Pointwise dimension for a class of measures on limit sets
aa r X i v : . [ m a t h . D S ] A ug Pointwise dimension for a class of measures on limit sets
Eugen Mihailescu
Abstract
We study the pointwise dimension for a new class of projection measures on arbitrary fractallimit sets without separation conditions. We prove that the pointwise dimension exists a.e. forthis class of measures associated to equilibrium states, and it is given by a formula in terms ofLyapunov exponents and a certain type of entropy. Thus these measures are exact dimensional.Self-conformal measures belong to the above class of measures, and this allows us to obtaina new geometric formula for their pointwise dimension. Thus for self-conformal measures weobtain also a geometric formula for their projection entropy.
Mathematics Subject Classification 2010:
Keywords:
Pointwise dimension; exact dimensional measures; densities; projection entropy;hyperbolic endomorphisms; self-conformal measures; Lyapunov exponents.
We study the pointwise dimension and the exact dimensionality for a new class of probability mea-sures on fractal limit sets of arbitrary conformal iterated function systems. We obtain a generalformula for the pointwise dimension of these new measures associated to equilibrium states. More-over we investigate the intricate process of interlacing of the generic iterates and the gaps consistingof non-generic iterates in a typical trajectory, and how they influence the local densities of thesemeasures and their pointwise dimensions. As a particular case we obtain a geometric formula forthe pointwise dimension of self-conformal measures for systems without separation conditions. Forself-conformal measures our geometric formula relates their projection entropy (defined in [4]), withthe average rate of growth for the generic number of overlappings in the limit set. Therefore ourresult gives a geometric interpretation for the pointwise dimension and for the projection entropyin the case of self-conformal measures, and allows to estimate these quantities more directly.The general setting is the following: let S = { φ i , i ∈ I } be an arbitrary finite iterated functionsystem of smooth conformal injective contractions of a compact set with nonempty interior V ⊂ R D .We do not assume any kind of separation condition for S . The limit set of S is then given by:Λ = ∪ ω ∈ Σ + I ∩ n ≥ φ ω ...ω n ( V ) , where Σ + I is the 1-sided symbolic space on | I | symbols, and ω = ( ω , . . . , ω n , . . . ) ∈ Σ + I arbitrary.Denote by [ ω . . . ω n ] the cylinder on the first n elements of ω , and by φ i ...i p := φ i ◦ . . . ◦ φ i p . The1hift σ : Σ + I → Σ + I is given by σ ( ω ) = ( ω , ω , . . . ) , ω ∈ Σ + I . Also let the canonical projection π : Σ + I → Λ , π ( ω ) := φ ω ω ... ( V )Now consider the endomorphism which associates a dynamical system to S ,Φ : Σ + I × Λ → Σ + I × Λ , Φ( ω, x ) = ( σω, φ ω ( x )) , ( ω, x ) ∈ Σ + I × ΛIf π is the projection on the first coordinate of Σ + I × Λ, then the following diagram is commutative,Σ + I × Λ Φ −→ Σ + I × Λ π ↓ ↓ π Σ + I σ −→ Σ + I (1)The endomorphism Φ has a type of hyperbolic structure, since it is expanding in the first coordinateand contracting in the second coordinate (due to the uniform contractions in S ).Consider a H¨older continuous potential ψ : Σ + I → R , and let µ + be the equilibrium measure of ψ onΣ + I . Define ˆ ψ := ψ ◦ π : Σ + I × Λ → R , which is also H¨older continuous. Thus since Φ is hyperbolic,there is a unique equilibrium state ˆ µ := µ ˆ ψ of ˆ ψ on Σ + I × Λ (as in [2], [7]). Hence π ∗ ˆ µ = µ + .Moreover we have the projection on the second coordinate, π : Σ + I × Λ → Λ , π ( ω, x ) = x The main focus of this paper is the measure π ∗ ˆ µ , which in general is different from thecanonical projection measure π ∗ ˆ µ . Let us denote these two measures on Λ by, ν := ( π ◦ π ) ∗ ˆ µ = π ∗ µ + , and ν := π ∗ ˆ µ (2)Since Φ n ( ω, x ) = ( σ n ω, φ ω n ...ω ( x )) reverses the order of ω , . . . , ω n in its second coordinate, andsince ˆ µ is Φ n -invariant, we call ν = π , ∗ ˆ µ an order-reversing projection measure . This is in contrastwith the construction of ν .Some important notions in Dimension Theory are those of lower/upper pointwise dimensionsof a measure, and the notion of exact dimensional measures (see for eg [14]). In general for aprobability Borel measure µ on a metric space X , the lower pointwise dimension of µ at x ∈ X isdefined as: δ ( µ )( x ) := lim inf r → log µ ( B ( x, r ))log r , and the upper pointwise dimension of µ at x ∈ X is: δ ( µ )( x ) := lim sup r → log µ ( B ( x, r ))log r If δ ( µ )( x ) = δ ( µ )( x ) then we call the common value the pointwise dimension of µ at x , denoted by δ ( µ )( x ). If for µ -a.e x ∈ X , the pointwise dimension δ ( µ )( x ) exists and is constant, we say that µ is exact dimensional . In this case there is a value α ∈ R s.t for µ -a.e x ∈ X , δ ( µ )( x ) := δ ( µ )( x ) = δ ( µ )( x ) = α projection entropy for an arbitrary σ -invariantprobability measure µ on Σ + I , namely h π ( σ, µ ) := H µ ( P| σ − π − γ ) − H µ ( P| π − γ ) , (3)where P is the partition with 0-cylinders { [ i ] , i ∈ I } of Σ + I , and γ is the σ -algebra of Borel sets in R d . If µ is ergodic, then it was shown in [4] that the measure π ∗ µ is exact dimensional on Λ, andthat for µ -a.e ω ∈ Σ + I , its pointwise dimension is given by: δ ( π ∗ µ )( πω ) = h π ( σ, µ ) − R Σ + I log | φ ω ( πσω ) | dµ ( ω ) (4)In our case, this implies that ν = π ∗ µ + is exact dimensional. Notice that in general, ν is notequal to ν . So the problem of the pointwise dimension of ν must be studied separately, and wedo this in the sequel.Let us denote the stable Lyapunov exponent of the endomorphism Φ with respect to ˆ µ by, χ s (ˆ µ ) := Z Σ + I × Λ log | φ ω ( x ) | d ˆ µ ( ω, x )We will use the Jacobian in the sense of Parry [12]; so let us consider the Jacobian J Φ (ˆ µ ) of aΦ-invariant measure ˆ µ on Σ + I × Λ (see also [3]). It is clear that for ˆ µ -a.e ( ω, x ) ∈ Σ + I × Λ, we have J Φ (ˆ µ ) ≥
