Multifractal analysis in non-uniformly hyperbolic interval maps
aa r X i v : . [ m a t h . D S ] F e b MULTIFRACTAL ANALYSIS IN NON-UNIFORMLYHYPERBOLIC INTERVAL MAPS
MA GUANZHONG, SHEN WENQIANG ∗ , YAO XIAO Abstract.
In this paper, we study the Hausdorff dimension ofthe generalized intrinsic level set with respect to the given ergodicmeausre in a class of non-uniformly hyperbolic interval maps withfinitely many branches.
Introduction
Let T : m S i =1 I i ⊂ [0 , → [0 ,
1] be a piecewise C map, where I i is theclosed interval for 1 ≤ i ≤ m such that int ( I i ) ∩ int ( I j ) = ∅ for anydistinct i and j . Here, int ( I i ) means the interior of I i . In this paper, weconsider the following class of non-uniformly hyperbolic interval maps, • T | I i : I i → [0 ,
1] is a surjective and continuously differentiablefor 1 ≤ i ≤ m. There is a unique x i ∈ I i such that T ( x i ) = x i for each i . • | T ′ ( x ) | > x
6∈ { x , . . . , x m } . Here, we also allow that | T ′ ( x i ) | > i ∈ { , , . . . , m } .If T ′ ( x i ) = 1 or T ′ ( x i ) = − i , we say that x i is a parabolic fixed point. The class of non-uniform hyperbolic maps includes theimportant example of Manneville-Pomeau map [21], T : [0 , → [0 , T x = x + x β (mod 1), where 0 < β <
1, see Fig 1.We define the repeller Λ of T byΛ := ( x ∈ m [ i =1 I i : T n ( x ) ∈ [0 , , ∀ n ≥ ) . We know that (Λ , T ) has a very natural Markov partition. Let S i be theinverse branch of T | I i : I i → [0 ,
1] for i = 1 , . . . , m . Let A = { . . . , m } ,Σ = A N and Σ n be the set of all n -blocks over A for any n ∈ N . Here we ∗ Corresponding author. ∗ , YAO XIAO Figure 1.
Maneville mappoint out N is the set of positive integers in this paper. Let σ : Σ → Σbe the shift map, σ (( ω n ) n ≥ ) = ( ω n ) n ≥ . Define the semi-conjugacy map π : Σ → [0 ,
1] by π ( ω ) := lim n →∞ S ω ◦ S ω ◦ · · · ◦ S ω n − ([0 , . It is easy to check that π (Σ) = Λ and π ◦ σ ( ω ) = T ◦ π ( ω ) . We remarkedthat π is a bijection map from Σ to Λ except for a countable manypoints.Let φ be a continuous function in C (Λ , R d ). We denote the n -thBirkhoff sum by S n φ ( x ) = P n − j =0 φ ( T j x ) and n -th Birkhoff average by A n φ ( x ) = n S n φ ( x ) for any x ∈ Λ. For any α ∈ R d , we defineΛ φ ( α ) := { x ∈ Λ : lim n →∞ A n φ ( x ) = α } and more generally,˜Λ φ ( α ) := { x ∈ Λ : lim k →∞ A n k φ ( x ) = α for some { n k } ∞ k =1 with lim k →∞ n k = ∞} . Since Λ φ ( α ) not only depends on α , but also on the continuous func-tion φ , it is natural to introduce a set which is intrinsic in some sense.We denote the set of all invariant measures by M (Λ , T ) and the setof all ergodic measures by E (Λ , T ). For any µ ∈ E (Λ , T ), we define the ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS3 intrinsic level setΛ µ := ( x ∈ Λ : ( A n ) ∗ δ x = 1 n n − X i =0 δ T i ( x ) ∗ −−−→ n →∞ µ ) be the set of all µ generic points, where ∗ −−−→ n →∞ stands for the conver-gence in the weak- ∗ topology. Similarly, we also define the generalizedintrinsic level set˜Λ µ := n x ∈ Λ : ( A n k ) ∗ δ x ∗ −−−→ k →∞ µ for some { n k } ∞ k =1 with lim k →∞ n k = ∞ o . The multifractal analysis for uniformly hyperbolic conformal dynam-ical system is well developed in past years, we refer the [18, 8, 7, 16, 17]for entropy spectrum and Birkhoff spectrum of level set and dimen-sion spectrum of Gibbs measure or weak Gibbs measure. The origi-nal method developed in [18] is based on the equilibrium state (Gibbsmeasure) in thermal-formalism method, which needs the further reg-ularity conditions on T . Later, this method was further developedin [7, 16, 17, 2], and the regularity conditions have been completelyremoved in the uniformly hyperbolic case.Recently, there has been a trend in understanding the multifractalanalysis beyond the uniformly hyperbolic dynamical system. However,up to now, there is still not a complete picture in the direction of non-uniformly