Existence of SRB measures for expanding maps with weak regularity

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1 Existence of SRB measures for expanding maps with weak regularity WEIGU LI School of Mathematical Sciences Peking University Beijing People s Republic of China weigu@sxx0.math.pku.edu.cn MEIRONG ZHANG Department of Mathematical Sciences Tsinghua University Beijing People s Republic of China mzhang@math.tsinghua.edu.cn Abstract. We show that any C 1+Dini expanding map f on any compact manifold admits a unique absolutely continuous invariant probability measure (a.c.i.p.m.) µ 0. Moreover, the system (f, µ 0 ) is exact and therefore is ergodic. From a nonexistence example of Góra and Schmitt [Ergod. Th. Dynam. Syst. 9 (1989), ], one knows that such a Dini regularity on the Jacobian of f is the weakest condition ensuring the existence of a.c.i.p.m. Project supported by the National Natural Science Foundation of China Mathematics Subject Classification. Primary 37D35; Secondary 37D20, 37C30. This paper was published in Far East Journal of Dynamical Systems, 2 (2000),

2 1. Introduction and main results. It is well-known that a C 1+α, 0 < α 1, (uniformly) expanding map f on a compact manifold M admits a unique absolutely continuous invariant probability measure (a.c.i.p.m.) µ 0 with respect to the Lebesque measure m. Moreover, the system (f, µ 0 ) is exact, which means that the correlations c n (ϕ, ψ) = (ϕ f n )ψdµ 0 ϕdµ 0 ψdµ 0 tend to zero for sufficiently regular observables ϕ, ψ. In fact, c n (ϕ, ψ) tend to zero exponentially when ϕ, ψ C β (M), 0 < β 1, are Hölder continuous. As a result, the central limit law holds for ϕ f n when ϕ C β (M). The measure µ 0 is necessarily an SRB (Sinai-Ruelle- Bowen) measure by the Birkhoff ergodic theorem. For these conclusions and related topics on stochastic aspects of differentiable dynamics, see, e.g., the monograph of Viana [21] and the expository article of Baladi [3]. It is also known that the existence and nonexistence of a.c.i.p.m. for a map f do depend upon the following two aspects. The first is the hyperbolicity of the systems f such as uniform expansivity ([1, 11, 18, 19, 21, 22]), uniform hyperbolicity ([6, 21]), almost hyperbolicity ([14, 15]), nonuniform hyperbolicity ([2, 4, 5, 15]) etc. The second aspect is the regularity of the Jacobian det Df(x). Some well-known regularity conditions are the smoothness of class C 2 of f ([1, 11, 17, 18]), the Hölder continuity of det Df(x) ([21] and the references therein), the bounded variation and bounded generalized variation of det Df(x) ([10, 13, 16, 19, 21]) etc. This paper deals with the second aspect, i.e., under what weak regularity on det Df(x), a C 1 (uniform) expanding map f does admit an a.c.i.p.m.? Let M be a compact connected Riemannian manifold without boundary. Let f : M M be a C 1 expanding map, i.e., we can choose and henceforth fix such a Riemannian metric on M that (1.1) Df(x) v σ 1 v for all x M and all v T x M, where σ is some constant less than 1. The induced metric and the Lebesque measure on M are denoted by d and m respectively. As M is compact, M has a finite diameter d(m) < and a finite total volume m(m) <. Such an expanding map f admits an a.c.i.p.m. if f is C 1+α, 0 < α 1, i.e., the modulus of continuity of det Df(x) : (1.2) { } ω(δ) := sup det Df(x) det Df(y) : x, y M, d(x, y) δ is of order O(δ α ) when δ 0. In 1985 Collet and Eckmann proved in [8] the existence of a.c.i.p.m. for a one-dimensional piecewise expanding map f when Df(x) has a weaker regularity (than the Hölder continuity): (1.3) ω(δ) = O(1/(1 + log δ ) γ ), where γ > 1. 2