1. From definition we have for ˆ µ -a.e ( ω, x ) ∈ Σ + I × Λ, J Φ (ˆ µ )( ω, x ) = lim r → ˆ µ (Φ( B (( ω, x ) , r )))ˆ µ ( B (( ω, x ) , r ))From the Chain Rule for Jacobians, J Φ n (ˆ µ )( ω, x ) = J Φ (ˆ µ )(Φ n − ( ω, x )) · . . . · J Φ (ˆ µ )( ω, x ) for n ≥ J Φ (ˆ µ )( · , · ), we havethat for ˆ µ -a.e ( ω, x ) ∈ Σ + I × Λ,log J Φ n (ˆ µ )( ω, x ) n −→ n →∞ Z Σ + I × Λ log J Φ (ˆ µ )( η, y ) d ˆ µ ( η, y ) (5)Ruelle introduced ([18], [19]) the notion of folding entropy F f ( ν ) of a measure ν invariant withrespect to an endomorphism f : X → X on a Lebesgue space X , as the conditional entropy H ν ( ǫ | f − ǫ ), where ǫ is the point partition of X and f − ǫ is the partition with the fibers of f . Infact from [12], [18] it follows that F f ( ν ) = R X log J f ( ν ) dν . Thus in our case for the Φ-invariantmeasure ˆ µ on Σ + I × Λ, we have F Φ (ˆ µ ) = Z Σ + I × Λ log J Φ (ˆ µ ) d ˆ µ (6)The main result of the current paper is Theorem 1, saying that the pointwise dimension of ν is related to the folding entropy of the lift measure ˆ µ . Recall that in general ν is not equal to ν := π ∗ ˆ µ , thus usually δ ( ν ) is not given by (4).3 heorem 1. Let S be a finite conformal iterated function system, and ψ be a H¨older continuouspotential on Σ + I with equilibrium measure µ + , and denote by ˆ µ the equilibrium measure of ψ ◦ π on Σ + I × Λ with respect to Φ . Denote by ν := π ∗ ˆ µ . Then for ν -a.e x ∈ Λ , δ ( ν )( x ) = F Φ (ˆ µ ) − h σ ( µ + ) χ s (ˆ µ ) In particular, the measure ν is exact dimensional on Λ . In our case the folding entropy turns out to be related to the overlap number of ˆ µ . Thenotion of overlap number o ( S , µ g ) for an equilibrium measure µ g of a H¨older continuous potential g : Σ + I × Λ → R was introduced in [11], and represents an average asymptotic rate of growth forthe number of µ g -generic overlaps of order n in Λ. Namely for any τ > τ -genericpreimages with respect to µ g having the same n -iterates as ( ω, x ),∆ n (( ω, x ) , τ, µ g ) := { ( η , . . . , η n ) ∈ I n , ∃ y ∈ Λ , φ ω n ...ω ( x ) = φ η n ...η ( y ) , | S n g ( η, y ) n − Z Σ + I × Λ g dµ ψ | < τ } , where ( ω, x ) ∈ Σ + I × Λ and S n g ( η, y ) is the consecutive sum of g with respect to Φ. Denote by b n (( ω, x ) , τ, µ g ) := Card ∆ n (( ω, x ) , τ, µ g )Then, in [11] we showed that the following limit exists and defines the overlap number of µ g , o ( S , µ g ) = lim τ → lim n →∞ n Z Σ + I × Λ log b n (( ω, x ) , τ, µ g ) dµ g ( ω, x )Moreover, there is a relation between the overlap number and the folding entropy of µ g , o ( S , µ g ) = exp( F Φ ( µ g )) (7)When µ g = µ is the measure of maximal entropy, we denote o ( S , µ ) by o ( S ) and call it thetopological overlap number of S . Notice that all preimages are generic for the measure of maximalentropy, so o ( S ) represents an asymptotic rate of growth of the total number of overlaps betweenthe n -iterates of type φ i ...i n (Λ), when i , . . . , i n ∈ I and n → ∞ .Combining (7) and Theorem 1 we obtain a formula for the pointwise dimension of ν in terms ofits overlap number: Corollary 1.
In the setting of Theorem 1, the pointwise dimension of ν satisfies for ν -a.e x ∈ Λ , δ ( ν )( x ) = exp( o ( S , ˆ µ )) − h σ ( µ + ) χ s (ˆ µ )In particular, o ( S , ˆ µ ) and thus δ ( ν ) can be easily computed above if for some m ≥
1, thereexists a constant number of overlaps between the sets of type φ i ...i m (Λ), modulo ˆ µ .4n important case is that of self-conformal measures , i.e π -projections of Bernoulli measureson Λ. To fix notation, given the system S and the probability vector p = ( p , . . . , p | I | ), let ν p bethe corresponding Bernoulli measure on Σ + I . Any Bernoulli measure ν p on Σ + I is the equilibriummeasure of some H¨older continuous potential ψ p . In this case the Bernoulli measure ν p is themeasure µ + from before. Thus we have the above construction and results. Then denote by ˆ µ p thelift of ν p to Σ + I × Λ obtained as the equilibrium measure of ψ p ◦ π , and by ν , p , ν , p , the associatedprojected measures ν , ν .For self-conformal measures (i.e projections ν , p = π ∗ ν p of Bernoulli measures ν p on Σ + I ), weshowed in [11] that the measures ν , p and ν , p are equal. Therefore, δ ( ν , p ) = δ ( ν , p ) . We obtainthen below a formula for the projection entropy h π ( σ, ν p ) of the measure ν , p in terms of its overlapnumber. Theorem 2.
In the setting of Theorem 1, let ν p be a Bernoulli measure on Σ + I , and let ν , p , ν , p be the associated projection measures on Λ . Then ν , p = ν , p , and for ν , p -a.e x ∈ Λ , δ ( ν , p )( x ) = δ ( ν , p )( x ) = exp( o ( S , ˆ µ p )) − h σ ( ν p ) χ s (ˆ µ p ) Moreover the projection entropy of ν p is determined by, h π ( σ, ν p ) = h σ ( ν p ) − exp( o ( S , ˆ µ p )) In particular the projection entropy of the measure µ of maximal entropy on Σ + I is obtained as: h π ( σ, µ ) = log | I | − exp( o ( S )) Recall the setting from Section 1, where ψ is a H¨older continuous potential on Σ + I , µ + is theequilibrium measure of ψ on Σ + I , and ˆ µ denotes the equilibrium measure µ ˆ ψ of ˆ ψ := ψ ◦ π onΣ + I × Λ. We consider the measurable partition ξ of Σ + I × Λ with the fibers of the projection π : Σ + I × Λ → Λ, and the associated conditional measures µ ω of ˆ µ = µ ˆ ψ defined for µ + -a.e ω ∈ Σ + I (see [16]); from above, µ + = π ∗ ˆ µ . For µ + -a.e ω ∈ Σ + I , the conditional measure µ ω is defined on π − ω = { ω } × Λ. It is clear that the factor space Σ + I × Λ /ξ is equal to Σ + I , and the correspondingfactor measure of ˆ µ satisfies, ˆ µ ξ ( A ) = ˆ µ ( A × Λ) = µ + ( A ) , for any measurable set A ⊂ Σ + I . Thus ˆ µ ξ = µ + . From the properties of the conditional measures,we obtain that for any borelian set E in Σ + I × Λ,ˆ µ ( E ) = Z π E ( Z { ω }× Λ χ E dµ ω ) dµ + ( ω ) = Z π E µ ω ( E ∩ { ω } × Λ) dµ + ( ω ) (8)For a Borel set A in Λ, we have for µ + -a.e ω ∈ Σ + I , µ ω ( A ) = lim n →∞ ˆ µ ([ ω . . . ω n ] × A ) µ + ([ ω . . . ω n ]) (9)5 otation. Two quantities Q , Q are called comparable Q ≈ Q , if there is a constant C >
C Q ≤ Q ≤ CQ In general the comparability constant C is independent of the parameters appearing in Q , Q . (cid:3) The above conditional measures µ ω are defined on { ω } × Λ, so actually they can be consideredas probability measures on Λ. In the next Lemma, we compare µ ω ( A ) with µ η ( A ), and show thatˆ µ has an “almost” product structure with respect to µ + and µ ω . Lemma 1.