hyperbolic dynamical system. The topological entropy ofthese types of sets have been studied in [19, 20, 23, 3, 25, 22]. The di-mension spectrum of Birkhoff ergodic limits in non-uniform hyperbolicdynamic systems has been done in [13, 9, 10, 1, 2, 12].In [11], Gelfert and Rams studied the dimension spectrum of Λ φ for φ = log | T ′ | by. For the general continuous function φ , Johansson,Jordan, ¨Oberg, and Pollicott established a formula of dim H Λ φ in [13].For any ergodic measure µ , we call µ is hyperbolic if the Lyapunovexponent of µ λ ( µ, T ) := Z log | T ′ | dµ > , otherwise µ is called to be parabolic. If µ is a parabolic measure, µ only supports on some parabolic fixed point under the setting of thispaper.Given a compact and T -invariant set K ⊂ Λ, we say that K is ahyperbolic set, if | T ′ ( z ) | > z ∈ K . The hyperbolic dimensionof Λ is defined bydim hypH Λ := sup { dim H K : K is a hyperbolic set in Λ } . MA GUANZHONG, SHEN WENQIANG ∗ , YAO XIAO One of the main goal in the study of non-uniformly hyperbolic dy-namical system is to recover the sufficient hyperbolicity to dominate orbalance the non-hyperbolic behaviour, we refer the reader to [4, 5].Now, we will state our main result of the Hausdorff dimension ofΛ µ = ( x ∈ Λ : 1 n n − X i =0 δ T i ( x ) ∗ −→ µ ) and˜Λ µ = n x ∈ Λ : ( A n k ) ∗ δ x ∗ −−−→ n →∞ µ for some { n k } ∞ k =1 such that lim k →∞ n k = ∞ o . The topological entropy of h top (Λ µ ) and h top ( ˜Λ µ ) can be deducedfrom [20, 6]. The following result proved by Pfister and Sullivan in[20], and Fan, Liao, and Peyri`ere [6] independently for the system withthe (almost) specification property. It is known that this property holdsin the non-uninformly hyperbolic interval map we considered here. Theorem A ([20, 6]) . Let m be an invariant measure in M (Λ , T ) .We have (1) h top ( ˜Λ m ) = h top (Λ m ) = h ( m, T ) . Especially, h top ( ˜Λ m ) = 0 if λ ( m, T ) = 0 . Johannes and Takahasi [14] study the Hausdorff dimension of Λ µ in the setting of non-uniformly hyperbolic interval maps even withinfinitely many branches recently. Theorem B ([14]) . Let m be an ergodic measure in M (Λ , T ) . Wehave (1) dim H Λ m = h ( m,T ) λ ( m,T ) if λ ( m, T ) > ; (2) dim H Λ m ≥ dim hypH Λ > if λ ( m, T ) = 0 . Moreover if T is a C -map, we have dim H Λ m = dim H Λ . To the best of our knowledge, the relation between dim H Λ m anddim H ˜Λ m is still lacking even in the setting of the finitely many branches.We have the following result in this direction. Theorem 1.
Let m be an ergodic measure in M (Λ , T ) . We have (1) dim H ˜Λ m = h ( m,T ) λ ( m,T ) if λ ( m, T ) > ; ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS5 (2) dim H ˜Λ m ≥ dim H Λ m ≥ dim hypH Λ > if λ ( m, T ) = 0 . Moreoverif T is a C -map, we have dim H ˜Λ m = dim H Λ m = dim H Λ . Remark 1.
One sees that there is a great difference on the size ofintrinsic level set of a invariant measure from the viewpoints of topo-logical entropy and Hausdorff dimension. In fact, the C regularitycondition is only used to ensure that dim hypH Λ = dim H Λ . Indeed, the assumption of the ergodicity is not necessary in Theorem1. This can be removed by the careful approximating arguments and wedo not address it in this paper.
By the same technique in the proof of Theorem 1, we can get thefollowing result.
Theorem 1 ′ . Let K be a compact and connected set in M (Λ , T ) and Λ K := { x ∈ Λ : Asym( { (A n ) ∗ δ x } ∞ n=1 ) = K } , where Asym( { (A n ) ∗ δ x } ∞ n=1 ) is the set of all accumulating points of { ( A n ) ∗ δ x } ∞ n =1 in M (Λ , T ) . Then we have (1) dim H Λ K = inf n h ( m,T ) λ ( m,T ) : m ∈ K, λ ( m, T ) > o if K has someinvariant measure µ with λ ( µ, T ) > ; (2) dim H Λ K = dim H Λ if λ ( m, T ) = 0 for any m ∈ K and T is a C map. Here we do not pursue this generalization in this paper. Motivatedby Theorem 1 and Theorem B, we have the following corollary imme-diately.
Corollary 1.
