) [ n] (1 c) k + (1 c) [ n] λ n [ n] 2 k=0 exp Cθ [ n] + (1 c) [ n] λ 0. ( In particular, we see that λ n 1 = O c)} from which the result follows. θ n 2 ), where θ 2 = max{θ, (1 The following is an easy consequence of the theorem and its proof. Corollary 5.4. Assume there exists c 1 > 0 and 0 < θ < 1 such that var n (ψ 1 ) c 1 θ n for all n 0 (i.e., ψ 1 is Hölder continuous) then there exists c 2 > 0 such that var n (ψ 2 ) c 2 θ n for all n 0. The same conclusion, using a different proof, appears in (3). References [1] R. Bowen. Ergodic Theory of Axiom A diffeomorphisms. Lecture Notes of Mathematics, Springer-Verlag, Berlin, [2] J.-R. Chazottes and E. Ugalde, Projection of Markov Measures May Be Gibbsian, J. Statist. Phys. 111 (2003), no. 5-6, [3] J.R. Chazottes and E. Ugalde: Preservation Of Gibbsianness Under Amalgamation Of Symbols, preprint.

12 12 M. POLLICOTT AND T. KEMPTON [4] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, New York, [5] P.Walters: Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, Berlin, Mark Pollicott, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. address: Thomas Kempton, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. address: