Monatsh Math (2010) 159:81–113 DOI 10.1007/s00605-009-0149-4
The multifractal analysis of Birkhoff averages for conformal repellers under random perturbations Lin Shu
Received: 21 May 2008 / Accepted: 4 August 2009 / Published online: 1 September 2009 © Springer-Verlag 2009
Abstract We study the stability of multifractal structures for dynamical systems under small perturbations. For a repeller associated with an expanding C 1+β conformal topological mixing map, we show that the multifractal structure of Birkhoff averages is stable under small random perturbations. Keywords Birkhoff average · Bundle random dynamical system · Conformal repeller · Level sets · Multifractal analysis · Random perturbation · Spectrum Mathematics Subject Classification (2000)
28A78 · 28D20 · 60G57
1 Introduction The present paper is devoted to the study of the stability of the multifractal structure of Birkhoff averages for dynamical systems under small perturbations. Let M be a compact metric space and let f be a continuous transformation on it. The pair (M, f ) is referred to as a dynamical system. Given a reference continuous funcn−1 φ( f i (x)) for x ∈ M, the so-called tion φ : M → Rd , the limit behavior of (1/n)i=0 Birkhoff averages, reflect the dynamics of the orbits {x, f (x), . . . , f n (x), . . .}x∈M . Classify points accordingly as n−1 1 φ( f i (x)) = α , α ∈ Rd . E(α) := x ∈ M : lim n→∞ n i=0
Communicated by K. Schmidt. L. Shu (B) LMAM, School of Mathematical Sciences, Peking University, 100871 Beijing, People’s Republic of China e-mail:
[email protected]
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The multifractal analysis of the Birkhoff average studies the dimension or entropy of the level sets {E(α)} (i.e., the spectrum) and hence provides finer information of different orbits. Historically, this problem was first considered by Besicovitch [5] and Eggleston [14] (see also [6]) for real numbers from [0, 1] with specified appearance frequencies of 0 or 1 in their binary representations; and was intensively studied in a recent decade for various systems (see, e.g., [3,18,31,34,37,39] and also [2,4,11,17,20,24,32]). It is well-known from Birkhoff ergodic theorem (cf. [38]) that for any ergodic measure, except for one α, the set E(α) is of measure zero. However, it has been verified for a lot of important dynamical systems (see, e.g., [3,17,20,32,37]) that the dimensions (or entropies) of E(α) s form a function which relates entropies and Lyapunov exponents. For instance, let 0 be a conformal repeller for a C 1+β topologically mixing map f (on a neighborhood U of 0 ) of a smooth Riemannian manifold M. It was showed in [20] that dim H E(α) = max
hµ( f ) : log Dx f dµ(x)
φ dµ = α ,
(1.1)
where h µ ( f ) is the measure theoretical entropy of µ, Dx f is the operator norm of the differential Dx f , and dim H is the Hausdorff dimension. For the completeness of the multifractal analysis of Birkhoff averages, one can also study the set of divergence points
n−1 1 D(φ) := x ∈ M : the limit φ( f i (x)) does not exist . n i=0
It is true for a lot of systems (see, e.g., [4,18,20,32]) that D(φ) is either empty or has the same dimension (or entropy) as that of the system. Given a dynamical system, on which the spectrum of Birkhoff averages forms a function in relation with entropies and Lyapunov exponents as in (1.1), it is interesting to know whether the subtle multifractal nature of the averages is preserved and close to the original one under small random perturbations of the system. We consider this problem for conformal repellers in the set up of bundle random dynamical systems (RDS). Let 0 be a conformal repeller for a C 1+β topologically mixing map f in an open neighborhood U of 0 with compact closure on a smooth Riemannian manifold M. Let U( f ) denote a small open neighborhood of f in the space of C 1 diffeomorphisms from U to the images in M. Put = +∞ −∞ U( f ) and let P be a probability measure on which is ergodic with respect to the left shift map ϑ of . If U( f ) is small, it was showed in [28] that for P -a.e. w ∈ , there corresponds a compact set w ⊂ U such that gw w = ϑw , where gw is the map at the position 0 of w. The set w can be regarded as a continuation of the set 0 for the perturbation generated by picking up each time, at random,
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a map from U( f ) for iteration. Let = ∪w∈ {w} × w and let G : → , (w, x) → (ϑw, gw (x)). The pair (, G) constitutes a bundle RDS. Let φ : U → Rd be continuous. Identify φ as a function on by letting φ(w, x) = φ(x). For w ∈ and α ∈ Rd , define 1 E w (α) = x ∈ w : lim φn (w, x) = α , n→∞ n n−1 φ(G i (w, x)). Denote by where φn (w, x) := i=0
1 Dw (φ) = x ∈ w : the limit φn (w, x) does not exist . n ( ) A family (g(·) : → U( f )) >0 is called a random perturbation of f if
( ) = f in probability with respect to the C 1 distance (cf. [9]). If U( f ) lim →0 gw ( ) ( ) generates a bundle RDS (( ) , G ( ) ) [28]. Hence E w (α), is small, each gw ( )
w∈
Dw (φ), the corresponding sets of E w (α), Dw (φ) for (( ) , G ( ) ) and φ, are well
o ( ) defined. Moreover, for α ∈ α : E(α) = Ø , the set E w (α) is non-empty P -a.e. o for small (see Lemma 3.2). (Here A denotes the interior of the set A in Rd .) Denote by C r (U, M)(r ≥ 1) the set of C r transformations from U to the images endowed with a C r norm · r . Define L 1 (, C r (U, M)) to be the collection of measurable maps g(·) : → C r (U, M) such that gw r dP(w) < ∞ and similarly for L ∞ (, C r (U, M)). Our main result, tersely speaking, is that the multifractal structure of Birkhoff average is stable under random perturbations. Theorem 1.1 Let (0 , f ) be a mixing C 1+β conformal repeller. Let φ : U → Rd be continuous. There exists a C1 neighborhood U( f ) ⊆ C 1 (U, M) such that for any ( ) bounded random perturbation g(·) of f in L ∞ (, C 1+β (U, M)) satisfying >0
( ) = f in L 1 (, C 1+β (U, M)), lim gw
→0
o ( ) lim →0 dim H E w (α) = dim H E(α), P a.s., on α : E(α) = Ø . The convergence is uniform on any of its compact subsets.
( ) (ii) lim →0 dim H (Dw (φ)) = dim H (0 ), P a.s., if α : E(α) = Ø is not a single point.
Remark that only for α belonging to the interior of the set α : E(α) = Ø , can ( ) we make sure that the set E w (α)
is non-empty for P -a.e. w when is small enough. The boundary points of the set α : E(α) = Ø are hence not in consideration in (i). (i)
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But estimations of corresponding dim H E w (α) for each can be obtained (see Theorem 1.2). Although U( f ) is in C 1 (U, M), the assumption of the perturbations in L ∞ (, C 1+β (U, M)) ensures some kind of random bounded distortion property (see Lemma 2.4), which is important to establish the existence of a pressure function later (Proposition 3.1). We point out that rather than the dimension of E w (α), the Hausdorff dimension of w was studied by Bogenschütz and Ochs in [9]. Although {gw } are no longer conformal, they were able to show that the geometric balls are comparable with random Bowen balls (cf. Sect. 2) in a weak sense [9, Lemma 3.3]. Based on that and the theories on classical relativized topological pressure functions (cf. [8,27]), they proved that dim H w is close to dim H 0 if the perturbation is small and established the stability property of dim H 0 . Our proof of Theorem 1.1 relies on a dedicate study of dim H E w (α). The main difficulty is that the sets {E w (α)} are sensitive to w and are neither compact nor invariant. Our estimation of dim H E w (α) is based on a relativized variational principle for Birkhoff averages (Proposition 3.5) and the construction of random Moran like sets. In the following, we give some details about the idea. A conformal repeller (0 , f ) has markov partitions of arbitrarily small size (cf. [23]) and hence is a factor of a subshift of finite type (SFT). Denote by A the SFT with the shift map T . If U( f ) is small, there exists a homeomorphism between 0 and w [28]. Hence the bundle RDS (, G) is a factor of the random SFT ( × A , ϑ × T ). Moreover, gw , in a small neighborhood of f , is almost conformal in the sense that the expansion rates can be bounded from below and above by two functions λ(w, x) and η(w, x) which are close to Dx f (Proposition 2.5). A bundle RDS (, G) satisfying the above conditions is referred to as an expanding almost conformal repeller which can be modelled by a mixing SFT. (Precise definitions will appear in Sect. 2.3.) Denote by F the σ -algebra on and B the Borel σ -algebra on M. Let φ be a bounded F ⊗ B-measurable Rd -valued function on . Assume that φ is equicontinuous in the sense that for any > 0, there exists δ > 0 such that for P-a.e. ω ∈ , |φ(ω, y) − φ(ω, y )| < if d(y, y ) < δ. Denote by MP () the set of G-invariant measures on which projects to P under () : φ dµ = α and I (φ) = the map (w, x) → w. Put M(α) = µ ∈ M P