3 There is also a famous example of Góra and Schmitt [12] of one-dimensional piecewise expanding map f such that f admits no a.c.i.p.m. while the modulus of continuity of Df(x) has the type (1.4) ω(δ) = K/(1 + log δ ) as δ 0. Generally speaking, the smoothness of class C 1 for f cannot guarantee the existence of a.c.i.p.m. In fact, Quas [20] proved in 1999 that there is a residual G σ subset of C 1 circle expanding maps such that any map in this subset admits no a.c.i.p.m. Our answer to the regularity ensuring the existence of a.c.i.p.m. is: Theorem A. Let f be a C 1 expanding map on a compact manifold M. If the modulus of continuity of det Df(x) (see (1.2)) satisfies the Dini condition, then f admits a unique a.c.i.p.m. µ 0. Here the Dini condition refers to the convergence of the following singular integral: 1 0 ω(s) ds <. s Note that Collet-Eckmann s condition (1.3) satisfies the Dini condition, while Góra-Schmitt s example (1.4) does not. As a result, Theorem A is best possible in the aspect of regularity conditions. result. As for the decay of correlations for the system (f, µ 0 ), we will prove the following exactness Theorem B. Let f be as in Theorem A. Then the system (f, µ 0 ) is exact. More precisely, if ϕ L 1 (m) and ψ C 0 (M), then lim n c n (ϕ, ψ) = 0. Our proofs for Theorems A and B are based on the convergence of the Perron-Frobenius operators [21]. Denote for short D(x) = det Df(x), x M, where the determinant is with respect to the specific Riemannian metric. Now the Perron-Frobenius operator associated with f, acting on some convenient space of functions ϕ : M R, is defined by (Lϕ)(y) = f(x)=y ϕ(x) det Df(x) = f(x)=y ϕ(x) D(x). The Perron-Frobenius operator L is the duality of the operator Uψ := ψ f in the following sense: (1.5) (Lϕ)ψdm = ϕ(u ψ)dm. It is well-known that the (nonnegative) fixed point of L is the density of an absolutely continuous measure which is invariant under f, and vice versa. Such a technique is nowadays conventional (see [9, 13, 21]). However, as systems considered in this paper have relatively 3

4 weak regularity, it seems that it is impossible to construct suitable cones of functions on which the Perron-Frobenius operator is a strict contraction with respect to the corresponding projective metric (as in the case of C 1+α systems). To overcome this, we have exploited more aspects of topological dynamical behavior of expanding maps. The paper is organized as follows. In Section 2, we construct spaces and cones of functions for the Perron-Frobenius operators. Some necessary estimations are also established. Theorem A is proved in Section 3. In Section 4, Theorem B is proved and some discussion on decay of correlations is given. 2. Construction of function spaces. Our basis for construction of all spaces of functions is the space C 0 (M) of all continuous functions on M. Note that C 0 (M) with the maximum norm C 0 is a Banach space. The Perron-Frobenius operator L leaves the convex cone of all positive continuous functions in C 0 (M) invariant. To normalize such a cone, we introduce the following convex set in C 0 (M): C + = { } ϕ C 0 (M) : min ϕ > 0 and ϕdm = 1. M M Note that C + is not complete with respect to C 0. However, C + is complete with respect to the projective metric θ + on C +, see [9, 21]. Explicitly, for any ϕ, ψ C +, the projective metric θ + (ϕ, ψ) is given by { } ψ(x) α + (ϕ, ψ) = min ϕ(x) : x M, { } ψ(x) β + (ϕ, ψ) = max ϕ(x) : x M, θ + (ϕ, ψ) = log β +(ϕ, ψ) α + (ϕ, ψ) = log max { ψ(x)ϕ(y) ϕ(x)ψ(y) : x, y M } Lemma 2.1 (C +, θ + ) is a complete metric space. Proof. This is essentially Proposition 2.6 in [21]. For later use we sketch here the proof. Suppose that {ϕ n } C + is a θ + -Cauchy sequence. It can be proved that there exists a constant K 0 1 such that (2.1) K 1 0 min ϕ n max ϕ n K 0 for all n. This implies that (2.2) ϕ m ϕ n C 0 K 0 (exp(θ + (ϕ m, ϕ n )) 1) for all m, n. So {ϕ n } is a Cauchy sequence in C 0 (M) because θ + (ϕ m, ϕ n ) 0 as m, n. As a result, {ϕ n } has a limit ϕ 0 in C 0 (M). It is obvious that ϕ 0 dm = 1. By (2.1), 4

5 min ϕ 0 K 1 0 > 0. Thus ϕ 0 C +. Furthermore, {ϕ n } is actually convergent to ϕ 0 in the space (C +, θ + ) because and (2.3) ϕ n (x) ϕ 0 (x) 1 = ϕ n(x) ϕ 0 (x) K 0 ϕ n ϕ 0 ϕ 0 (x) C 0, x M, θ + (ϕ n, ϕ 0 ) = log max x,y M ϕ n (x) ϕ 0 (y) ϕ 0 (x) ϕ n (y) log 1 + K 0 ϕ n ϕ 0 C 0 1 K 0 ϕ n ϕ 0 C 0 0. By (1.5), L leaves C + invariant and acts on C + affinely. As L maps C + into itself, L does not expand the projective metric θ +, i.e., θ + (Lϕ, Lψ) θ + (ϕ, ψ), ϕ, ψ C +, cf. [21]. Thus L : (C +, θ + ) (C +, θ + ) is continuous. As for the Hölder continuous functions of exponents 0 < α 1, we introduce some Hölder-like continuous functions on M. Definition 2.1 Let h : R + = [0, ) R + be a nondecreasing continuous function such that h(0) = 0. We call a function ϕ : M R is Hölder continuous with respect to h, or simply, h-hölder continuous, if sup { ϕ(x) ϕ(y) /h(d(x, y)) : x, y M, 0 < d(x, y) δ 0 } <. Remark 2.1 Although the h-hölder continuity in above definition is locally defined, compactness and connectedness of M show that the h-hölder continuity is actually global on M. In fact, one can prove as in [21] that there exists a constant A = A(δ 0, M, h) > 0 such that if ϕ is h-hölder continuous as in Definition 2.1 then (2.4) sup { ϕ(x) ϕ(y) /h(d(x, y)) : x, y M, x y} A sup { ϕ(x) ϕ(y) /h(d(x, y)) : x, y M, 0 < d(x, y) δ 0 } <. Denote by C h (M) the collection of all h-hölder continuous functions on M. Then C h (M) is a linear subspace of C 0 (M). It is easy to check that if ϕ, ψ C h (M) then ϕψ C h (M), and if ϕ C h (M) satisfies ϕ(x) 0 for all x then 1/ϕ is also in C h (M). Thus C h (M) is actually an algebra. Definition 2.2 We call a continuous nondecreasing function h : R + R +, h(0) = 0, a Dini-function if the following singular integral is convergent: (2.5) 1 0 h(s) ds <. s 5

6 Such a terminology comes from Fourier analysis [23]. Example 2.1 Let 0 < α 1. The function l α (t) = t α satisfies (2.5) and is thus a Dinifunction. In this case, C lα (M) is just the usual space C α (M) of α-hölder continuous functions on M. Example 2.2 For any integer k 1 and any number p > 1, define a function 0, if t = 0, l k,p (t) = 1 (log [k] (1/t)) p k 1, i=1 log[i] (1/t) if 0 < t 1, where log [k] s = log log s, s 1. }{{} k The function l k,p is a Dini-function because in this case the singular integral (2.5) corresponds to the following convergent one: by (2.6) 0 ( t log [k] (1/t) dt ) p k 1 i=1 log[i] (1/t) = ds ( ) ( ) ( ) p <. s log [1] s log [k 1] s log [k] s Next let h be a Dini-function. For any given 0 < ν < 1, define a function h ν : R + R + h ν (t) = ν h(ν i t). i=1 Such a definition is simply based on the following equality: (2.7) h ν (νt) + νh(νt) h ν (t). The convergence of h ν (t) in (2.6) is only ensured by (2.5) because h ν (t) = ν h(ν i t) i=1 ν h(ν s t)ds 0 ν t h(u) = du <. log ν 0 u A lower bound for h ν (t) can also be given similarly. In fact one has (2.8) ν νt log ν 0 h(u) u du h ν(t) ν t log ν 0 h(u) u du. It is obvious that h ν (t) is increasing with respect to t or ν. Note that h ν is not necessarily a Dini-function. 6

7 Example 2.3 Let h = l α in Example 2.1. Then h ν (t) = (ν 1+α /(1 ν α ))t α. Thus C h ν (M) is independent of ν and is just the space C α (M). When h is the l k,p in Example 2.2, by (2.8) one has ν h ν (t) log ν = t 0 ν (p 1) log ν l k,p (u) du u 1 ( log [k] (1/t) ) p 1 for 0 < t 1. Similarly Thus h ν (t) is of order h ν (t) ν (p 1) log ν ν (p 1) log ν 1 ( log [k] (1/νt) 1 ( log [k] (1/t) ) p 1 ) p 1. as t 0+. The space C h ν (M) is also independent of ν in this case and is given by { C h ν (M) = C [k,p] (M) := ϕ C 0 (M) : ( ) p 1 } log [k] (1/d(x, y)) ϕ(x) ϕ(y) <, sup 0<d(x,y) δ 0 where δ 0 > 0 is a small constant. Let now f : M M be an C 1 expanding map. Assume that (1.1) holds for some constant σ < 1. Then f expands locally the metric d: There exists a constant δ 0 > 0 such that (2.9) d(f(x), f(y)) σ 1 d(x, y), x, y M with d(x, y) δ 0. Here the constant σ may be a little bit larger than that in (1.1). It follows from (1.1) that D(x) σ 1 for all x. Suppose further that the modulus of continuity of D(x) := det Df(x) is a Dini-function. So there is some Dini-function h such that (2.10) sup{ D(x) D(y) /h(d(x, y)) : x, y M, 0 < d(x, y) δ 0 } 1. We construct a family of convex sets {C hν } 0<ν<1 as follows. Let C hν = {ϕ C + : log ϕ(x) satisfies condition (2.11)}, where condition (2.11) means that log ϕ C h ν (M) and (2.11) sup{ log ϕ(x) log ϕ(y) /h ν (d(x, y)) : x, y M, 0 < d(x, y) δ 0 } 1. It is obvious that C hν is a convex subset for each 0 < ν < 1. We will prove that the family {C hν } of convex sets has the following properties: 7

8 ) (P1) C hν ( C hν = {0}. (P2) If 0 < ν < ν < 1 then C hν C h ν C +. (P3) C hν is compact in (C +, θ + ) for each 0 < ν < 1. (P4) L maps C hν to itself whenever σ ν < 1. (P5) The convex set C hν spans the space C hν (M) for each 0 < ν < 1. (P3). It is easy to prove properties (P1) and (P2). Now we prove the compactness property Lemma 2.2 Property (P3) is satisfied for all 0 < ν < 1. Proof. Let {ϕ n } be any sequence in C hν. From (2.4) in Remark 2.1, sup{ log ϕ n (x) log ϕ n (y) /h ν (d(x, y)) : x, y M, x y} A <, where A is a constant independent of n. As a result, Since ϕ n dm = 1, one has max ϕ n / min ϕ n exp(ah ν (diam M)) <. min ϕ n 1/m(M) max ϕ n, n 1. These show that there exists a constant K 1 > 0 such that K 1 1 min ϕ n max ϕ n K 1, n 1. So the sequence {ϕ n } is bounded in the space C 0 (M). By the definition of C hν, {ϕ n } is also equi-continuous on M. Now the Ascoli-Arzela theorem shows that {ϕ n } has a subsequence converging to some ϕ 0 in C 0 (M). The limit function ϕ 0 satisfies min ϕ 0 K 1 1 > 0 and is in C hν because C hν is a closed subset of C 0 (M). Furthermore, {ϕ n } is actually convergent to ϕ 0 in the space (C +, θ + ), cf. (2.3). This proves that C hν is a compact subset of (C +, θ + ). Let now α ν, β ν and θ ν = log β ν /α ν be the corresponding objects in the projective metric for C hν. Explicitly, for any ϕ 1, ϕ 2 C hν, { ϕ2 (x) α ν (ϕ 1, ϕ 2 ) = inf ϕ 1 (x), exp (h } ν(d(x, y))) ϕ 2 (x) ϕ 2 (y) exp (h ν (d(x, y))) ϕ 1 (x) ϕ 1 (y) : x, y M, 0 < d(x, y) δ 0, { ϕ2 (x) β ν (ϕ 1, ϕ 2 ) = sup ϕ 1 (x), exp (h } ν(d(x, y))) ϕ 2 (x) ϕ 2 (y) exp (h ν (d(x, y))) ϕ 1 (x) ϕ 1 (y) : x, y M, 0 < d(x, y) δ 0. Now we prove the invariance property (P4). Lemma 2.3 Property (P4) is satisfied for all σ ν < 1. 8

9 Proof. Let y 1, y 2 M be such that d(y 1, y 2 ) δ 0. Denote f 1 (y j ) = {x j1,, x jk }, j = 1, 2, where k = #f 1 (y) is finite and is independent of y M. By the expansivity (2.9) one has (2.12) because d(y 1, y 2 ) δ 0. d(x 1i, x 2i ) σd(y 1, y 2 ), 1 i k, Let ϕ C hν. Then Lϕ C 0 (M) and min Lϕ > 0 because f is onto M. By (1.5), Lϕ satisfies also Lϕdm = 1. Suppose now that ν [σ, 1). Then (Lϕ)(y 1 ) = = k i=1 k i=1 k i=1 k i=1 k i=1 ϕ(x 1i ) D(x 1i ) ϕ(x 2i ) exp [log ϕ(x 1i ) log ϕ(x 2i )] exp [log D(x 2i) log D(x 1i )] D(x 2i ) ϕ(x 2i ) D(x 2i ) exp [log ϕ(x 1i) log ϕ(x 2i )] exp [ν D(x 2i ) D(x 1i ) ] ϕ(x 2i ) D(x 2i ) exp [h ν(d(x 1i, x 2i )) + νh(d(x 1i, x 2i ))] ϕ(x 2i ) D(x 2i ) exp [h ν(νd(y 1, y 2 )) + νh(νd(y 1, y 2 ))] = (Lϕ)(y 2 ) exp [h ν (d(y 1, y 2 ))], where the inequalities 1/D(x) σ ν, (2.12) and d(x 1i, x 2i ) σd(y 1, y 2 ) νd(y 1, y 2 ), the monotonicity of h ν (t) and the equality (2.7) are used. satisfies This proves that ψ(y) := (Lϕ)(y) log ψ(y 1 ) log ψ(y 2 ) h ν (d(y 1, y 2 )) when 0 < d(y 1, y 2 ) δ 0, i.e., ψ = Lϕ C hν. Lemma 2.4 Property (P5) is satisfied for all 0 < ν < 1. Proof. Let ϕ C h ν (M). Denote B := sup{ ϕ(x) ϕ(y) /h ν (d(x, y)) : 0 < d(x, y) δ 0 } <. Let ϕ ± = ε( ϕ ± ϕ) + 1, where ε > 0 is sufficiently small. Let γ ± > 0 be constants such that γ ± ϕ ± dm = 1. Then γ ± ϕ ± C +. Moreover, if 0 < d(x, y) δ 0, then log(γ ± ϕ ± (x)) log(γ ± ϕ ± (y)) /h ν (d(x, y)) 9