Let m be an ergodic measure and T is C in the settingabove. Then we have dim H ˜Λ m = dim H Λ m . It is proved that in [7] that dim H ˜Λ m = dim H Λ m holds for the C conformal repeller. We do not known whether this holds or not in theuniformly hyperbolic case under the C condition. For this direction,we have the following result. Theorem 2.
Let m be an ergodic measure and assume that T has onlyone parabolic fixed point p in [0 , . Then we have dim H ˜Λ m = dim H Λ m . MA GUANZHONG, SHEN WENQIANG ∗ , YAO XIAO If T has more than 1 fixed points, we do not know how to controlthe persistent recurrence behaviors between (or among) the multipleparabolic fixed points in the C regularity condition.In the proof of Theorem 1, we also recover a complete proof of The-orem B by using our framework for the reader’s convenience. In [14], aproof of Theorem B was already given even in the framework of infin-itely branches by a series of delicate approximation arguments. Here,we will give a direct proof in our simple setting, which is inspired by[6, 13].This paper is organised as follows. In Section 2, some some basicpreliminaries and results are collected. In Section 3, we will prove theupper bound of Theorem 1. In Section 4, we will make some effort togive a unified framework of geometric Moran construction in dimen-sion 1, which may be of independent interest. We believe that thisframework can be used to deal with many problems in multifractalanalysis in non-uniformly hyperbolic dynamical system. In Section 5,we will prove the lower bound for Theorem 1. Finally, Theorem 2 willbe proved in Section 6. 2. Preliminaries
In this section, we will collect some basic facts. For any ω ∈ Σ, wedenote D n ( ω ) = max {| x − y | : x, y ∈ S ω ◦ S ω · · · S ω n − ([0 , } .Let g ( x ) = log | T ′ ( x ) | for any x ∈ Λ and G ( ω ) = log | T ′ ( π ( ω )) | forany ω ∈ Σ. And we know that g ( π ( ω )) = G ( ω ) for ω ∈ Σ. For any f ∈ C (Λ , R ), we denote F : Σ → R by(2) F ( ω ) = f ( π ( ω )) . For any n ∈ N , we define(3) var n ( f ) := sup {| f ( π ( ω ) − f ( π ( τ )) | : ω = τ , . . . , ω n − = τ n − } . The following lemma shows a relation between the logarithm growthof D n ( ω ) and the Birkhoff average of G ( ω ). Lemma 1. [13]
Under the setting above, we have lim n →∞ sup ω ∈ Σ (cid:12)(cid:12)(cid:12) − log D n ( ω ) n − A n G ( ω ) (cid:12)(cid:12)(cid:12) = 0 . In our following discussion, we also need the following fact.
Lemma 2. [13]
Let f be a continuous function on Λ . Then we have lim n →∞ n var n (f) = 0 . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS7
Since Λ is a compact set, there exists a countable dense subset { f n } ∞ n =1 in C (Λ , R ) and we can assume f n n ∈ N withoutloss of generality. Let M (Λ) be the set of all probability measures. Weintroduce a metric d on M (Λ) by d ( µ, ν ) := ∞ X n =1 | R f n dµ − R f n dν | n k f n k . And the topology induced by the metric d on M (Λ) is compatible withthe weak- ∗ topology. Lemma 3.
Let m be an invariant measure in M (Λ , T ) and Λ m,k := { x ∈ Λ : lim n →∞ A n f i ( x ) = Z f i dm, for ≤ i ≤ k } . Then we have Λ m = ∞ \ k =1 Λ m,k . The proof of Lemma 3 is rather simple, we will omit the proof here.The existence of parabolic fixed points has a important impact on thesize of m -generic set. Assume that T has l + 1 different parabolic fixedpoints { p i , ≤ i ≤ l } and we denote M p (Λ , T ) := ( l X i =0 λ i δ p i : λ i ≥ l X i =0 λ i = 1 ) . It is obvious that M p (Λ , T ) is a l -dimensional simplex in M (Λ , T ) and M p (Λ , T ) = { µ ∈ M (Λ , T ) : λ ( µ, T ) = 0 } . Lemma 4.