φ dµ : µ ∈ MP () . Theorem 1.1 will follow from a stability theorem for relativized pressure functions of the level sets (Proposition 3.7) and Theorem 1.2 Let (, G) be an expanding almost conformal repeller which can be modelled by a mixing SFT. Let φ be a bounded F ⊗ B measurable function such that φ(w, ·) ∈ C(U, Rd ), P-a.e. and w → φ(w, ·) is equi-continuous. Then there exists a set with P() = 1 such that for any w ∈
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(i)
α ∈ Rd : E w (α) = Ø = I (φ) and for any α ∈ I (φ), sup
µ∈M(α)
h(µ|P) log η(w, x) dµ
≤ dim H (E w (α)) ≤ sup
µ∈M(α)
(ii)
85
h(µ|P) , log λ(w, x) dµ
(1.2)
where h(µ|P) denotes the relativized entropy of µ with respect to P. if I (φ) has more than one point, then sup
µ∈MP ()
h(µ|P) log η(w, x) dµ
≤ dim H (Dw (φ)) ≤ sup
µ∈MP ()
h(µ|P) . log λ(w, x) dµ
In the case (w , gw ) is a shift space on m symbols and λ(w, x) = η(w, x) = m, Theorem 1.2 reduces to [21, Theorem 1.3]. However, the above Theorem can not be achieved by lifting the problems to the random SFT ( × A , ϑ × T ) and applying [21, Theorem 1.3]. Partially due to the lack of regularity assumption (e.g., Hölder continuity) on φ, the set M(α) may not contain any ergodic elements and hence the estimations in (1.2) can not be interpreted as bounds for the local dimension of measures. Moreover, the lifting map varies on w and is not one to one and, besides, gw is not conformal. This makes a direct calculation of the dimension of the projection of an invariant measure impossible. In the case (, G) is a general random subshift and λ(w, x) = η(w, x) = Constant, Kifer [25, Theorem 5.1] proved (1.2) for α belongs to the interior of I (φ) if φ is a potential function of some random Gibbs measure. His approach by the thermodynamic formalism for random shifts, however, is not enough to deal with the boundary points of I (φ) and does not work for general φ. An analogue of Kifer’s result was also obtained by Fan [16] (see also Fan and Shieh [19]) in the setting of infinite products through a large deviation approach, and some further study was given by Barral, Coppens and Mandelbrot in [1] for the multiplicative martingale measures. We point out that for small perturbations of a conformal repeller, the new bundle RDS (, G) has some sort of Markov partitions on inherited from the original system [28]. However, these partitions no longer give good geometric coverings as usual. This is because we have no control on adjacent points of w and things are even worse after non-conformal iterations. So, we can not establish (1.2) by constructing Moran sets using these partitions as in the deterministic case (cf. [20]). To remove the above difficulties, we construct Moran sets using random Bowen balls (see Sect. 2) directly on w ’s according to the dynamics of w. This combines and extends the ideas used in [21] and [20]. Nevertheless, random Bowen balls on w are not so easy to handle compared to cylinder sets in random shifts spaces or Markov blocks in the deterministic case. For any w ∈ and δ > 0, n ∈ N, consider 1 F(α; w, n, δ) := x ∈ w : φn (w, x) − α < δ . n
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Let γ = λ−1 or η−1 . For > 0, n ∈ N and s ∈ [0, +∞), define f (α, s; w, n, , δ) := sup
S ∈S x∈S
γn (w, x)s ,
n−1
s; w, n, , δ) denotes the colwhere γn (w, x) := i=0 γ ◦ G i (w, x) and S = S(α, lection of maximal (w; n, )-separated sets in F(α; w, n, δ) (see Sect. 2). Based on two technical Moran constructions, we are able to show that there exists an upper semi-continuous concave function on P(·, s) on I (φ) such that
lim lim lim inf
→0 δ→0 n→∞
1 1 log f (α, s; w, n, , δ) = lim lim lim sup log f (α, s; w, n, , δ) →0 δ→0 n n→∞ n = P(α, s).
Then by using an idea of Misiurewicz’s proof of the variational principle [30] and that of Bogenschütz’s proof of the conditional variational principle for random dynamical systems [8], we establish the relativized variational principle P(α, s) = sup
µ∈M(α)
h(µ|P) + s
log γ (w, x) dµ , ∀ s ∈ [0, +∞),
(1.3)
from which we derive that (α) = sup
µ∈M(α)
h(µ|P) − log γ (w, x) dµ
is the unique s such that P(α, s) = 0. Again by using a construction of random Moran set in E w (α), we give the lower bound estimation in (1.2) replacing γ by η−1 . The upper bound comes after a box counting argument using properties of P(α, s) for γ = λ−1 . The proof of (ii) of the theorem will follow from another construction of random Moran set. We note that the stability property of the relativized pressure function P(·, s) (Proposition 3.7) provides a finer version of the stability of the classical relativized pressure function for bundle RDS (cf. [9, Proposition 2.3]). It will be derived from the upper semi-continuity and concavity of the function P(·, s) and the variational principle in (1.3). The paper is arranged as follows. Section 2 is on some basic notions and properties of the systems concerned. In Sect. 3, we introduce the relativized pressure function P(α, s), establish the variational principle (1.3) and show the stability theorem for it. The proof of the main theorem will appear in the last section. 2 Preliminary We provide some basic notions and properties of bundle random dynamical systems (RDS), almost conformal repellers, and Moran structures.
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2.1 Bundle random dynamical system Let (, F, P) be a countably generated probability space with an invertible transformation ϑ on it. Let M be a compact polish space with Borel σ -algebra B. Denote by K(M) the collection of all compact subsets of M endowed with the Hausdorff topology. Let = ∪w∈ {w} × w be a bundle over with w ∈ K(M) depending measurably on w. Let {gw : w → ϑw }w∈ be a family of continuous onto maps which makes the map G : → , (w, x) → (ϑw, gw (x)) measurable. The couple (, G) is called a bundle random dynamical system (over (, F, P, ϑ) or simply ). The sets w , w ∈ , are the fibers of the bundle RDS (, G). Denote by PP ( × M) the space of probability measures on × M which projects (under the map (w, x) → w) to P on . Let L 1 (, C(M)) denote the space of measurable in w and continuous in x functions φ(w, x) on × M such that φ = sup |φ(w, x)| dP(w) < ∞. x∈M
(Here C(M) is the set of continuous real functions on M.) For µ, µn ∈ PP ( × M), n = 1, 2, . . ., µn is said to converge to µ if limn→∞ φ dµn = φ dµ for any φ ∈ L 1 (, C(M)). This convergence introduces a weak* topology for PP ( × M), which makes it compact and metrizable [13, Theorem 4.38]. Denote by PP () the collection of measures of PP ( × M) with support on . Let MP () be the set of G-invariant elements of PP (). Then both PP () and MP () are non-empty compact convex subsets of PP (× M) (cf. [12]). Denote by E P () the set of G-ergodic measures in MP (). It coincides with the extreme points of MP (). 2.2 Relativized entropy Let µ ∈ MP (). For any finite measurable partition ξ of M, denote by ξˆ := {( × A) ∩ : A ∈ ξ }. Let ˆ = {{w} × w : w ∈ } be the partition of into fibers. Define the relativized entropy of µ with respect to ξ as n−1 1 −k ˆ (2.1) h(µ|P, ξ ) := lim Hµ G ξ |ˆ , n→∞ n k=0
where Hµ (η|ζ ) is the canonical conditional entropy function of η with respect to ζ for a measure µ and ∨ denotes the join of partitions (cf. [38]). Define the relativized entropy of µ (with respect to P) by h(µ|P) := sup h(µ|P, ξ ), ξ
where the supremum is taken over all finite measurable partitions of M.
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For any µ ∈ MP (), there is a probability measure Q µ on E P (), the so-called ergodic decomposition of µ, such that φ dµ = φ dν d Q µ (ν), ∀ φ ∈ L 1 (, C(M)). E P ()
Resembling the Jacobs theorem (cf. [38, Theorem 8.4]) for the deterministic case, we have Proposition 2.1 Let (, G) be a bundle RDS. For µ ∈ MP (), let Q µ be its ergodic decomposition. Then h(µ|P) = E P () h(ν|P) d Q µ (ν) (both sides could be ∞). Proof Let ξ = {A1 , . . . , Am } be a finite partition of M. We first show h(µ|P, ξ ) = h(ν|P, ξ ) d Q µ (ν).