10 = log(ϕ ± (x)) log(ϕ ± (y)) /h ν (d(x, y)) = 1 η ± ϕ ± (x) ϕ ± (y) /h ν (d(x, y)) 2ε η ± ϕ(x) ϕ(y) /h ν (d(x, y)) 2εB, where η ± = η ± (x, y) is between ϕ ± (x) and ϕ ± (y) and therefore η ± 1. If one takes ε 1/(2B), then ψ ± := γ ± ϕ ± C hν. Consequently, ϕ = 1 2ε (ψ +/γ + ψ /γ ) span (C hν ). This proves (P5). 3. Existence of a.c.i.p.m. After properties (P1) (P5) for convex sets {h ν } and the Perron-Frobenius operator L were established in Section 2, we can now give the existence of a.c.i.p.m. Theorem A is contained in the following result. Theorem 3.1 Let f : M M be a C 1 expanding map. Assume that D(x) = det Df(x) satisfies for some Dini-function h that (3.1) A 0 := sup{ D(x) D(y) /h(d(x, y)) : x, y M, 0 < d(x, y) δ 0 } <. Then (i) f admits a unique a.c.i.p.m. µ 0. Moreover, the density ϕ 0 = dµ 0 /dm is continuous, strictly positive and ϕ 0 has the regularity that ϕ 0 C h σ (M), i.e., (3.2) sup{ ϕ(x) ϕ(y) /h σ (d(x, y)) : x, y M, 0 < d(x, y) δ 0 } <. (ii) The system (f, µ 0 ) is ergodic. Proof. We give the proof in several steps. Existence. We prove the existence of a.c.i.p.m. µ 0 = ϕ 0 m. Without loss of generality, assume that the constant A 0 in (3.1) is 1. Otherwise one may replace the Dini-function h by A 0 h. So condition (2.10) is satisfied for h. Recall that L : (C +, θ + ) (C +, θ + ) is a continuous operator. From (P3), the set C hσ is a compact convex subset of (C +, θ + ). By (P4), L maps C hσ into itself. Now the Schauder fixed point theorem shows that L has at least one fixed point ϕ 0 in C hσ which gives an a.c.i.p.m. µ 0 = ϕ 0 m of f. In fact, let ϕ C hσ be any given function. Then the sequence (3.3) n 1 1 L j ϕ n j=0 10

11 has a subsequence converging in C + to some ϕ 0 C hσ. Now such a ϕ 0 is the desired fixed point of L. Note that the regularity (3.2) holds because ϕ 0 C hσ C h σ (M). Uniqueness. ϕ 0, ψ 0 C hσ We prove at the present stage that L has only one fixed point in C hσ. Let be fixed points of L. Then L n ϕ 0 = ϕ 0 and L n ψ 0 = ψ 0. As a result, α + (L n ϕ 0, L n ψ 0 ) = α + (ϕ 0, ψ 0 ), β + (L n ϕ 0, L n ψ 0 ) = β + (ϕ 0, ψ 0 ), n 0. On the other hand, let x 0 M be such that Then ϕ 0 (x) β + (ϕ 0, ψ 0 ) = max x M ψ 0 (x) = ϕ 0(x 0 ) ψ 0 (x 0 ). β + (ϕ 0, ψ 0 ) = β + (L n ϕ 0, L n ψ 0 ) = Ln ϕ 0 (x 0 ) L n ψ 0 (x 0 ) f = n (y)=x 0 ϕ 0 (y)/ det Df n (y) f n (y)=x 0 ψ 0 (y)/ det Df n (y) f n (y)=x 0 β + (ϕ 0, ψ 0 )ψ 0 (y)/ det Df n (y) f n (y)=x 0 ψ 0 (y)/ det Df n (y) = β + (ϕ 0, ψ 0 ). This implies that (3.4) ϕ 0 (y) = β + (ϕ 0, ψ 0 )ψ 0 (y) for all y in the inverse orbit: O (x 0 ) := {y M : there exists some integer n 0 such that f n (y) = x 0 }. As f is expanding, O (x 0 ) is dense in the whole manifold M. Now (3.4) implies that ϕ 0 (y) = β + (ϕ 0, ψ 0 )ψ 0 (y) for all y M because ϕ 0 and ψ 0 are continuous. Consequently β + (ϕ 0, ψ 0 ) = 1 because ϕ 0 dm = ψ 0 dm = 1. Therefore ϕ 0 = ψ 0. Ergodicity. We prove ergodicity of the system (f, µ 0 ). Let A M be any invariant set under f and χ A be its characteristic function. Then χ A f j = χ A for all j 0. Let ϕ be any given function in C hσ. As in the proof of the existence, for any sequence {n k } of integers with n k, the sequence (3.5) 1 n k n k 1 j=0 L j ϕ 11