There exists a constant
C > so that for µ + -a.e ω, η ∈ Σ + I and any Borel set A ⊂ Λ , C µ η ( A ) ≤ µ ω ( A ) ≤ Cµ η ( A ) Moreover for any Borel sets A ⊂ Σ + I , A ⊂ Λ and for µ + -a.e ω ∈ Σ + I , we have: C µ + ( A ) · µ ω ( A ) ≤ ˆ µ ( A × A ) ≤ Cµ + ( A ) · µ ω ( A ) In particular there is a constant
C > such that, for µ + -a.e ω ∈ Σ + I and any Borel set A ⊂ Λ , C µ ω ( A ) ≤ ν ( A ) ≤ Cµ ω ( A ) Proof.
First recall formula (9) for the conditional measure µ ω . From the Φ-invariance of ˆ µ ,ˆ µ ([ ω . . . ω n ] × A ) = X i ∈ I ˆ µ ([ iω . . . ω n ] × φ − i A ) (10)Now we can cover the set A with small disjoint balls (modulo ˆ µ ), so it is enough to consider sucha small ball B = A ⊂ Λ. The general case will follow then from this.Recall also that for any i , . . . , i n ∈ I, n ≥ φ i ...i n denotes the composition φ i ◦ . . . ◦ φ i n . We have aBounded Distortion Property, since we work with smooth conformal contractions φ i , i.e there existsa constant C > x, y, n, i , . . . , i n , we have | φ ′ i ...i n ( x ) | ≤ C | φ ′ i ...i n ( y ) | . Since thecontractions φ i are conformal, let i , . . . , i p ∈ I such that φ − i p . . . φ − i B = φ − i ...i p B is a ball B ( x , r )of a fixed radius r . In this way we inflate B along any backward trajectory i = ( i , i , . . . ) ∈ Σ + I upto some maximal order p ( i ) ≥
1, so that φ − i ...i p ( i ) B contains a ball of radius C r and it is containedin a ball of radius r , for some fixed constant C independent of B, i . Then by using successivelythe Φ-invariance of ˆ µ , relation (10) becomes:ˆ µ ([ ω . . . ω n ] × B ) = X i ∈ I ˆ µ (cid:0) [ i p ( i ) . . . i ω . . . ω n ] × φ − i ...i p ( i ) B (cid:1) (11)Without loss of generality one can assume that φ − i ...i p ( i ) B is a ball of radius r . Notice that the set[ i p ( i ) . . . i ω . . . ω n ] × φ − i ...i p ( i ) B is the Bowen ball [ i p ( i ) . . . i ω . . . ω n ] × B ( x , r ) for Φ. Since ˆ µ isthe equilibrium state of ψ ◦ π , and since P Φ ( ψ ◦ π ) = P σ ( ψ ) := P ( ψ ), we have:ˆ µ ([ i p ( i ) . . . i ω . . . ω n ] × φ − i ...i p ( i ) B ) ≈ exp( S n + p ( i ) ψ ( i p ( i ) . . . i ω . . . ω n ) − ( n + p ( i )) P ( ψ )) ≈≈ µ + ([ ω . . . ω n ]) · ˆ µ ([ i p ( i ) . . . i ] × φ − i ...i p ( i ) B ) , (12)6here the comparability constant above does not depend on B, i , . . . , i p ( i ) , n . If we choose anotherfinite sequence ( η , . . . , η n ) ∈ I n , then we can take again for any i ∈ Σ + I , the same indices i , . . . , i p ( i ) such that φ − i ...i p ( i ) B is a ball of radius r , thus,ˆ µ ([ i p ( i ) . . . i η . . . η n ] × φ − i ...i p ( i ) B ) ≈ exp( S n + p ( i ) ψ ( i p ( i ) . . . i η . . . η n ) − ( n + p ( i )) P ( ψ )) ≈ µ + ([ η . . . η n ]) · ˆ µ ([ i p ( i ) . . . i ] × φ − i ...i p ( i ) B ) , (13)where the comparability constant does not depend on B, i , . . . , i p ( i ) , n . But the cover of A withsmall balls of type B and the above process of inflating these balls along prehistories i to balls ofradius r , can be done along any trajectories ω, η . Thus by (11) and using the uniform estimates(12), (13) and (9), we obtain that there exists a constant C > µ + -a.e ω, η ∈ Σ + I ,1 C µ η ( A ) ≤ µ ω ( A ) ≤ Cµ η ( A ) (14)Thus from (14) and the desintegration formula (8) for ˆ µ , it follows that there exists a constant(denoted also by C ) such that for all Borel sets A ⊂ Σ + I , A ⊂ Λ,1
C µ + ( A ) · µ ω ( A ) ≤ ˆ µ ( A × A ) ≤ Cµ + ( A ) · µ ω ( A )For the final statement, recall that ν = π ∗ ˆ µ , so ν ( A ) = ˆ µ (Σ + I × A ). Then we use the last displayedformula to obtain a constant C >
0, such that for µ + -a.e ω ∈ Σ + I and any Borel set A ⊂ Λ, we have1
C µ ω ( A ) ≤ ν ( A ) ≤ Cµ ω ( A )Now we prove Theorem 1, i.e. the formula for the pointwise dimension of ν . Proof of Theorem 1.
First we prove the upper estimate for the pointwise dimension of ν . For any n ≥ , ( ω, x ) ∈ Σ + I × Λ, Φ n ( ω, x ) = ( σ n ω, φ ω n ...ω ( x )). From Birkhoff Ergodic Theorem applied to the Φ-invariantmeasure ˆ µ , it follows that for ˆ µ -a.e ( ω, x ) ∈ Σ + I × Λ,1 n log | φ ′ ω n ...ω | ( x ) −→ n X i ∈ I Z [ i ] × Λ log | φ ′ i ( x ) | d ˆ µ ( ω, x ) = χ s (ˆ µ )On the other hand, from the Chain Rule for Jacobians, Birkhoff Ergodic Theorem and the formulafor folding entropy F Φ (ˆ µ ), it follows that for ˆ µ -a.e ( ω, x ) ∈ Σ + I × Λ,1 n log J Φ n (ˆ µ )( ω, x ) −→ n F Φ (ˆ µ )Thus for a set of ( ω, x ) ∈ Σ + I × Λ of full ˆ µ -measure,1 n log | φ ′ ω n ...ω | ( x ) −→ n X i ∈ I Z [ i ] × Λ log | φ ′ i ( x ) | d ˆ µ ( ω, x ) = χ s (ˆ µ ) , and 1 n log J Φ n (ˆ µ )( ω, x ) −→ n F Φ (ˆ µ )7e now want to prove that the Jacobian J Φ n (ˆ µ )( ω, x ) depends basically only on ω , . . . , ω n , i.ethere exists a constant C > n ≥
1, and ˆ µ -a.e ( η, x ) ∈ [ ω . . . ω n ] × Λ,1
C J Φ n (ˆ µ )( η, x ) ≤ J Φ n (ˆ µ )( ω, x ) ≤ CJ Φ n (ˆ µ )( η, x ) (15)In order to prove this, notice that if r >
0, and p > diam [ ω . . . ω n + p ] = r , then J Φ n (ˆ µ )( ω, x ) = lim r → ,p →∞ ˆ µ (Φ n ([ ω . . . ω n + p ] × B ( x, r ))ˆ µ ([ ω . . . ω n + p ] × B ( x, r )) (16)In our case, Φ n ([ ω . . . ω n + p ] × B ( x, r )) = [ ω n +1 . . . ω n + p ] × φ ω n ...ω B ( x, r ). If η ∈ [ ω . . . ω n ],Φ n ([ η . . . η n + p ] × B ( x, r )) = [ η n +1 . . . η n + p ] × φ ω n ...ω B ( x, r )But from Lemma 1 there exists a constant C > µ + -a.e ω ∈ Σ + I , and any n, p ≥ C µ + ([ ω n +1 . . . ω n + p ]) µ ω ( φ ω n ...ω B ( x, r )) ≤ ˆ µ ([ ω n +1 . . . ω n + p ] × φ ω n ...ω B ( x, r )) ≤≤ Cµ + ([ ω n +1 . . . ω n + p ]) µ ω ( φ ω n ...ω B ( x, r )) , (17)and similarly for ˆ µ ([ η n +1 . . . η n + p ] × φ ω n ...ω B ( x, r )). Hence in view of (16) and (17), we have onlyto compare the following quantities, µ + ([ ω n +1 . . . ω n + p ]) · µ ω ( φ ω n ...ω B ( x, r )) µ + ([ ω . . . ω n + p ]) · µ ω ( B ( x, r )) and µ + ([ η n +1 . . . η n + p ]) · µ ω ( φ ω n ...ω B ( x, r )) µ + ([ η . . . η n + p ]) · µ ω ( B ( x, r ))However recall that η ∈ [ ω . . . ω n ], thus there exists a constant K > | S n ψ ( η . . . η n . . . ) − S n ψ ( ω . . . ω n . . . ) | ≤ K, (18)since ψ is H¨older continuous and σ is expanding on Σ + I . The same argument also implies that S n + p ψ ( ω . . . ω n + p . . . ) is determined in fact only by the first n + p coordinates (modulo an additiveconstant). Since µ + is the equilibrium measure of ψ on Σ + I , then µ + ([ ω n +1 . . . ω n + p ]) µ + ([ ω . . . ω n + p ]) ≈ exp( S p ψ ( ω n +1 . . . ω n + p . . . ) − pP ( ψ ))exp( S n + p ψ ( ω . . . ω n + p ) − ( n + p ) P ( ψ )) and ,µ + ([ η n +1 . . . η n + p ]) µ + ([ η . . . η n + p ]) ≈ exp( S p ψ ( η n +1 . . . η n + p . . . ) − pP ( ψ ))exp( S n + p ψ ( η . . . η n + p ) − ( n + p ) P ( ψ )) , (19)where the comparability constant does not depend on n, p, ω, η . But we have: S n + p ψ ( ω . . . ω n + p . . . ) = S n ψ ( ω . . . ω n + p . . . ) + S p ψ ( ω n +1 . . . ω n + p . . . ). And similary for S n + p ψ ( η . . . η n + p . . . ). Therefore,using (16), (17), (18) and (19), we obtain the Jacobians inequalites in (15).Let us take now, for any n > ε >
0, the Borel set in Σ + I × Λ: A ( n, ε ) := { ( ω, x ) , (cid:12)(cid:12) log | φ ′ ω n ...ω ( x ) | n − χ s (ˆ µ ) (cid:12)(cid:12) < ε, | log J Φ n (ˆ µ )( ω, x ) n − F Φ (ˆ µ ) | < ε, and | S n ψ ( ω ) n − Z ψdµ + | < ε } Then from Birkhoff Ergodic Theorem, for any ε >
0, ˆ µ ( A ( n, ε )) −→ n →∞
1. From (15), if ( ω, x ) ∈ A ( n, ε )and η ∈ [ ω . . . ω n ], then ( η, x ) ∈ A ( n, ε ), so for any δ > µ + ( { ω ∈ Σ + I , ν ( π ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ)) > − δ } ) −→ n
18e have from (15) that [ ω . . . ω n ] × π A ( n, ε ) ⊂ A ( n, ε ). Thus from Lemma 1, for n > n ( δ ),ˆ µ ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ) > C (1 − δ ) · ˆ µ ([ ω . . . ω n ] × Λ) (20)Let now r n := 2 ε | φ ′ ω n ...ω ( x ) | , for ( ω, x ) ∈ A ( n, ε ) ∩ [ ω . . . ω n ] × Λ. With y = φ ω n ...ω ( x ) wehave ν ( B ( y, r n )) ≥ ν ( φ ω n ...ω ( π ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ)), and then since ν = π ∗ ˆ µ , we obtain ν ( B ( y, r n )) ≥ ˆ µ (Σ + I × π (Φ n ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ))) ≥ ˆ µ (Φ n ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ))But Φ n is injective on the cylinder [ ω . . . ω n ] × Λ since φ j are injective. Thus we can apply theJacobian formula for the measure of the Φ n -iterate in the last term of last inequality above, ν ( B ( y, r n )) ≥ ˆ µ (Φ n ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ)) = Z A ( n,ε ) ∩ [ ω ...ω n ] × Λ J Φ n (ˆ µ ) d ˆ µ ≥ exp (cid:0) n ( F Φ (ˆ µ ) − ε ) · ˆ µ ( A ( n, ε ) ∩ [ ω . . . ω n ] × Λ (cid:1) (21)Recall that µ + ([ ω . . . ω n ]) = ˆ µ ([ ω . . . ω n ] × Λ). Then, from (21) and (20) we obtain: ν ( B ( y, r n )) ≥ exp( n ( F Φ (ˆ µ ) − ε ))(1 − δ ) Cµ + ([ ω . . . ω n ]) ≥ C (1 − δ ) e n ( F Φ (ˆ µ ) − ε ) exp( S n ψ ( ω ) − nP ( ψ )) , (22)where C is independent of n, ω, x, y , and P ( ψ ) := P σ ( ψ ). Since µ + is the equilibrium state of ψ , P ( ψ ) = h ( µ + ) + Z ψ dµ + But from the definition of A ( n, ε ), for any ( ω, x ) ∈ A ( n, ε ), we have e n ( χ s (ˆ µ )+ ε ) ≥ r n = 2 ε | φ ω n ...ω ( x ) | ≥ e n ( χ s (ˆ µ ) − ε ) , hence ,n ( χ s (ˆ µ ) + ε ) ≥ log r n ≥ n ( χ s (ˆ µ ) − ε ) (23)From (22), (23) and the above formula for pressure, we obtain that for ν -a.e y ∈ Λ, δ ( ν )( y ) = lim r → log ν ( B ( y, r ))log r ≤ F Φ (ˆ µ ) − h ( µ + ) χ s (ˆ µ ) = h ( µ + ) − F Φ (ˆ µ ) | χ s (ˆ µ ) | , (24)which proves the upper estimate for the (upper) pointwise dimension of ν .Now we prove the more difficult lower estimate for the pointwise dimension of ν . Define forany m ≥ ε >
0, the following Borel set in Σ + I × Λ,˜ A ( m, ε ) := (cid:8) ( ω, x ) ∈ Σ + I × Λ , | n log | φ ′ ω n ...ω ( x ) | − χ s (ˆ µ ) | < ε, and | n log J Φ n (ˆ µ )( ω, x ) − F Φ n (ˆ µ ) | < ε, and | n S n ψ ( ω ) − Z ψdµ + | < ε, ∀ n ≥ m (cid:9) We know from Birkhoff Ergodic Theorem that ˆ µ ( ˜ A ( m, ε )) → m
1, for any ε >
0. So we obtain ν ( π ˜ A ( m, ε )) → m
1. But Φ n ([ i . . . i n ] × Λ) = Σ + I × φ i n ...i Λ, and from the Φ-invariance of ˆ µ , we have9 µ (Φ n ([ i . . . i n ] × Λ) ≥ ˆ µ ([ i . . . i n ] × Λ). Moreover ˆ µ ([ i . . . i n ] × Λ) >
0, since ˆ µ is the equilibriummeasure of ψ ◦ π and [ i . . . i n ] × Λ is an open set. In conclusion, ν ( φ i n ...i Λ) = ˆ µ (Σ + I × φ i n ...i Λ) = ˆ µ (Φ n ([ i . . . i n ] × Λ) ≥ ˆ µ ([ i . . . i n ] × Λ) > m ([ i . . . i m ] × Λ) = Σ + I × φ i m ...i Λ, and ν = π ∗ ˆ µ , we have: ν ( φ i m ...i Λ) = ˆ µ (Σ + I × φ i m ...i Λ) = ˆ µ (Φ m ([ i . . . i m ] × Λ)Notice that a small ball B ( x, r ) can intersect many sets of type φ i m ...i Λ, for various m -tuples( i , . . . , i m ) ∈ I m , and these image sets may also intersect one another. Thus when estimating ν ( B ( x, r )), all of these sets must be considered; it is not enough in principle to consider only oneintersection B ( x, r ) ∩ φ i m ...i Λ. However, we know from (25) that ν ( φ i m ...i Λ) >
0, thus from theBorel Density Theorem (see [14]), it follows that for ν -a.e x ∈ φ i m ...i Λ, and for all 0 < r < r ( x ), ν ( B ( x, r ) ∩ φ i m ...i Λ) ν ( B ( x, r )) > / ν , the intersection B ( x, r ) ∩ φ i m ...i Λ contains atleast half of the ν -measure of the ball B ( x, r ). This hints to the fact that it is enough to consideronly one “good” image set of type φ i m ...i Λ. Then since ν ( π ˜ A ( m, ε )) → m
1, we can consider only ν ( φ i m ...i ( π ˜ A ( m, ε ))), which can be estimated using the Jacobian J Φ m (ˆ µ ) and the genericity ofpoints in ˜ A ( m, ε ) with respect to the functions log J Φ m (ˆ µ ) and log | φ ′ i m ...i | . Then we repeat thisargument whenever the iterate of a point belongs to the set of generic points ˜ A ( m, ε ). Howevernot all iterates of a point belong to this set, but it will be shown by a delicate estimate that”most” of them hit ˜ A ( m, ε ). For the iterates which do not belong to ˜ A ( m, ε ), we use a differenttype of estimate. Then we will repeat and combine these two types of estimates, by an interlacingprocedure.We now proceed with the full detailed proof:For any integer m >
1, consider the Borel set ˜ A ( m, ε ) defined above. Then for any α > m ( α ) > m > m ( α ), we have:ˆ µ ( ˜ A ( m, ε )) > − α (26)Let us fix such an integer m > m ( α ). Then from Birkhoff Ergodic Theorem applied to Φ m and χ ˜ A ( m,ε ) , we have that for ˆ µ -a.e ( ω ′ , x ′ ) ∈ Σ + I × Λ,1 n Card { ≤ k ≤ n, Φ km ( ω ′ , x ′ ) ∈ ˜ A ( m, ε ) } −→ n →∞ ˆ µ ( ˜ A ( m, ε ))Hence, there exists an integer n ( α ) and a Borel set D ( α ) ⊂ Σ + I × Λ, with ˆ µ ( D ( α )) > − α , suchthat for ( ω ′ , x ′ ) ∈ D ( α ) and n ≥ n ( α ), we have:1 n Card { ≤ k ≤ n, Φ mk ( ω ′ , x ′ ) ∈ ˜ A ( m, ε ) } > − α (27)10n other words a large proportion of the iterates of points ( ω ′ , x ′ ) in D ( α ), belong to the set ofgeneric points ˜ A ( m, ε ). So in the Φ m -trajectory ( ω ′ , x ′ ) , Φ m ( ω ′ , x ′ ) , . . . , Φ nm ( ω ′ , x ′ ), there are atleast (1 − α ) n iterates in ˜ A ( m, ε ).For arbitrary indices i , . . . i m ∈ I , let us define now the Borel set in Λ, Y ( i , . . . , i m ) := φ i ...i m π ( ˜ A ( m, ε ))Consider first the intersection of all these sets, namely T i ,...,i m ∈ I Y ( i , . . . , i m ). Then take the inter-sections of these sets except only one of them, so consider sets of type T ( j ,...,j m ) =( i ,...,i m ) Y ( j , . . . , j m ) \ Y ( i , . . . , i m ), for all ( i , . . . , i m ) ∈ I m . Then we consider the intersections of all the sets Y ( j , . . . , j m )excepting two of them, namely the intersections of type T ( j ,...,j m ) / ∈{ ( i ,...,i m ) , ( i ′ ,...,i ′ m ) } Y ( j , . . . , j m ) \ (cid:0) Y ( i , . . . , i m ) ∪ Y ( i ′ , . . . , i ′ m ) (cid:1) , for all the m -tuples ( i , . . . , i m ) , ( i ′ , . . . , i ′ m ) ∈ I m . We continuethis procedure until we exhaust all the possible intersections of type \ ( j ,...,j m ) / ∈J Y ( j , . . . , j m ) \ [ ( i ,...,i m ) ∈J Y ( i , . . . , i m ) , for some arbitrary given set J of m -tuples from I m . Notice that in this way, by taking all the subsets J ⊂ I m , we obtain by the above procedure mutually disjoint Borel sets (some may be empty).Denote these mutually disjoint nonempty sets obtained above by Z ( α ; m, ε ) , . . . , Z M ( m ) ( α ; m, ε ).Now if for some 1 ≤ i ≤ M ( m ), we know that ν ( Z i ( α ; m, ε )) >
0, then from the Borel DensityTheorem (see [14]), there exists a Borel subset G i ( α ; m, ε ) ⊂ Z i ( α ; m, ε ), with ν ( G i ( α ; m, ε )) ≥ ν ( Z i ( α ; m, ε ))(1 − α ), and there exists r i ( α ; m, ε ) >
0, such that for any x ∈ G i ( α ; m, ε ), ν ( B ( x, r ) ∩ Z i ( α ; m, ε )) ν ( B ( x, r )) > − α, (28)for any 0 < r < r i ( α ; m, ε ). Now define the Borel subset of Λ, G ( α ; m, ε ) := M ( m ) [ i =1 G i ( α ; m, ε )Now from the construction of the mutually disjoint sets Z i ( α ; m, ε ), it follows that, X ≤ i ≤ M ( m ) ν ( Z i ( α ; m, ε )) = ν ( ∪ i ,...,i m ∈ I Y ( i , . . . , i m )) (29)But from definition of Y ( i , . . . , i m ) and the disjointness of different m -cylinders, we have that, ∪ i ,...,i m ∈ I Y ( i , . . . , i m ) = ∪ i ,...,i m ∈ I π (Φ m ([ i m . . . i ] × π ( ˜ A ( m, ε )))) == π ( ∪ i ,...,i m ∈ I Φ m ([ i m . . . i ] × π ˜ A ( m, ε ))) = π (Φ m (Σ + I × π ˜ A ( m, ε )))However since ν = π ∗ ˆ µ , and using the Φ-invariance of ˆ µ and (26), it follows that: ν (cid:0) π (Φ m (Σ + I × π ˜ A ( m, ε ))) (cid:1) = ˆ µ (Σ + I × π (Φ m (Σ + I × π ˜ A ( m, ε )))) ≥ ˆ µ (Φ m (Σ + I × π ˜ A ( m, ε ))) ≥≥ ˆ µ (Σ + I × π ˜ A ( m, ε )) ≥ ˆ µ ( ˜ A ( m, ε )) > − α ν ( [ i ,...,i m ∈ I Y ( i , . . . , i m )) > − α (30)Hence from (29) and (30), we obtain: ν (cid:0) [ ≤ i ≤ M ( m ) Z i ( α ; m, ε ) (cid:1) > − α (31)But the Borel sets Z i ( α ; m, ε ) , ≤ i ≤ M ( m ) are mutually disjoint, and from (28) we know that G i ( α ; m, ε ) ⊂ Z i ( α ; m, ε ) and for 1 ≤ i ≤ M ( m ), ν ( G i ( α ; m, ε )) ≥ ν ( Z i ( α ; m, ε ))(1 − α )Hence from the definition of G ( α ; m, ε ) = S ≤ i ≤ M ( m ) G i ( α ; m, ε ) and from (31), ν ( G ( α ; m, ε )) > (1 − α ) > − α (32)Denote now the following intersection set by, X ( α ; m, ε ) := G ( α ; m, ε ) ∩ π ˜ A ( m, ε )Then from the above estimate for ν ( G ( α ; m, ε )) and using (26), we obtain: ν ( X ( α ; m, ε )) > − α (33)Now by applying the same argument as in (27) to the set ˜ A ( m, ε ) ∩ (cid:0) Σ + I × X ( α ; m, ε ) (cid:1) , we obtainthat there exists a Borel set ˜ D ( α ; m, ε ) ⊂ ˜ A ( m, ε ) ⊂ Σ + I × Λ, withˆ µ ( ˜ D ( α ; m, ε )) > − α, (34)and such that for any pair ( i, x ′ ) ∈ ˜ D ( α ; m, ε ), at least a number of (1 − α ) n of the points π ( i, x ′ ), π Φ m ( i, x ′ ) , . . . , π Φ nm ( i, x ′ ) belong to X ( α ; m, ε ). Moreover any point ζ ∈ X ( α ; m, ε ) satisfiescondition (28) for all 0 < r < r i ( α ; m, ε ) if ζ ∈ Z i ( α ; m, ε ), for some 1 ≤ i ≤ M ( m ). So denote r m ( α, ε ) := min ≤ i ≤ M ( m ) r i ( α ; m, ε )Consider now ( i, x ′ ) ∈ ˜ D m ( α, ε ), and denote by x = φ i nm ...i ( x ′ ) = π Φ nm ( i, x ′ ). So we have thefollowing backward trajectory of x with respect to Φ m determined by the sequence i from above, x, φ i ( n − m ...i ( x ′ ) , . . . , φ i m ...i ( x ′ ) , x ′ , (35)and denote these points respectively by x, x − m , . . . , x − ( n − m , x − nm = x ′ . To see the next argument,assume for simplicity that the first preimage of x in this trajectory, namely x − m = φ i ( n − m ...i ( x ′ )12elongs to X ( α ; m, ε ). Then from (28), Lemma 1, and the genericity of J Φ m on ˜ A ( m, ε ), ν ( B ( x, r )) ≤ − α ν ( B ( x, r ) ∩ φ i nm ...i ( n − m π ˜ A ( m, ε )) == 11 − α ˆ µ (Σ + I × ( B ( x, r ) ∩ φ i nm ...i ( n − m π ˜ A ( m, ε ))) ≤≤ − α ˆ µ (Φ m ([ i ( n − m . . . i nm ] × ( π ˜ A ( m, ε ) ∩ ( φ i nm ...i ( n − m ) − B ( x, r )))) == 11 − α Z [ i ( n − m ...i nm ] × ( π ˜ A ( m,ε ) ∩ ( φ inm...i ( n − m ) − B ( x,r )) J Φ m (ˆ µ ) d ˆ µ ≤≤ C − α e m ( F Φ (ˆ µ )+ ε ) µ + ([ i ( n − m...i nm ]) · ν (( φ i nm ...i ( n − m ) − B ( x, r )) ≤≤ C − α e m ( F Φ (ˆ µ )+2 ε − h ( µ + )) ν (( φ i nm ...i ( n − m ) − B ( x, r )) , where the last inequality follows since µ + is the equilibrium state of ψ and P ( ψ ) = h µ + + R ψdµ + .Thus we obtain from above the following estimate on the measure of B ( x, r ), ν ( B ( x, r )) ≤ C − α · e m ( F Φ (ˆ µ ) − h ( µ + )+2 ε ) · ν (( φ i nm ...i ( n − m ) − B ( x, r )) (36)This argument can be repeated until we reach in the above backward trajectory of x (35), apreimage which is not in X ( α ; m, ε ). Denote then by k ≥ k for which x − mk / ∈ X ( α ; m, ε ), and assume that the above process is interrupted for k ′ indices, namely { x − mk , x − m ( k +1) , . . . , x − m ( k + k ′ − } ∩ X ( α ; m, ε ) = ∅ . Then denote y := x − mk , and let us esti-mate ν (( φ i nm ...i ( n − k m ) − B ( x, r )). By definition of the projection