(2.2)
E P ()
Let = ∞ 0 {1, . . . , m} be the symbolic space on m symbols with the shift map T . Define π : → × by letting ∞ π(w, x) = (w, (τi )i=0 ) if g(w, i)x ∈ Aτi , for x ∈ w ,
where g(w, i) = gw ◦ gϑw · · · ◦ gϑ i−1 w for w ∈ , i ∈ N and g(w, 0) = id. The map π satisfies π ◦ G = (ϑ × T ) ◦ π . Hence for µ ∈ MP (), πˆ µ := µ ◦ π −1 satisfies h(µ|P, ξ ) = h(πˆ µ|P, η),
(2.3)
where η = {[1], . . . , [m]} is the canonical partition of according to the first symbol of the points. Let Q µ be the ergodic decomposition of µ. Then Q µ πˆ −1 is a corresponding ergodic decomposition of πˆ µ. We claim that the map µ → h( µ|P, η) is upper semi-continuous on M(×) (the set of invariant probability measures of × ). Let {ζi }i∈N be an increasing sequence of finite partitions of such that the σ algebra generated by ζi increases to F (such a sequence exists because F is countably generated). Let ζˆi = {A × : A ∈ ζi }. By (2.1) and basic properties of the entropy function H µ (·) (cf. [38]), we have ⎞ j−1 1 h( µ|P, η) = inf inf inf H (ϑ × T )−k η (ϑ × T )−l ζˆi ⎠. µ⎝ n i j n l=0 k=0 ⎛
n−1
(2.4)
j−1 −k −l ˆ For A ∈ ∨n−1 k=0 (ϑ × T ) η ∨ ∨l=0 (ϑ × T ) ζi , its characteristic function belongs to 1 L (, C()). Hence for each n, i, j, the map ⎛ ⎞ j−1 n−1 (ϑ × T )−k η (ϑ × T )−l ζˆi ⎠
µ → H µ⎝ l=0 k=0
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is continuous. We conclude from (2.4) that µ → h( µ|P, η), as the infimum of a sequence of continuous maps, is upper semi-continuous. Let E( × ) be the set of ergodic measures on × . Note that Q µ πˆ −1 is a corresponding ergodic decomposition of πˆ µ. Applying Choquet’s theorem (cf. [35]) to µ → h( µ|P, η) gives
h(ν|P, η) d Q µ πˆ −1 (ν).
h(πˆ µ|P, η) = E(×)
This immediately implies (2.2) by (2.3). ∞ be a sequence of decreasing finite measurable partitions of M Finally, let {ξi }i=1 with limi→∞ diam ξi = 0. Then h(µ|P) = limi→∞ h(µ|P, ξi ) (cf. [29, Theorem 4.7]). The proposition follows by using the monotone convergence theorem on the measure Q µ and applying (2.2) for h(µ|P, ξi ). By using this conditional version of the ergodic decomposition of entropy, exactly the same argument as in [40, p. 535] gives Proposition 2.2 Let φ ∈ L 1 (, C(M)). For any µ ∈ MP (), let α = φ dµ. k p = 1 and Then for any > 0, there exists k ∈ N, pi ≥ 0, i < k with i=1 i k {µi }i≤k ⊆ E P () such that µ = i=1 pi µi satisfies φ d µ|P) > h(µ|P) − . µ − α < and h( Another equivalent definition of (2.1) will be useful in later proofs (see Proposition 3.5). For µ ∈ MP (), (, µ) becomes a Lebesgue space. Hence there exists a random measure w → µw (unique P -a.e.) such that i) for each Borel set B of M, w → µw (B) is measurable; ii) for P almost every w ∈ , µw is a Borel probability measure on w ; and iii) φ(ω, x) dµ(ω, x) = φ(ω, x) dµω (x) dP(ω)
w
for every bounded measurable φ : → R (see [13, Proposition 3.6] for a proof). The above random measure ω → µω is often named as the disintegration of µ (with respect to P). For any finite measurable partition ξ of M, we have by [8, Theorem 2.2] that n−1 1 −1 h(µ|P, ξ ) = lim Hµw g(w, i) ξ , n→∞ n i=0
where g(w, i) = gw ◦ gϑw · · · gϑ i−1 w for w ∈ , i ∈ N and g(w, 0) = id. For any w ∈ and n ∈ N, let dnw (x, y) = max {d(g(w, k)x, g(w, k)y) : 0 ≤ k ≤ n − 1} , ∀ x, y ∈ w ,
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be the nth Bowen metric on fiber w . For > 0, n ∈ N, two points x, y ∈ w are said to be (w; n, )-separated if dnw (x, y) ≥ . A subset S of w is called (w; n, )separated if any different two points of it are (w; n, )-separated. Call S maximal if it has maximal cardinality. The Bowen balls around x ∈ w are referred to
Bw (x; n, ) := y ∈ w : dnw (x, y) ≤ , > 0, n ∈ N. Resembling the Brin-Katok formula of entropy in the deterministic case [7, Theorem 1], Proposition 2.3 (cf. [26]) Let (, G) be a bundle RDS. For any µ ∈ E P (), let {µw }w∈ be its disintegration with respect to P. Then for P-a.e. w 1 log µw (Bw (x; n, )) n 1 = − lim lim sup log µw (Bw (x; n, )), µw -a.e. →0 n→∞ n
h(µ|P) = − lim lim inf →0 n→∞
(2.5)
2.3 Almost conformal repellers A bundle RDS (, G) is called an almost conformal repeller (cf. [9]) if there exist measurable functions λ, η : → R+ , and ι : → R+ with log λ, log η ∈ L 1 (, C(M)), logι ∈ L ∞ (), and λ(w, ·), η(w, ·) Hölder continuous in x with uniform exponent β, 0 < β ≤ 1 such that λ(w, x) − ι(w)d(x, y)β ≤
d(gw x, gw y) ≤ η(w, x) + ι(w)d(x, y)β d(x, y)
for w ∈ and x = y ∈ w . An almost conformal repeller is expanding if min(w,x)∈ λ(w, x) > 1. Lemma 2.4 (cf. [9, Lemma 3.3]) Let (, G) be an expanding almost conformal repel and C (depends on 0 ) such that for P -a.e. ler. There exist positive constants 0 , R, R w ∈ , (i) for x, y ∈ w , d(g(w, n)x, g(w, n)y) ≤ 0 for n ∈ N implies x = y. R (ii) Bw (x; n + 1, 0 ) ⊆ B(x, λn (w,x) ), for x ∈ w , n ∈ N.
R ) ⊆ Bw (x; n, ), ∀ 0 < < 1, x ∈ w , n ∈ N. (iii) B(x, ηn (w,x) (iv) for x, y ∈ w with y ∈ Bw (x; n, 0 ),
C −1 ≤
λn (w, x) ≤ C, ∀ n ∈ N λn (w, y)
n−1 and similarly for ηn (w, x). (Here λn (w, x) := i=0 λ ◦ G i (w, x).)
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Proof (i)–(iii) is a reformulation of [9, Lemma 3.3] in our setting using the expanding assumption. For (iv), let λ0 be such that min(w,x)∈ λ(w, x) > λ0 > 1. Then by standard distortion estimations and (ii),
| log λn (w, x) − log λn (w, y)| ≤ const
n−1
d(g(w, i)x, g(w, i)y)β
i=0
≤ const
β 0
≤ const
β 0
+
n−2 i=0
Rβ i λn−1−i (ϑ w, g(w, i)x)β
n−2 + (λ0−(n−1−i) R)β
i=0
≤ const ( 0 , R, λ0 , β) . The same argument applies to ηn (w, x).
Let A be a matrix with entries 0 or 1. Let A be a one sided subshift of finite type (SFT) with the left shift map T . We say that a bundle RDS (, G) can bemodelled by a mixing SFT if there exist a mixing SFT ( A , T ) and a bundle of surjective uniformly continuous maps {h w : A → w }w∈ such that the map π(w, τ ) = (w, h w (τ )) for w ∈ , τ ∈ A satisfies π(ϑw, T τ ) = (ϑw, gw ◦ h w (τ )). Let C 1 (U, M) be the set of all C 1 maps from U to M equipped with the compact open topology. Let Emb1 (U, M) be the Borel subset of C 1 (U, M) whose elements are diffeomorphisms from U to the images. Let U( f ) denote an open neighborhood of f in Emb1 (U, M). Then, Proposition 2.5 ([28, Theorem 1.1]) Let 0 be a conformal repeller for a C 1+β topological mixing map f (on a neighborhood U of 0 ). Then (i)
(ii)
(iii)
there is an open neighborhood U( f ) of f in Emb1 (U, M) such that any bun∞ 1+β (U, M)) is an dle RDS (, G) over = +∞ −∞ U( f ) with g ∈ L (, C expanding almost conformal repeller which can be modelled by a mixing SFT. for any given > 0, one can shrink the U( f ) above such that for each w ∈ and x ∈ 0 , there is a unique xw ∈ w such that d( f n (x), g(w, n)xw ) ≤ for all n ≥ 0. the map h w : 0 → w sending x to xw is a homeomorphism. It is equi-continuous in the sense that for each > 0, there is δ > 0 such that d(x, y) < δ h w y) < for x, y ∈ 0 and w ∈ . implies d( h w x,
Consequently, lots of problems related to small perturbations of conformal repellers can be reduced to that for expanding almost conformal repellers modelled by a mixing SFT.
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2.4 Non-homogenous Moran sets We modify a technique used in [20] (see also [17,22,33]) for the estimation of the Hausdorff dimension of non-invariant subsets from below. Let {ak }∞ k=1 be a sequence of positive integers. Let
D = ∪k≥0 Dk with D0 = Ø ; Dk = {j = ( j1 , . . . , jk ) : 1 ≤ ji ≤ ai , 1 ≤ i ≤ k} ; D∞ = {( j1 , j2 , j3 , . . .) : 1 ≤ ji ≤ ai , i ≥ 1} . Let {i : i ∈ D} be a collection of subsets of a compact metric space M. We say that the collection {i : i ∈ D} forms a strong Moran structure if (i) (ii) (iii)
i0 ···ik j ⊆ i0 ···ik for j ≤ ak+1 . diam i0 ···ik → 0 as k → ∞. (separation condition) there exists {ri : i ∈ Dk } , k ∈ N and c0 > 0 such that ri0 ···ik → 0 as k → ∞ and B(i ∩ F, c0 ri ) ∩ B(j ∩ F, c0 rj ) = Ø,
(iv)
where B(A, r ) = ∪x∈A B(x, r ), B(x, r ) is the usual ball for any set A ⊆ M, r > 0, and F = ∩k>0 ∪i∈Dk i . there exist positive constants C1 and C2 such that C1
r i j rij rij ≤ ≤ C2 ri r i ri
for all ij = i j, where i, i ∈ Dm , ij, i j ∈ Dn , m ≤ n. Given a strong Moran structure {i : i ∈ D}, the limit set F is called a nonhomogenous Moran set satisfying strong separation condition (or a strong Moran set for short). Roughly speaking, the limit set F is a cantor like set obtained by digging (inductively) level sets {i : i ∈ Dk } in each step k ∈ N from those sets in step k − 1. Different sets of the same level are separated from each other (see condition iii)); each set i with i = i 0 · · · i k has a diameter of approximately ri ; and moreover, the proportion of the next (or finite step) descendants in i , measured by diameters, is independent of the i ∈ Dk chosen (see condition iv)). Furthermore, if the digging process is some how smooth in the sense that the diameters of the next generations of each level set does not drop dramatically, we can give a lower bound of the Hausdorff dimension of the set F. For this, let ρk =
123
min
(i 1 ···i k )∈Dk
ri1 ···ik , ri1 ···ik−1
Mk =
max
(i 1 ···i k )∈Dk
ri1 ···ik .
The multifractal analysis of Birkhoff averages for conformal repellers
Lemma 2.6 Let F =
k>0
i∈Dk
93
i be a strong Moran set satisfying
log ρk = 0. k→∞ log Mk
(2.6)
lim
Then dim H F ≥ lim inf k→∞ sk , where sk satisfies the equation i∈Dk risk = 1. Proof We follow the idea used in [20, Proposition 3.1]. Let s < lim inf k→∞ sk . For each k ∈ N, let Bk be the σ -algebra generated by the cylinders [i], i ∈ Dk and define µk ([i]) =
ris
s. i∈Dk ri
It is easy to verify using (iv) that there is C > 0 such that for any l < k and for i ∈ Dl , C −1 < µk ([i])/µl ([i]) < C. Let µ be some cluster point of {µk }∞ k=1 in the weak* topology of the probability measure space on M. Then µ satisfies C −1 <
µ([i]) < C, ∀ l > 0 and i ∈ Dl . µl ([i])
(2.7)
Let be the projection map from D∞ to F by letting (i 1 i 2 · · · ) = ∩n>0 i1 ···in . Let
µ = µ ◦ −1 . We claim that for any t < s,
µ(B(x, r )) ≤ Cr t , for r small, µ ≥ t by the mass distribution for every x ∈ F. That will give dim H (F) ≥ dim H principle of measures (cf. [15]). Then we are done since t < lim inf k→∞ sk is arbitrary. By (iii), ri0 ···ik → 0 as k → ∞. Since s < lim inf k→∞ sk , there exists k0 such that for any k ≥ k0 , i∈Dk ris > 1. For i ∈ Dk with k ≥ k0 , we have
µ(i ) ≤ C
ris s i∈Dk ri
≤ Cris ,
where C is the constant involved in (2.7). Let i = i 1 i 2 · · · i k · · · be the coding of x ∈ F such that x ∈ i1 ···ik , k ∈ N (the coding is unique due to (iii)). Let r > 0 and let k be the largest integer such that c0 ri1 ···ik+1 ≤ r < c0 ri1 ···ik , where c0 is as in (iii). Then B(x, r ) ∩ F ⊆ B(i1 ···ik , c0 ri1 ···ik ) ∩ F ⊆ i1 ···ik ∩ F. Hence
µ(B(x, r ) ∩ F) ≤ µ(i1 ···ik ∩ F) ≤ Cris1 ···ik ≤ Crit1 ···ik+1 , for k large by (2.6), ≤ Cr t .
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3 A relativized pressure function for level sets In this section we construct the relativized pressure function P(α, s) and show the variational principle in (1.3). Let (, G) be a bundle RDS which can be modelled by a mixing SFT ( A , T ). Let φ : → Rd be a bounded F ⊗ B measurable function such that φ(w, ·) ∈ C(U, Rd ), P-a.e. and w → φ(w, ·) is equi-continuous. Let γ : → R+ be a measurable function satisfying 0 < γ0 = min(w,x)∈ γ (w, x) < max(w,x)∈ γ (w, x) < 1 and (iv) of Lemma 2.4. 3.1 The definition of P(α, s) Let I (φ) = consider
φ dµ : µ ∈ MP () . For α ∈ I (φ), w ∈ and δ > 0, n ∈ N, 1 φn (w, x) − α < δ . n
F(α; w, n, δ) := x ∈ w :
s; w, n, , δ) be the collection of maximal For > 0 and n ∈ N, let S = S(α, + (w; n, )-separated sets in F(α; w, n, δ). Define for s ∈ R+ 0 := R ∪ {0} f (α, s; w, n, , δ) := sup
S ∈S x∈S
γn (w, x)s .
(3.1)
Let 1 log f (α, s; w, n, , δ); n 1 P w (α, s; ) = lim lim sup log f (α, s; w, n, , δ). δ→0 n→∞ n P w (α, s; ) = lim lim inf δ→0 n→∞
(3.2)
Observe that P w (α, s; ) (or P w (α, s; )) is a decreasing function on . Let P w (α, s) := lim P w (α, s; ) and P w (α, s) := lim P w (α, s; ). →0
→0
(3.3)
The main proposition we will show in this subsection is Proposition 3.1 There exists an upper semi-continuous concave function on P(·, s) on I (φ) such that for P -a.e. w ∈ , P w (α, s) = P w (α, s) = P(α, s), ∀ α ∈ I (φ). The idea to prove this proposition will be similar to [21, Proposition 3.8] (for the case (, G) = ( × , ϑ × T ), γ = m −1 , where (, T ) denotes the shift space on m symbols). Some tedious calculation which can be achieved similarly as in [21] is thus
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95
omitted. Nevertheless, the point of difference is that the Moran constructions involved here are on w using separated random Bowen balls and P(·, ·) is for general γ . We begin with two simple lemmas concerning the well-definement of the limits above. Lemma 3.2 There ⊆ with P() = 1 such that for any
exists a measurable set w ∈ , we have α ∈ Rd : E w (α) = Ø = I (φ).
(w, τ ) = φ(w, h w (τ )) for (w, τ ) ∈ Proof Lift φ to ( × A , ϑ × T ) by letting φ
w (α) be the corresponding fiber level sets for φ
. Then I (φ
) = I (φ) × A . Let E
w (α) = Ø. The conclusion follows by that for E
w (α) and E w (α) = Ø if and only if E by [21, Proposition 3.1]. Lemma 3.3 For any > 0, there exist two functions P(α, s; ) and P(α, s; ) from + I (φ) × R+ 0 to [0, ∞] such that for any (α, s) ∈ I (φ) × R0 , P(α, s; ) = P w (α, s; ) and P(α, s; ) = P w (α, s; ), P − a.e.
s; ϑw, n, , δ). Let n ≥ Proof For ϑw such that Lemma 3.2 holds, let S ⊆ S(α, 2φ/δ, where φ = sup(w,x)∈ |φ(w, x)|. By choosing one point from the primage −1 (x)), we get an (w; n + 1, )-separated set S ⊆ F(α; w, of each x ∈ S (i.e. gw n + 1, 2δ) with y∈S
γn+1 (w, y)s ≥ γ0s
γn (ϑw, x)s .
x∈S
This gives P w (α, s; ) ≥ P ϑw (α, s; ) (or for P) and hence the Lemma by ergodic theory. We fix some more notations. Let ∗A := {τ |n : τ ∈ A , n ∈ N}, where τ |n is the first n symbols of τ . Let p ∈ N be such that for any σ, σ ∈ ∗A , there is ς ∈ ∗A such that σ ς σ ∈ ∗A . Write n = n + p for n ∈ N. Given a sequence {τk }k∈N of ∗A , denote by τ k the extension of τk with a word of length p to make τ 1 · · · τ k · · · ∈ A . Lemma 3.4 Let P(α, s) = lim →0 P(α, s; ) and similarly for P(α, s). The functions P(α, s), P(α, s) coincide on I (φ) × R+ 0 . (Denote by P(α, s) the common function.) Proof Let (α, s) ∈ I (φ) × R+ 0 . It suffices to show for ε > 0, P w (α, s) ≥ P(α, s) − 3ε a.e. First pick up the w’s. Let a > 0 be small enough to be specified later. By (3.2) and (3.3) there are 1 , δ1 such that the set 1 A0 : = w ∈ : lim sup log f (α, s; w, n, , δ) > P(α, s) − ε/2, n→∞ n ∀ < 1 , δ < δ1
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has measure 1 − a/2. Fix < 1 , δ < δ1 . By uniform continuity of π (the map from × A to ), there exists N0 ∈ N such that for τ, τ ∈ A , τ | N0 = τ | N0 implies d(π(w, τ ), π(w, τ )) < /3 and |φ(π(w, τ )) − φ(π(w, τ ))| < δ.
(3.4)
Choose N1 , N2 with N2 > N1 > N 0 (N 0 = N0 + p) such that the set A := w ∈ : f (α, s; w, n, , δ) > en(P(α,s)−ε) for some n ∈ [N1 , N2 ] has measure P(A) > 1 − a. Denote by
n−1 1 H := w ∈ : lim χ A (ϑ i w) = P(A) , n→∞ n i=0
where χ A is the characteristic function of A. Then P(H ) = 1 by Birkhoff average theorem. For each w
∈ H , we claim 1 f (α, s; w
, n, , 4δ) ≥ Const · en(P(α,s)−3ε) for n large. 3
(3.5)
Step 1: We construct a sequence of integers {lk }∞
by induction. Start with l1 . k=1 for w If w
∈ A, let l1 = p; otherwise let l1 ∈ [N1 + N 0 , N2 + N 0 ] be such that f (α, s; w
, l, , δ) ≥ el(P(α,s)−ε) for l = l1 − N 0 . Suppose l1 , . . . lk−1 has been constructed for some k ≥ 2. We proceed as follows: if ϑ l1 +···+lk−1 w
∈ A, let lk = p; otherwise let lk ∈ [N1 + N 0 , N2 + N 0 ] be such that
, l, , δ) ≥ el(P(α,s)−ε) for l = lk − N 0 . f (α, s; ϑ l1 +···+lk−1 w In this way we obtain a sequence {lk }∞
. k=1 for w Step 2: We associate with each lk a set Tk of ∗A . For lk = p, let Tk = {ς } for some
; lk − N 0 , )-separated ς ∈ ∗A with length p. Otherwise, let Sk be some (ϑ l1 +···+lk−1 w l +···+l 1 k−1 subset of F(ϑ w
, lk − N 0 , δ) such that x∈Sk
γlk −N 0 (ϑ l1 +···+lk−1 w
, x)s ≥ e(lk −N 0 )(P(α,s)−ε) .
Such Sk exists by our choice of lk for w
. For each x ∈ Sk , choose a symbolic reprew, τ ) = x and let sentation τ = τ (x) ∈ A with π(
Tk = τ (x)|lk − p : x ∈ Sk .
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By our choice of N0 and Sk , we see that x → τ (x)|lk − p is a bijection between Sk and Tk . Step 3: We obtain a set of point x ∈ w
by specifying its symbolic sequence in A . Let T = {τ = τ 1 · · · τ k · · · : τk ∈ Tk , k ∈ N}
(3.6)
and let T = {π( w, τ ) : τ ∈ T }. In the following we claim for n large, there is some subset S ⊆ T such that (i) (ii)
S is some ( w ; n, /3)-separated set of F( w, n, 4δ); x∈S γn ( w , x)s ≥ en(P(α,s)−3ε) .
Clearly, this will imply (3.5) and finish the proof of the lemma. Step 4: We choose the set S. Let n > l1 . There is a unique k such that l1 + · · · + lk ≤ n < l1 + · · · + lk+1 . Let S be a subset of T with maximal cardinality such that π( w , τ ) = π( w , τ ) ∈ S implies τ1 · · · τk = τ1 · · · τk . Points in S are ( w ; n, /3)-separated due to the choices of Sk ’s and N0 . Let N3 be such n−1 χ A > 1 − a for n ≥ N3 . To see S ⊆ F( w, n, 4δ), we simply repack that (1/n)i=0 w , π( w , τ )) according to τk ’s (see (3.6)) and then conclude the summation in φn ( |φn ( w , π( w , τ )) − nα| ≤ 4nδ, for n ≥ max {N3 , 2N2 φ/δ} , using (3.4) and the fact that Sk ⊆ F(ϑ l1 +···+lk−1 w
, lk − N 0 , δ). Step 5: Finally we show (ii) and clarify a in the construction of the set A. We have by step 2 and assumption of γ (·, ·) that sp#{l j = p } s(n− j≤k l j ) γ0
x∈S γn ( w, x)s ≥ Cγ0
γl j
l j > p x∈S j
−N 0 (ϑ l1 +···+l j−1 w
, x)s γ0s N 0 s(npa+N 0 k+N2 ) ≥ Cγ0 e(l j −N 0 )(P(α,s)−ε) l j>p
≥ Ce ≥ Ce
N N n(spa+ N 0 + n2 ) 1
N
e
n(1− N 0 )(P(α,s)−ε) 1
n(P(α,s)−3ε)
if we take a < ε/(3sp), N1 > max 3N 0 /ε, (P(α, s) − ε)N 0 /ε and n > N2 ε/3. Proof of Proposition 3.1 We first show the upper semi-continuity of P(·, s). Let α ∈ I (φ) and ε > 0 be given. By Lemma 3.4, there are δ1 and 1 > 0 such that the set 1 A := w ∈ : lim inf log f (α, s; w, n, δ, ) < P(α, s) + ε, for δ < δ1 , < 1 n→∞ n
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has positive measure. Let α ∈ I (φ) be such that |α − α| ≤ δ1 /2. For any w ∈ A, δ < δ1 /2, < 1 , we see that f (α , s; w, n, δ, ) ≤ f (α , s; w, n,
δ1 , ) ≤ f (α, s; w, n, δ1 , ). 2
It follows that P w (α , s) ≤ P(α, s) + ε on A. Hence P(α , s) ≤ P(α, s) + ε since A has positive measure. For the concavity of P(α, s), it suffices to show for α, β ∈ I (φ) and ε > 0 Pw
α+β ,s 2
≥
1 1 P(α, s) + P(β, s) − 3ε 2 2
(3.7)
for a set of w with positive measure. Relabel {α, β, (α + β)/2} as {α1 , α2 , α3 }. We first pick up the w’s. By definition of P(α, s), there are δ2 , 2 such that the set 1 A0 = w ∈ : | lim inf log f (αi , s; w, n, δ, ) n→∞ n −P(αi , s)| < ε for δ < δ2 , < 2 , i ≤ 3 has measure greater than 1 − a/2 for some small a to be fixed later. Fix δ < δ2 /4, < 3 2 . As in the proof of Lemma 3.4, choose N0 such that (3.4) holds. Then choose a sufficient large l > N 0 (N 0 = N0 + p) such that the set A := w ∈ : f (αi , s; w, l, δ, ) > el(P(αi ,s)−ε) , i ≤ 2 has measure at least 1 − a. Denote by n−1 1 i(l+N 0 ) χ A (ϑ w) = P(A) . H := w ∈ A0 : lim n→∞ n
i=0
Then P(H ) = P(A0 ) > 1 − a by Birkhoff ergodic theorem. We claim lim inf n→∞ n1 log f (α3 , s; w, n, 4δ, 13 ) ≥ 21 i≤2 P(αi , s) − 2ε for w ∈ H . This will imply (3.7) by Lemma 3.3. Let Si , i = 1, 2 be some (w; l, )-separated subset of F(αi ; w, l, δ) such that
γl (w, x)s > el(P(αi ,s)−ε) .
x∈Si
For each i ≤ 2, choose for each x ∈ Si a symbolic representation τ = τ (x) ∈ A with π(w, τ ) = x. Then by our choice of N0 , x = x ∈ Si is equivalent to τ |l+N0 = τ |l+N0 .
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Define a sequence of sets Ti of ∗A by ⎧
⎪ τ (x)|l+N0 : x ∈ S1 , ⎪ ⎪ ⎨
τ (x)|l+N0 : x ∈ S2 , Ti = ⎪ ⎪ ⎪ ⎩ {ς } with |ς | = l + N0 ,
if ϑ i(l+N 0 ) w ∈ A, i odd; if ϑ i(l+N 0 ) w ∈ A, i even; otherwise.
Let T = {τ = τ 1 · · · τ k · · · : τk ∈ Tk , k ∈ N} and let T = {π( w, τ ) : τ ∈ T }. Then by essential the same procedure as in Lemma 3.4, we can find a small a such that for each n large, there is some subset S ⊆ T such that (i) (ii)
S is some (w; n, /3)-separated set of F(α3 ; w, n, 4δ); 1 x∈S γn (w, x)s ≥ en( 2 i≤2 P(αi ,s)−2ε) .
This proves the claim and finishes the proof of the lemma.
3.2 A variational principle for P(α, s) In this subsection we show
Proposition 3.5 P(α, s) = supµ∈M(α) h(µ|P) + s log γ (w, x) dµ for (α, s) ∈ I (φ) × R+ 0. Proof of “≥” First assume µ ∈ M(α) is ergodic. Let A be the set of w satisfying (2.5) such that limn→∞ n1 log γn (w, x) = log γ (w, x) dµ, and limn→∞ n1 log φn (w, x) = α hold µw -a.e. Then P(A) = 1. Let w ∈ A. For any ε > 0, k ∈ N, there are 1 , N0 such that the set
k = {x ∈ w : µw (Bw (x; n, )) ≤ e−n(h(µ|P)−ε) ; A γn (w, x) ≥ en(
log γ (w,x) dµ−ε)
|φn (w, x) − nα| <
;
1 n, ∀ < 1 , n ≥ N0 } k
(3.8)
has µw measure greater than 1 − 1/2k+1 . Let < 1 , n > N0 . By the first inequality
k has cardinality greater than of (3.8), any maximal (w; n, 2 )-separated set S of A 1 n(h(µ|P)−ε) . Moreover, 2e
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γn (w, x)s ≥
x∈S
1 exp n h(µ|P) + s log γ (w, x) dµ − (s + 1)ε . 2
This clearly implies lim inf n→∞
1 1 f (α, s; w, n, 2 , ) ≥ h(µ|P)+s n k
k . log γ (w, x) dµ − (s + 1)ε, ∀ w ∈ A
= ∩k A
> 1/2. For any w ∈ A,
we have
k . Then P( A) Let A 1 1 P w (α, s) = lim lim lim inf f (α, s; w, n, 2 , ) →0 k→∞ n→∞ n k ≥ h(µ|P) + s log γ (w, x) dµ − (s + 1)ε. So P(α, s) ≥ h(µ|P) + s log γ (w, x) dµ by Lemma 3.4 since ε > 0 is arbitrary. Next let µ ∈ M(α) be arbitrary. Let ε > 0. By upper semi-continuity of P(α, s), there is 0 < ε < ε with ε < ε such that |α − α | < ε implies |P(α, s) − P(α , s)| < ε. k pi µi satisApproximate µ with a linear combination of ergodic measures µ = i=1 fying Proposition 2.2 for ε . Let αi = φ dµi and α = i≤k pi αi . Then |α −α| < ε . We have ε + P(α, s) ≥ P(α , s) ≥
k i=1
≥
k
pi P(αi , s) pi h(µi |P) + s log γ (w, x) dµi
i=1
= h( µ|P) + s ≥ h(µ|P) + s
log γ (w, x) d µ log γ (w, x) dµ − 2ε.
Since ε > 0 is arbitrary, this shows P(α, s) ≥ h(µ|P) + s µ ∈ M(α).
log γ (w, x) dµ for
Proof of “≤” It suffices to show for each > 0, there is a µ ∈ M(α) and a partition ξ of M satisfying P(α, s; ) ≤ h(µ|P, ξ ) + s log γ (w, x) dµ. In the construction of µ, we follow an idea of Misiurewicz’s proof of the variational principle [30] and that of Bogenschütz’s proof of the conditional variational principle for random dynamical systems [8].
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First we choose two sequences {δk } and {n k }. Write P = P(α, s; ) for simplicity. By Lemma 3.3, for any k ∈ N, there exists δk > 0 such that P
1 1 1 > 1 − k+1 . w : lim inf log f (α, s; w, n, , δk ) ≥ P − n→∞ n k 2
Choose n k ∈ N such that P
1 1 1 >1− k. w: log f (α, s; w, n k , , δk ) ≥ P − nk k 2
(3.9)
We may assume δk ↓ 0 and n k ↑ ∞. Also it is of no harm to assume P > 1/k. Next, we define a multifunction. For q ∈ N, let K(M q ) denote the space of compact subsets of M q . For k ∈ N, define q (·, n k , δk ) : → K(M q ) by letting q (w, n k , k ) := {(y1 , . . . , yq ) : yi ∈ F(α; w, n, δk ), i ≤ q; dnw (yi , y j ) ≥ if i = j; q
γn k (w, yi )s ≥ en k (P−1/k) }.
i=1
Let p(w, n k , δk ) be the smallest cardinality of any (w; n k , )-separated set S of F(α; w, n, δk ) satisfying
γn k (w, y)s ≥ en k (P−1/k) .
y∈S
Fix y ∈ w . For each w ∈ , define
(w, n k , δk ) :=
p × M × M × . . . ,
if p = p(w, n k , δk ) ≥ 1;
{ y} × M × M × . . . ,
otherwise.
Each (w, n k , δk ) is a closed nonempty subset of M ∞ , which is a complete separable metric space. Moreover, the graph of (·, n k , δk ) is measurable in F ⊗ B ∞ . Hence the Selection Theorem [10, Theorem III.30] applies and there exists a measurable map (w, n k , δk ) for all w ∈ . ping (·, n k , δk ) : → M ∞ such that (w, n k , δk ) ∈ Put p(w,n S(w, n k , δk ) :=
i=1
{ y} ,
k ,δk )
πi (w, n k , δk ),
if p(w, n k , δk ) ≥ 1; otherwise,
where πi denotes the projection from M ∞ to the i-th component.
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We proceed to construct a measure µ ∈ MP () as follows. For each w ∈ , construct two probability measures νwk , µkw on w by νwk := µkw
1 #S(w, n k , δk )
δ(w,y) ,
y∈S (w,n k ,δk )
n k −1 1 := νϑk −i w ◦ g(ϑ −i w, i)−1 . nk i=0
Here δy denotes the point mass at y and # means cardinality. Assume for some subse k quence k j , µwj converges to some measure µw on M. Then {µw }w∈ is a family of conditional measures of some probability measure µ of MP () (cf. [12, Theorem 4]). We show µ ∈ M(α). For each k ∈ N, we calculate that φ(w, x) w
dµkw (x)
n k −1 1 dP(w) = φ(ϑ i w, g(w, i)x) dνwk dPϑ i (w) nk i=0 w
n k −1 1 = φ(ϑ i w, g(w, i)x) dνwk dP(w) nk i=0 w 1 = φn (w, x) dνwk (x) dP(w). (3.10) nk k w
By our choice of n k and δk , we get from (3.9) that 1 1 k φn k (w, x) d νw (x) dP(w) − α ≤ k−1 φ + δk . 2 n w k Combining this with (3.10), then replacing k by k j in (3.10) and letting j tend to infinity, we get φ(w, x) dµw dP(w) = α. w
It follows that φ dµ = α since {µw }w∈ is the disintegration of µ with respect to P. Finally, we estimate h(µ|P) + s log γ (w, x) dµ and show it is greater than P. Choose a finite partition ξ = {ξ1 , . . . ξm } of M such that diam ξi < and Pr M µ(∂ξi ) = 0 for 1 ≤ i ≤ q, where Pr M µ denotes the projection of µ to M. Hence µw (∂ξi ) = 0, P-a.e. For any k ∈ N and w ∈ , notice that each element of ξ n k contains at most one point from S(w, n k , δk ), we have 1 s log γ (w, x) dνwk (x) + Hνwk (ξ n k ) ≥ n k (P − ), if p(w, n k , δk ) ≥ 1. (3.11) k
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By (3.9), this is true for a set of w with P measure greater than 1 − 1/2k . Fix q ∈ N, a standard argument by repacking ξ n k (cf. [38, Theorem 8.6 ] or [8, Theorem 6.1]) yields q Hνwk (ξ ) ≤ nk
n k −1
Hνwk ◦g(w,i)−1 ξ q + 2q 2 log m.
i=0
Hence
n k −1 1 1 2q 1 nk Hνwk (ξ ) dP(w) ≤ · log m Hνwk ◦g(w,i)−1 (ξ q ) dP ◦ ϑ i (w) + nk q nk nk i=0 2q 1 H k (ξ q ) dP(w) + log m. ≤ q µw nk
On the other hand, we have by (3.11) that 1 s nk
dνnwk (x)
log γ (w, x) 1 1 P− . ≥ 1− k 2 k
dP(w) +
1 H k (ξ n k ) dP(w) n k νw
Writing k j instead of k and letting j tend to ∞, we obtain nk j 1 kj P ≤ lim H k (ξ ) dP(w) + s log γ (w, x) dµw (x) dP(w) j→∞ n k j νwj 1 q ≤ Hµ (ξ ) dP(w) + s log γ (w, x) dµ. q w Letting q tend to ∞ gives P(α, s) ≤ h(µ|P, ξ ) + s log γ (w, x) dµ.
3.3 The stability of P(α, s)
Let (, G), φ, γ be as above. Let φ ( ) >0 be a family of bounded, equi-continuous (in w) measurable functions from to Rd with lim φ ( ) − φ = 0.
→0
Denote by P ( ) (·, s) the relativized pressure function corresponding to φ ( ) . We will analyze the limit behavior of P ( ) (·, s). We need one more lemma from convex analysis theory: Lemma 3.6 ([41, Theorem 2.2.22]) Let X be a Banach space and C ⊂ X be an open convex set. Consider ( f i )i∈I a nonempty family of continuous convex functions from
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C into R. If ( f i (x))i∈I is bounded for every x ∈ C then for every x ∈ C there exist rx , L x > 0 such that Ux := {y : y − x ≤ rx } ⊂ C and | f i (y) − f i (z)| ≤ L x y − z for all y, z ∈ Ux and all i ∈ I (i.e. ( f i )i∈I is locally equi-Lipschitz on C). Proposition 3.7 For any s ≥ 0, lim →0 P ( ) (α, s) = P(α, s) for α ∈ I (φ)o . Moreover, the convergence is uniform on any compact subset of I (φ)o . Proof First notice that for any µ ∈ MP (), φ ( ) dµ − φ dµ ≤ φ ( ) − φ → 0, as → 0. Hence each α ∈ I (φ)o belongs to I (φ ( ) ) for small. It is of no harm to assume each P ( ) (·, s) is well-defined on I (φ)o . Let α ∈ I (φ)o , we first show lim sup →0 P ( ) (α, s) ≤ P(α, s). By upper semicontinuity of P, for any ε > 0, there exists δ > 0 such that P( α , s) ≤ P(α, s) + ε, ∀ α ∈ B(α, δ), where B(α, δ) is the usual ball in Rd centered at α with radius δ. For small, each α) µ ∈ M ( ) (α) := µ ∈ MP () : φ ( ) dµ = α corresponds to an element of M( for some α ∈ B(α, δ). Hence, we have P ( ) (α, s) ≤ P( α , s) ≤ P(α, s) + ε As ε > 0 is arbitrary, we have the desired inequality. Conversely, we apply Lemma 3.6 for P ( ) (·, s) >0 . Note that every upper semicontinuous concave function is continuous in the interior of its domain for any locally convex topological vector spaces (cf. [36, Theorem 10.1]). We have by Proposition 3.1 that P ( ) (α, s) are continuous on I (φ)o . The set I (φ)o is open and convex. Hence by Lemma 3.6, there exists B(α, δ) for some δ > 0 and L α > 0 such that α , s) − P ( ) ( α , s)| ≤ L α | α − α | for α, α ∈ B(α, δ). |P ( ) (
(3.12)
For > 0 small, µ ∈ M(α) corresponds to an element of M ( ) (α ( ) ) for some α ( ) ∈ B(α, δ) which converges to α. So, we have by (3.12) that P(α, s) ≤ P ( ) (α ( ) , s) ≤ P ( ) (α, s) + L α |α − α ( ) |. As |α ( ) − α| → 0, we have P(α, s) ≤ lim inf →0 P ( ) (α, s). Finally, let K be a compact subset of I (φ)o . Suppose the convergence lim →0 P ( ) (α, s) = P(α, s) is not uniform on K . Then there exists ε > 0, αn and n ↓ 0 such that |P ( n ) (αn , s) − P(αn , s)| ≥ ε, ∀ n ∈ N.