12 has a subsequence converging in the space (C +, θ + ) to some fixed point of L in C hσ. As f has a unique fixed point ϕ 0 in C hσ, the sequence (3.5) has a subsequence converging in (C +, θ + ) to ϕ 0. As a result, the sequence (3.3) itself converges in (C +, θ + ) to ϕ 0, i.e., One then has (3.6) θ + ( 1 n n 1 j=0 lim 1 n n L j ϕ, ϕ 0 ) 0 as n. n 1 j=0 L j ϕ ϕ 0 C 0 = 0, see (2.2) in the proof of Lemma 2.1. This shows that (3.7) Note that ϕdm = 1. Then = 1 n ( 1 n n 1 ( 1 n n 1 j=0 n 1 j=0 L j ϕ ϕ 0 )χ A dm 0. L j ϕ ϕ 0 )χ A dm ( )( ) (L j ϕ)χ A dm χ A dµ 0 ϕdm j=0 = 1 n 1 ( ϕ(χ A f j )dm χ A dµ 0 )ϕdm n j=0 ( = ϕχ A dm χ A dµ 0 )ϕdm [ = χ A χ A dµ 0 ]ϕdm. Now (3.7) shows that [ χ A (3.8) χ A dµ 0 ]ϕdm = 0 for all ϕ C hσ. By the linearity, (3.8) holds for all ϕ span (C hσ ) = C h σ (M), cf. Lemma 2.4. Thus we get from (3.8) that χ A (x) = χ A dµ 0 = µ 0 (A) m-a.e. x. Consequently, µ 0 (A) is either 0 or 1, and µ 0 is an ergodic measure. Uniqueness revisited. As we have proved that µ 0 is equivalent to m and (f, µ 0 ) is ergodic, it can be proved as in [21] that f has a unique a.c.i.p.m. In fact, let µ be an invariant probability measure such that µ << m. Then µ << µ 0. Since µ 0 is ergodic, one has µ = µ 0. In Theorem 4.1 of the next section, we will prove that the system (f, µ 0 ) is actually exact. 12

13 4. Decay of correlations. In this section, we give the proof of Theorem B. Namely we will prove that correlations (4.1) c n (ϕ, ψ) = ( )( ) (ϕ f n )ψdµ 0 ϕdµ 0 ψdµ 0 always decay to zero as n for all continuous observables ϕ, ψ C 0 (M). As a result, the system (f, µ 0 ) is exact. It is well-known that if c n (ϕ, ψ) has further decay rates such as exponential rate, then the central limit law holds for {ϕ f n } as a sequence of random variables. It thus is an important problem to examine the decay rates for correlations. We refer to [3, 7, 14, 21] for this topic. As µ 0 = ϕ 0 m in Theorem B is absolutely continuous, the following equalities c n (ϕ, ψ) = (ϕ f n )ψdµ 0 ϕdµ 0 ψdµ 0 = (ϕ f n )(ψϕ 0 )dm ϕϕ 0 dm ψϕ 0 dm [ [ ] (4.2) = ϕ L n (ψϕ 0 ) ψϕ 0 dm]ϕ 0 dm show that the decay of correlations can be estimated by the convergence of the iterates of Perron-Frobenius operator L: (4.3) c n (ϕ, ψ) [ L n C (ψϕ 0 ) ψϕ 0 dm] ϕ 0 ϕ 0 L 1 (m). Therefore the most important matter is to examine the decay of the C 0 -norms of (4.4) [ D n (ψ) := L n ψ ψdm] ϕ 0. When ψ are in convex sets C hν, the decay of D n (ψ) can be controlled using the quantities (4.5) Θ n (ψ) := θ + (L n ψ, ϕ 0 ), cf. (2.2) in the proof of Lemma 2.1. Theorem B is contained in the following result. Theorem 4.1 Let f be as in Theorem A and D(x) = det Df(x) satisfies (3.1) for some Dini-function h. Then for any ϕ L 1 (m) and any ψ C 0 (M), one has (4.6) Proof. (4.7) lim c n(ϕ, ψ) = 0. n By (4.2) (4.4), we see that (4.6) can be achieved by proving that lim L n C ψ ϕ 0 = 0 0 n for all ψ C 0 (M). 13