measure ν , ν (( φ i nm ...i ( n − k m ) − B ( x, r )) = ˆ µ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r ))By definition of k , k ′ , we have that x − m ( k + k ′ ) ∈ X ( α ; m, ε ). By repeating the estimate in (36),we obtain an upper estimate for ν ( B ( x, r )), ν ( B ( x, r )) ≤ ( C − α ) k · e mk ( F Φ (ˆ µ ) − h ( µ + )+2 ε ) · ν (( φ i nm ...i ( n − k m ) − B ( x, r )) (37)Now on the other hand from the Φ-invariance of ˆ µ and the definition of ν , ν (( φ i nm ...i ( n − k m ) − B ( x, r )) = ˆ µ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )) == ˆ µ ( φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )))= X j p ∈ I, ≤ p ≤ mk ′ ˆ µ ([ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ))(38)Recall now that the set of non-generic points satisfies,ˆ µ ((Σ + I × Λ) \ ˜ A ( m, ε )) < α We now compare the ˆ µ -measure of the set of generic points with respect to the ˆ µ -measure, withthe ˆ µ -measure of the set of non-generic points. There are 2 cases:13) If,ˆ µ (cid:0) ˜ A ( m, ε ) \ Φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )) (cid:1) <
12 ˆ µ (Φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r ))) , then non-generic points have more mass than the generic points, hence, ν (( φ i nm ...i ( n − k m ) − B ( x, r )) = ˆ µ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )) < α << α →
0, this case is then straightforward.b) If,ˆ µ (cid:0) ˜ A ( m, ε ) ∩ Φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )) (cid:1) ≥
12 ˆ µ (Φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r ))) , then using also (38) we obtain: ν (( φ i nm ...i ( n − k m ) − B ( x, r )) = ˆ µ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r )) == ˆ µ (Φ − mk ′ (Σ + I × ( φ i nm ...i ( n − k m ) − B ( x, r ))) ≤ X j generic ˆ µ (cid:0) ˜ A ( m, ε ) ∩ (cid:0) [ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ) (cid:1)(cid:1) (39)But for two generic histories j, j ′ , i.e. for j, j ′ ∈ π ( ˜ A ( m, ε )), we know that for any ( ω, z ) ∈ ˜ A ( m, ε ) T (cid:0) [ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ) S [ j ′ . . . j ′ mk ′ ] × φ − j ′ mk ′ ...j ′ ( φ i nm ...i ( n − k m ) − B ( x, r ) (cid:1) , J Φ mk ′ (ˆ µ )( ω, z ) ∈ (cid:0) e mk ′ ( F Φ (ˆ µ ) − ε ) , e mk ′ ( F Φ (ˆ µ )+ ε ) (cid:1) . Hence since Φ mk ′ ( ˜ A ( m, ε ) T [ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ) ⊂ φ i nm ...i ( n − k m ) − B ( x, r )and using the above estimate on the Jacobian of ˆ µ with respect to Φ mk ′ , it follows that there existsa constant factor C > µ -measures of the two preimage type sets˜ A ( m, ε ) \ [ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ) , and ˜ A ( m, ε ) \ [ j ′ . . . j ′ mk ′ ] × φ − j ′ mk ′ ...j ′ ( φ i nm ...i ( n − k m ) − B ( x, r ) , corresponding to generic j, j ′ , belongs to the interval (cid:0) C − e − mk ′ ε , Ce mk ′ ε (cid:1) .Notice also that there are at most d mk ′ sets of type [ j . . . j mk ′ ] × φ − j mk ′ ...j ( φ i nm ...i ( n − k m ) − B ( x, r ),where d := | I | . The maximality assumption for k ′ implies that φ − i m ( n − k ...i m ( n − k − k ′ ( y ) ∈ X ( α ; m, ε ),where y := x − mk . Thus using the above discussion and (39), we obtain: ν (( φ i nm ...i ( n − k m ) − B ( x, r )) ≤ Cd mk ′ (1+ ε log d ) · ν ( φ − i m ( n − k − ...i m ( n − k − k ′ ( φ i nm ...i ( n − k m ) − B ( x, r ))(40)Now, recall we concluded above that φ − i m ( n − k ...i m ( n − k − k ′ ( y ) ∈ X ( α ; m, ε ), and then we canapply again the argument from (37) along the sequence i until we reach another preimage of x X ( α ; m, ε ); after that we apply again the argument from (40), and so on,until reaching the nm -preimage of x , namely x − nm = φ − i nm ...i ( x ). Recall from (35) that x ′ := x − nm .Consider now r > n = n ( r ) be chosen such that due to the conformalityof φ i , i ∈ I , one can assume that: φ − i nm ...i B ( x, r ) = B ( x − nm , r ) = B ( x ′ , r ) (41)On the backward m -trajectory (35) of x determined by i above, recall that we denoted by k ′ the length of the first maximum “gap” consisting of preimages of x which are not in X ( α ; m, ε ).Now let us denote in general the lengths of such maximal “gaps” in this trajectory (35), consist-ing of consecutive preimages which do not belong to X ( α ; m, ε ), by k ′ , k ′ , . . . . More precisely,we have x − jm ∈ X ( α ; m, ε ) , ≤ j ≤ k −
1, followed by x − mk , . . . , x − m ( k + k ′ − / ∈ X m ; then x − m ( k + k ′ ) , . . . , x − m ( k + k ′ + k − ∈ X m ( α, ε ), followed by x − m ( k + k ′ + k ) , . . . , x − m ( k + k ′ + k + k ′ − / ∈ X ( α ; m, ε ); then, x − m ( k + k ′ + k + k ′ ) , . . . , x − m ( k + k ′ + k + k ′ + k − ∈ X ( α ; m, ε ), and so on.Denote thus by k , . . . , k p ( n ) and k ′ , . . . , k ′ p ′ ( n ) all the integers obtained by the above procedure,corresponding to the sequence (35). Clearly we have k + k ′ + . . . + k p ( n ) + k ′ p ( n ) = n Also p ′ ( n ) is equal either to p ( n ) or to p ( n ) −