123
(3.13)
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Since K is compact, we may assume αn → α ∈ K ⊂ I (φ)o . Then by (3.12) |P ( n ) (αn , s) − P(α, s)| ≤ L α |αn − α| + |P ( n ) (α, s) − P(α, s)| → 0 as n goes to infinity, which contradicts (3.13) since limn→∞ P(αn , s) = P(α, s). For each α ∈ I (φ), let (α) = sup
µ∈M(α)
h(µ|P) . − log γ (w, x) dµ
Notice that Lemma 3.8 For s, t ∈ R+ 0 with s < t, we have C3 (s − t) ≤ P(α, s) − (s − t), where C = maxx∈w log γ (w, x) dP(w) and C4 = P(α, t) ≤ C 4 3 minx∈w log γ (w, x) dP(w). Proof We only show the direction “≤”. It is easy to deduce from (3.1) that 1 1 f (α, s; w, n, , δ) 1 · log ≤ log max γn (w, x) x∈w s−t n f (α, t; w, n, , δ) n ≤
n−1 1 max log γ (ϑ i w, x) x∈ϑ i w n i=0
The conclusion follows by applying ergodic theorem for maxx∈w log γ (w, x) on . Hence we have by Proposition 3.5 that Lemma 3.9 For α ∈ I (φ), the number (α) is the unique s such that P(α, s) = 0. Let ( ) (·) be the corresponding function of (·) for φ ( ) . As a consequence of Proposition 3.7 and Lemma 3.8, we see that Proposition 3.10 lim →0 ( ) (α) = (α) on I (φ)o . The convergence is uniformly on any compact subset of I (φ)o . 4 Proof of the main result We first show Theorem 1.2. As mentioned in Sect. 3.1, the ideas involved to construct Moran sets are similar to that of [21]. Nevertheless, the construction using random Bowen balls on w is more complicated than using cylinders on symbolic spaces.
be as in Lemma 2.4. Let Let (, G) be a bundle RDS as in Theorem 1.2. Let 0 , R, R γ (·, ·) be as in Sect. 3. Lemma 4.1 (cf. [21, Lemma 3.13]) The map α → (α) is upper semi-continuous on I (φ) and hence there exists a countable subset I (φ) ⊆ I (φ) such that for any α ∈ ∞ ⊆ I (φ) with limi→∞ αi = α and limi→∞ (αi ) = (α). I (φ), there exists {αi }i=1
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Proof of (i) of Theorem 1.2 We first show the upper bound. Let γ (w, x) = λ(w, x)−1 in the definition of P(α, s) and let s > (α). By Lemma 3.8, P(α, s) < 0. Let w ∈ be such that Proposition 3.1 holds. Fix < 0 /4, where 0 is in Lemma 2.4. Note that P w (α, s; ) is decreasing on . Then there exist ε > 0 and δ > 0 such that lim sup n→∞
1 log f (α, s; w, n, , δ) < −ε. n
Hence there is a number N = N (w, ε, δ) such that f (α, s; w, n, , δ) < e−εn for
w, k, δ) = ∩n≥k F(α; w, n, δ). We see that n > N . Let F(α; E w (α) ⊆
∞ !
w, k, δ). F(α;
(4.1)
k=1
w, k, δ). Let n ≥ max {k, N } and let Sn be some maximal (w; n, ) k := F(α; Fix F separated set of F(α; w, n, δ) such that
γn (w, x)s = f (α, s; w, n, , δ) < e−εn .
(4.2)
x∈Sn
k . The set of Bowen balls {Bw (x; n, 2 ) : x ∈ Sn , n ≥ k} forms an open cover of F Moreover, by Lemma 2.4 we have diamBw (x; n, 2 ) < R
n−1 max γ (w, x)
(w,x)∈
→0
as n tends to infinity. Using (4.2) and Lemma 2.4, we estimate that
diamBw (x; n, 2 )s ≤ R s γ0−s
n≥k x∈Sn
γn (w, x)s ≤ R s γ0−s
n≥k x∈Sn
e−εn < ∞
n≥k
k ) ≤ s. Hence by (4.1), we have dim H E w (α) ≤ supk dim H This implies dim H ( F
( Fk ) ≤ s. To estimate the lower bound, let γ (w, x) = η(w, x)−1 in the definition of P(α, s). Let I (φ) ⊆ I (φ) be as in Lemma 4.1 for (α) and let S be a countable dense subset of Rd containing (α) : α ∈ I (φ) . Let N0 ∈ N be such that for τ, τ ∈ A , τ | N0 = τ | N0 implies d(π(w, τ ), π(w, τ )) < /4. Let δk = 1/k. Choose {Nk }∞ k=1 to be a sequence of integers greater than N0 such that τ | Nk = τ | Nk implies |φ(π(w, τ )) − φ(π(w, τ ))| < δk .
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We first pick up w’s. For α ∈ I (φ), s ∈ S and k, j ∈ N, denote Aα,s;k, j := w ∈ : f (α, s; w, n, , δk ) > en(P(α,s)−δk ) for all n ≥ j . Let H denote the set of all w such that n−1 1 χ Aα,s;k, j (ϑ pr +q (w)) = P(Aα,s;k, j ), ∀α ∈ I (φ), s ∈ S, k, j, p, q ∈ N. n→∞ n
lim
r =0
Then P(H ) = 1 by Birkhoff ergodic theorem. Let w ∈ H and let β ∈ I (φ), we will show dim H E w (β) ≥ (β) by proving dim H E w (β) ≥ s for any s < (β) with s ∈ S. Let {αi } ⊆ I (φ) be such that limi→∞ αi = β, limi→∞ (αi ) = (α). Fix s < (β) with s ∈ S. We have by Lemma 3.3 that P
1 = 1. w ∈ : lim inf log f (αk , s; w, n, , δk ) ≥ P(αk , s) n→∞ n
(4.3)
Therefore, we can choose a sequence of integers {n k } ↑ ∞ with n k > 2 Nk such that for any k ∈ N, the set Aαk ,s;k,n k := w ∈ : f (αk , s; w, n, , δk ) ≥ en(P(αk ,s)−δk ) for n ≥ n k has measure greater than 1 − 2−k . We may assume en k (P(αk ,s)−δk ) > 1. Hence Ak := Aαk ,s;k,n k ⊆ {w ∈ : f (αk , s; w, n, , δk ) > 1 for n ≥ n k } .
(4.4)
Fix w ∈ H , we have by construction of H and (4.3) that n−1 1 χ Ak (ϑ (n k +N k )t+q w) = P(Ak ) > 1 − 2−k , ∀ k, q ∈ N. n→∞ n
lim
(4.5)
t=0
We will construct a Moran set T ⊆ E w (β) and show dim H T > s by Lemma 2.6. Let Sk ⊆ F(αk , s; w, n k , δk ) be some maximal (w; n k , )-separated set such that
γn k (w, x)s > 1.
(4.6)
x∈Sk
The existence of Sk is guaranteed by the definition of P(α, s) and (4.4). For each x ∈ Sk , we fix one symbolic representation τ = τ (x) such that π(w, τ ) = (w, x). By our choice of {Nk }∞ k=0 , we see that x → τ (x)|n k +Nk is a bijection between Sk and the set
Tk := τ (x)|n k +Nk : x ∈ Sk .