14 The proof is based on the following important property for the operator L: If ϕ, ψ C 0 (M) with min ϕ > 0, min ψ > 0, then (4.8) (4.9) max x M min x M Lϕ (x) max Lψ x M Lϕ (x) min Lψ x M ϕ ψ (x), ϕ ψ (x). Let us first prove (4.7) for ψ C hσ. Let ψ n = L n ψ C hσ. Set ψ n (x) M n = max x M ϕ 0 (x) and ψ n (x) m n = min x M ϕ 0 (x). It follows from (4.8) and (4.9) that M n decreases and m n increases when n increases. Let M = lim n M n, m = lim n m n. Since C hσ is compact in (C +, θ + ), for any sequence n k, the sequence L n kψ has a subsequence L n k j ψ converging in (C +, θ + ) to some ψ 0 C hσ : For any m N, one has lim ψ n j kj = ψ 0. max Lm ψ 0 ϕ 0 = max lim j ψ m+n kj ϕ 0 (4.10) = lim j max ψ m+n kj ϕ 0 = M = max ψ 0 ϕ 0. = lim j M m+nkj In the following we proceed the proof as for that of the uniqueness in Theorem 3.1. Let x 0 M be such that For any m N, it follows from (4.10) that M = max ψ 0 ϕ 0 = ψ 0(x 0 ) ϕ 0 (x 0 ). M = ψ 0(x 0 ) ϕ 0 (x 0 ) = max Lm ψ 0 (x 0 ) ϕ 0 (x 0 ) f = m (y)=x 0 ψ 0 (y)/ det Df m (y) f m (y)=x 0 ϕ 0 (y)/ det Df m (y) f m (y)=x 0 M ϕ 0 (y)/ det Df m (y) f m (y)=x 0 ϕ 0 (y)/ det Df m (y) = M. This implies that ψ 0 (y) = M ϕ 0 (y), for all y O (x 0 ). 14

15 As a result, one has ψ 0 (y) = M ϕ 0 (y), y M because of the continuity of ψ 0, ϕ 0 and of the density of O (x 0 ) in M. As ψ 0 dm = ϕ0 dm = 1, we have M = 1 and ψ 0 = ϕ 0. We have proved that any sequence {L n kψ} has a subsequence converging in (C +, θ + ) to ϕ 0. Consequently, the sequence {L n ψ} itself converges in (C +, θ + ) to ϕ 0. It now follows from (2.2) that (4.7) holds when ψ C hσ. Now we prove (4.7) for general ψ C +. For any ε > 0, choose functions ψ ± C h σ (M) such that (4.11) ψ ψ ψ +, ψ + ψ C 0 < ε 2. Applying (4.7) to functions ψ ± / ψ ± dm C hσ, one has lim L n ψ ± n Thus there exists N 1 such that (4.12) [ ψ ± dm] ϕ 0 C 0 = 0. [ ψ dm 2] ε [ ϕ 0 < L n ψ L n ψ + < ψ + dm + 2] ε ϕ 0, n N. It now follows from (4.11) and (4.12) that [1 ε]ϕ 0 < [ ψ dm 2] ε [ ϕ 0 < L n ψ L n ψ L n ψ + ψ + dm + 2] ε ϕ 0 < [1 + ε]ϕ 0. This implies that L n ψ ϕ 0 C 0 < ε ϕ 0 C 0, n N. Namely, (4.7) holds for all ψ C +. Finally, since (4.7) is linear in ψ, one then knows that (4.7) holds for all ψ span(c + ) = C 0 (M). This proves the theorem. When an expanding map f is C 1+α 0 for some 0 < α 0 1, the condition (3.1) is satisfied for h(t) = l α (t) = at α, where a > 0 and 0 < α α 0. In this case, it can be proved that the operator L maps (C hν, θ ν ), σ ν < 1, into C hλν, where 0 < λ < 1 is some constant independent of ν. As C hλν has finite diameter in (C hν, θ ν ), L maps (C hν, θ ν ) (σ ν < 1) into itself in a contraction way, cf. [21]. One thus has (4.13) θ + (L n ψ, ϕ 0 ) θ ν (L n ψ, ϕ 0 ) Λ n θ ν (ψ, ϕ 0 ) for all n 0 and all ψ C hν, where Λ (0, 1) is some constant. Now (4.3), (4.4), (4.5) and (4.13) show that the correlations c n (ϕ, ψ) in (4.1) have an exponential decay in the spaces C α (M), 0 < α α 0. We recover in this way the well-known result for C 1+α 0 expanding maps. 15

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