1. Then from the properties of ˜ D m ( α, ε ) in (34), k + . . . + k p ( n ) ≥ n (1 − α ) , and k ′ + k ′ + . . . + k ′ p ′ ( n ) ≤ αn (42)We will apply (37) for the generic preimages x, . . . , x − m ( k − from the trajectory (35), then we applythe estimate in (40) for the nongeneric preimages x − mk , . . . , x − m ( k + k ′ − , then again apply (37)for x − m ( k + k ′ ) , . . . , x − m ( k + k ′ + k − , followed by (40) for x − m ( k + k ′ + k ) , . . . , x − m ( k + k ′ + k + k ′ − ,and so on. Hence applying succesively the estimates (37) and (40), and recalling (41) and thebound on the combined length of gaps k ′ + k ′ + . . . + k ′ p ′ ( n ) ≤ αn from (42), we obtain: ν ( B ( x, r )) ≤ ( C − α ) n · d αmn (1+ ε log d ) · e ( F Φ (ˆ µ ) − h ( µ + )+2 ε ) · mn (1 − α ) ν ( B ( x ′ , r )) (43)But χ s (ˆ µ ) is the stable Lyapunov exponent of ˆ µ , i.e χ s (ˆ µ ) = R Σ + I × Λ log | φ ′ ω ( x ) | d ˆ µ ( ω, x ). From (41)it follows that r = r n = r | φ ′ i nm ...i ( x ′ ) | , and recall that ( i, x ′ ) ∈ ˜ D m ( α, ε ). Therefore, e nm ( χ s (ˆ µ ) − ε ) ≤ r n ≤ e nm ( χ s (ˆ µ )+ ε ) (44)Denote by ˜ X ( α ; m, ε ) := π ˜ D m ( α, ε ). Then since ν := π ∗ ˆ µ and ˆ µ ( ˜ D m ( α, ε )) > − α , we obtain, ν ( ˜ X ( α ; m, ε )) = ˆ µ (Σ + I × ˜ X ( α ; m, ε )) ≥ ˆ µ ( ˜ D m ( α, ε )) ≥ − α From (43) and (44) it follows that by taking possibly C ′ = 2 C >
1, then for every x ∈ ˜ X ( α ; m, ε )and for a sequence r n →
0, the following estimate holds: ν ( B ( x, r n )) ≤ ( C ′ − α ) n · d αmn (1+ ε log d ) · r (1 − α ) · F Φ(ˆ µ ) − h ( µ +)+2 εχs (ˆ µ ) n ν ( B ( x, r n )) ≤ (1 − α ) log r n · F Φ (ˆ µ ) − h ( µ + ) + 2 εχ s (ˆ µ ) + n log C ′ − α + 3 αmn · log d (1 + 2 ε log d ) (45)But by using (44) and dividing in (45) by log r n , one obtains:log ν ( B ( x, r n ))log r n ≥ (1 − α ) F Φ (ˆ µ ) − h ( µ + ) + 2 εχ s (ˆ µ ) + log C ′ − α m ( χ s (ˆ µ ) + ε ) + (1 + 2 ε log d ) 3 α log dχ s (ˆ µ ) + ε (46)Now let us take some arbitrary radius ρ >
0, and assume that for some integer n , we have r n +1 ≤ ρ ≤ r n , where r n is defined at (44). Then ν ( B ( x, ρ )) ≤ ν ( B ( x, r n )), thereforelog ν ( B ( x, ρ ))log ρ ≥ log ν ( B ( x, r n ))log ρ But log ρ ≥ log r n +1 , hence ρ ≤ r n +1 <
0, hence from above,log ν ( B ( x, ρ ))log ρ ≥ log ν ( B ( x, r n ))log r n +1 ≥ log ν ( B ( x, r n )) c + log r n , (47)since r n +1 > cr n , for some constant c independent of n . Thus by letting ρ →
0, and using (46) and(47), we obtain the following lower estimate for the lower pointwise dimension of ν : δ ( ν )( x ) ≥ (1 − α ) F Φ (ˆ µ ) − h ( µ + ) + 2 εχ s (ˆ µ ) + log C ′ − α m ( χ s (ˆ µ ) + ε ) + 3 α (1 + 2 ε log d ) log dχ s (ˆ µ ) + ε But α → m → ∞ , and then ν ( ˜ X ( α ; m, ε )) →
1. Thus from the last displayed estimate, itfollows that for ν -a.e x ∈ Λ, δ ( ν )( x ) ≥ F Φ (ˆ µ ) − h ( µ + ) + 2 εχ s (ˆ µ )Since ε is arbitrarily small, it follows from the above lower estimate and (24), that for ν -a.e x ∈ Λ, δ ( ν )( x ) = F Φ (ˆ µ ) − h ( µ + ) χ s (ˆ µ ) (cid:3) Proof of Theorem 2.
We proved in [11] that if µ + is a Bernoulli measure ν p on Σ + I , then the corresponding projections ν , p and ν , p are equal. Hence they have the same pointwise dimensions. Also recall from [4] that ν , p is exact dimensional, thus also ν , p is exact dimensional. Thus, by using the formula (4) andTheorem 1, we obtain the expression for h π ( σ, ν p ) in terms of the overlap number of ˆ µ p , i.e, h π ( σ, ν p ) = h σ ( ν p ) − exp( o ( S , ˆ µ p ))In case ν p is the measure of maximal entropy µ on Σ + I , we have h ( µ ) = log | I | , and we obtain h π ( σ, µ ) = log | I | − exp( o ( S )). (cid:3)
16n particular, we obtain a geometric formula for the pointwise dimension of the
Bernoulli con-volution ν λ in terms of its overlap number, for all λ ∈ ( , ν λ is obtained as the projection of the measure of maximalentropy ˆ µ onto the limit set Λ λ of the system S λ given by the two contractions φ ( x ) = λx − , φ ( x ) = λx + 1 on R (see [13]). When λ ∈ ( , S λ has overlaps, and its limit set Λ λ is the interval I λ = [ − − λ , − λ ]. The measure ν λ is the unique self-conformal measure satisfying ν λ = ν λ ◦ φ − + ν λ ◦ φ − . It is clear that in this case, if ν (
12 12 ) is the Bernoulli measure on Σ +2 given by the probability vector ( , ), and if ˆ µ is the measure of maximal entropy on Σ +2 × Λ, then ν λ = π ∗ ν ( , ) = π ∗ ˆ µ Thus ν λ is exact dimensional, and by applying Theorem 2 we obtain: Corollary 2.
For any λ ∈ ( , , the Hausdorff dimension of the Bernoulli convolution ν λ satisfies: δ ( ν λ ) = δ ( ν λ )( x ) = log o ( S λ ) | log λ | , for ν λ − a.e x Acknowledgements:
The author thanks Professor Yakov Pesin for interesting discussionsduring a visit at Pennsylvania State University. During work on this article Eugen Mihailescuwas supported by grant PN III-P4-ID-PCE-2016-0823 from UEFISCDI. He also acknowledges thesupport of Institut des Hautes ´Etudes Sci´entifiques, Bures-sur-Yvette, France, for a stay when partof this paper was done.
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