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Fix ς ∈ A . Let {m k }∞ k=1 be a sequence diverging to ∞ fast to be determined in the sequel. For k ∈ N and i ≤ m k , let Tki =
Tk , if ϑ N (k, j) w ∈ Ak ;
ς |n k +Nk , otherwise,
where N (k, j) = q
T = τ = τ 11 · · · τ 1m 1 · · · τ k1 · · · τ km k · · · : τki ∈ Tki , k ∈ N, i ≤ m k , We make the convention that the extensions are uniquely determined by adjacent words. We determine {m k }∞ k=1 so that T = {π(w, τ ) : τ ∈ T } ⊆ E w (β) n k = n k + Nk for simplicity. First choose m 1 such that and dim H (T ) ≥ s. Write m 1 −1 1 n1 t χ A1 (ϑ w) > 1 − 2−1 and m1 t=0
1 l
l−1
n 2 t+q χ A2 (ϑ w) > 1 − 2−2 , ∀ l ≥
t=0
m1 , 0 ≤ q ≤ n 2 − 1.
n2
Suppose m 1 , · · · , m k−1 have been constructed well. By (4.5), we can choose m k such that
n k+1 m k > max 2m k−1 , 2 m k −1 1 n k t+ k−1 njm j j=1 χ Ak (ϑ w) > 1 − 2−k , and mk t=0
1 l
l−1 t=0
n k+1 t+q χ Fk+1 (ϑ w) ≥ 1 − 2−k , ∀ l ≥
mk , 0 ≤ q ≤ n k+1 − 1.
n k+1
In this way, we can choose {m k }∞ k=1 well. Now relabel the sequence T11 · · · T1m 1 · · · Tk1 · · · Tkm k · · ·
∞ as Tk∗ k=1 . Denote by n ∗k the corresponding length of words in Tk∗ and δk∗ the corresponding δ j ’s. Let τ = τ 1 τ 2 · · · τ k · · ·, τ = τ 1 τ 2 · · · τ k · · · ∈ T . One easily verifies that: (i) (ii)
If τ1 · · · τk = τ1 · · · τk , then dnw (π(w, τ ), π(w, τ )) ≥ 21 for n = n ∗1 + · · · n ∗k ; If τ1 · · · τk = τ1 · · · τk , then dnw (π(w, τ ), π(w, τ )) ≤ 14 for n = n ∗1 + · · · n ∗k ;
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(iii)
109
If n ∗1 + · · · n ∗k ≤ n < n ∗1 + · · · n ∗k+1 for k large, then |φn (π(w, τ )) − nβ| ≤ 4δk∗ .
With (iii), we see that T ⊆ E w (β). With (i), (ii), we will show that T forms a strong Moran set and hence Lemma 2.6 applies and gives dim H (T ) ≥ s. To fit T into the model introduced in Sect. 2.4, let
Dk = j = ( j1 , j2 , . . . , jk ) : 1 ≤ ji ≤ #Ti ∗ , 1 ≤ i ≤ k . Repack τ ∈ A as τ = τ 1 · · · τ k · · · so that each τ k has length n ∗k . For each j = ( j1 , j2 , . . . , jk ) ∈ Dk , let j := π(w, τ ) : τ 1 · · · τ k = τ j1 · · · τ jk , τ ∈ A
where τ jk denotes the jk -th element of Tk∗ . It is clear that j : j ∈ Dk satisfies (i), (ii) of the requirement of a strong Moran structure and T =
" !
j .
k>0 j∈Dk
Let rj = maxx∈j ∩T γl (w, x), where l = n ∗1 + · · · n ∗k if j ∈ Dk . Then by (i) above and Lemma 2.4, we have 1 1
B j ∩ T , Rrj ∩ B j ∩ T , Rrj = Ø, if j = j . 4 4 Furthermore, by (ii) above, we have dnw (x, x ) < /4 for x, x ∈ j ∩ T . Hence by (iv) of Lemma 2.4, we have that rj : j ∈ D satisfies the requirement (iv) of a
strong Moran set. By applying Lemma 2.6, we have dim H T ≥ lim inf k→∞ ti , where ti satisfies j∈Di rjti = 1. What left is to show lim inf k→∞ ti ≥ s. Suppose otherwise, there is tk < s for k arbitrarily large. Let i j be the trace back position of k in the sequence {i j}i∈N, j≤m i . Write i j ≤ i j if i j is before i j in lexical order. Then by using (4.5), (4.6) and the fact that n i > 2 Ni for all i, we see that for k large 1 < C −k γ0sl
i j ≤i j,ϑ N (i
, j)
w∈Ai x∈Si
γn i (w, x)s ≤
rjtk = 1,
j∈Dk
where l = kp + i j Ni + ϑ N (i , j ) w∈ A n i . This is a contradiction. i
For dim H Dw (φ), we follow the idea in [20]. For (w, x) ∈ , let Aφ (w, x) be the set of accumulation points of n1 φn (w, x). For α, β ∈ I (φ), let
w (α, β) = x ∈ w : α, β ∈ Aφ (w, x) . E
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w (α) for E
w (α, β). We have When α = β, simply write E Lemma 4.2 There is a set with P() = 1 such that for any w ∈ and α, β ∈ I (φ), min
sup
α ∈{α,β} µ∈M(α )
h(µ|P) log η(w, x) dµ
w (α, β)) ≤ dim H ( E ≤ min
sup
α ∈{α,β} µ∈M(α )
h(µ|P) . log λ(w, x) dµ (4.7)
w (α, β) ⊆ Proof Let be as in the construction for (i) of Theorem 1.2. Observe that E
E w (α ) for α = α or β. The same argument as there for the upper bound gives
w (α, β) ≤ dim H E
w (α ) ≤ dim H E
sup
µ∈M(α )
h(µ|P) . log λ(w, x) dµ
For the lower bound, let s < L.H.S. of (4.7). Let {αk }∞ k=1 be such that α2k = in the construction in the proof of α, α2k−1 = β for k ∈ N. Using this {αk }∞ k=1
w (α, β) with dim H (T ) ≥ s. The (i) of Theorem 1.2 gives a strong Moran set T ⊆ E lemma is proved. Proof of (ii) of Theorem 1.2 It suffices to show the lower bound since clearly dim H (Dw (φ)) ≤ dim H (w ) ≤
sup
µ∈MP ()
h(µ|P) , log λ(w, x) dµ
where the second inequality holds by letting φ ≡ 0 in (i) of Theorem 1.2. Let γ (w, x) = η(w, x)−1 in the definition of P(α, s). By Lemma 4.1, there is some α ∈ I (φ) such that (α) = sup (α ). α ∈I (φ)
Let β ∈ I (φ) be another point that differs from α. For ρ ∈ (0, 1), let αρ = (1 − ρ)α + ρβ. For any ε > 0, let µ1 , µ2 ∈ MP () be such that
h(µ2 |P) ε ε h(µ1 |P) ≥ (α) − , ≥ (β) − . 2 2 log η(w, x) dµ1 log η(w, x) dµ2
Then we have (αρ ) ≥
123
h (((1 − ρ)µ1 + ρµ2 )|P) ≥ (1 − ρ)(α) + ρ(β) − 2ε log η(w, x)d((1 − ρ)µ1 + ρµ2 )
The multifractal analysis of Birkhoff averages for conformal repellers
111
for ρ small. So by Lemma 4.2,
w (αρ , α) ≥ (1 − ρ)(α) + ρ(β) − 2ε. dim H (Dw (φ)) ≥ dim H E Letting ρ → 0, the lower bound follows since ε > 0 is arbitrary.
Proof of Theorem 1.1 The conformal repeller (0 , f ) can be considered as an almost conformal repeller (, G) := ( × 0 , ϑ × f ) with λ(w, x) = η(w, x) = Dx f . Let φ : U → Rd be continuous. It can be regarded as a random function on by setting φ(w, x) := φ(x). ( ) For each > 0, denote by h w the homeomorphism in Proposition 2.5 that sending ( ) ( ) x ∈ 0 to xw ∈ w . Then φ can be lifted to be a random function on by letting φ ( ) (w, x) = φ(h ( ) w (x)), ∀ (w, x) ∈ × 0 . Denote by I (φ ( ) ) =
φ ( ) dµ : µ ∈ MP () and
φ ( ) dµ = α , ∀ α ∈ I (φ ( ) ). M ( ) (α) = µ ∈ MP () : Note that lim →0 h ( ) w = id in probability [9, Proposition 4.2]. Hence lim φ − φ ( ) = lim
→0
→0
sup |φ(x) − φ(h ( ) w (x))| dP(w) = 0.
x∈0
Apply Proposition 3.10 for γ = λ−1 gives lim
sup
→0 µ∈M ( ) (α)
h(µ|P) log λ(w, x) dµ
= (α) = dim H E(α)
(4.8)
and the convergence is uniform on any subset K of I (φ)o . compact ( ) of f as in the assumption, (( ) , G ( ) ) For any small random perturbation gw >0 are expanding almost conformal repellers ([9, Theorem 4.3]) with ( ) ( ) (x) and λ( ) (w, x) := (Dgw (x))−1 −1 . η( ) (w, x) := Dgw
Moreover, they can be modelled by a mixing SFT. Denote by ( ) (α) Ew
= x∈
( ) w
n−1 1 ( ) : lim φ(g (w, i)x) = α . n→∞ n i=0
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( )
( )
( )
Note that h w is a homeomorphism satisfying gw ◦ h w = h ϑw ◦ f . By Theorem 1.2, sup µ∈M ( ) (α)
h(µ|P) log η( ) (w, h w x) dµ
( ) (α) ≤ dim H E w
≤
sup µ∈M ( ) (α)
h(µ|P) ( ) log λ (w, h w x) dµ
(4.9) ( )
( )
As lim →0 gw = f in L 1 (, C 1+β (U, M)) and lim →0 h w = id in probability, we see that log λ( ) (w, h w x) dµ − log λ(w, x) dµ → 0, as → 0, uniformly for µ ∈ MP () and similarly for η( ) . So, (i) of Theorem 1.1 follows by (4.8).
Note that if the convex set I (φ) = α : E(α) = Ø has more than two points, so does I (φ ( ) ) for small. Hence (ii) of Theorem 1.1 is a direct consequence of ( ) Theorem 1.2 using the fact that lim →0 dim H w = dim H 0 in probability (see [9, Theorem 4.3], this can also be seen from (4.9) by letting φ ≡ 0). Acknowledgments The author thanks Prof. Peidong Liu for valuable discussions on random perturbations and Prof. Dejun Feng for sharing insights on multifractal analysis. This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).
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