Let µ be a hyperbolic measure in M (Λ , T ) and M µ,k := (cid:26) ν ∈ M (Λ , T ) : Z f i dν = Z f i dµ for ≤ i ≤ k (cid:27) . Then we have M µ,k ∩ M p (Λ , T ) = ∅ for k sufficiently large. Proof of Lemma 4
We assume that there exists a sequence { k i } ∞ i =1 ,such that µ k i ∈ M µ,k i ∩ M p (Λ , T ) . It follows that Z f j dµ k i = Z f j dµ MA GUANZHONG, SHEN WENQIANG ∗ , YAO XIAO for any j ≤ k i , which implies that d ( µ k i , µ ) ≤ k i and lim i →∞ µ k i = µ. Recalling that λ ( µ k i , T ) = 0 for any i ∈ N , we get λ ( µ, T ) = 0, whichcontradicts the fact that µ is hyperbolic. (cid:3) Proof for the upper bound
Recall that g ( x ) = log | T ′ ( x ) | for any x ∈ Λ. We define I n ( ω ) := S ω ◦ S ω ◦ · · · ◦ S ω n − ([0 , ω ∈ Σ. For any δ > ǫ ∈ (0 , δ ), α = ( α , . . . , α k ) ∈ R k and n ∈ N , let G k ( α, δ ; n, ǫ ) be the set of all cylinders in Σ n such that forany [ ω ] n ∈ G k ( α, δ ; n, ǫ ) there exists x ∈ I n ( ω ) satisfying the followingproperties(1) (cid:12)(cid:12)(cid:12) n S n φ i ( x ) − α i (cid:12)(cid:12)(cid:12) < ǫ for 1 ≤ i ≤ k ;(2) A n g ( x ) ≥ δ − ǫ .We define the two pressure like functions g k ( α, δ ; s, n, ǫ ) and g ∗ k ( α, δ ; s, n, ǫ )associated to G k ( α, δ ; n, ǫ ) as follows,(4) g k ( α, δ ; s, n, ǫ ) := X [ ω ] n ∈ G k ( α,δ ; n,ǫ ) (diam I n ( ω )) s and(5) g ∗ k ( α, δ ; s, n, ǫ ) := X [ ω ] n ∈ G k ( α,δ ; n,ǫ ) sup x ∈ I n ( ω ) exp( − sS n g ( x ))Let(6) g k ( α, δ ; s ) := lim ǫ → lim sup n →∞ n log g k ( α, δ ; s, n, ǫ )and(7) g k ( α, δ ; s ) := lim ǫ → lim inf n →∞ n log g k ( α, δ ; s, n, ǫ ) . Similarly, we define g ∗ k ( α, δ ; s ) and g ∗ k ( α, δ ; s ). Then we have the fol-lowing Lemma. Lemma 5.
Under the setting above, we have (8) g k ( α, δ ; s ) = g k ( α, δ ; s ) = g ∗ k ( α, δ ; s ) = g ∗ k ( α, δ ; s ) for any s ≥ . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS9
Proof.
The proof can be essentially deduced from [7]. (cid:3)
We denote the common limit above by g k ( α, δ ; s ). Recalling thatthere exists a dense set { f n } ∞ n =1 in C (Λ , R ) and f n n ∈ N .Denote F k : Λ → R k by(9) F k ( x ) = ( f ( x ) , . . . , f k ( x )) . And for any µ ∈ M (Λ , T ), we define(10) Z F k dµ := (cid:18) Z f dµ, . . . , Z f k dµ (cid:19) . For any α = ( α , . . . , α k ), we introduce the norm k · k ∞ on R k by(11) k α k ∞ := sup ≤ i ≤ k | α i | . Proposition 1.
Let m be an invariant measure in M (Λ , T ) such that λ ( m, T ) > and α = R F k dm ∈ R k . For any δ ∈ (0 , λ ( m, T )) and τ ∈ (0 , δ ) , we have (12) g k ( α, δ ; s ) ≤ sup { h ( µ, T ) − sλ ( µ, T ) : µ ∈ X ( α, δ, k, τ ) } , where X ( α, δ, k, τ ) := (cid:26) µ ∈ M (Λ , T ) : (cid:13)(cid:13)(cid:13)(cid:13)Z F k dµ − α (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ τ, λ ( µ, T ) ≥ δ − τ (cid:27) . Proof.
Let a = g k ( α, δ ; s ). We have a = lim ǫ → lim sup n →∞ n log X [ ω ] n ∈ G k ( α,δ ; n,ǫ ) sup x ∈ I n ( ω ) exp( − sS n g ( x )) . We first introduce a probability measure ˜ µ n on Σ n . For any [ ω ] n ∈ G k ( α, δ ; n, ǫ ), we define˜ µ n ( ω ) := sup x ∈ I n ( ω ) exp( − sS n g ( x )) P [ ω ] n ∈ G k ( α,δ ; n,ǫ ) sup x ∈ I n ( ω ) exp( − sS n g ( x ))Now, we construct a n -Bernoulli measure (˜ µ n ) ⊗ on Σ, which is justthe products of countably many copies of (Σ n , ˜ µ n ). Here, we identifythe probability spaces ((Σ n ) N , (˜ µ n ) N ) with (Σ , (˜ µ n ) ⊗ ) by the followingisomorphism Θ : ((Σ n ) N , (˜ µ n ) N ) → (Σ , (˜ µ n ) ⊗ )(13) Θ( θ , θ , . . . , θ n , . . . ) = ( θ θ . . . θ n . . . ) ∗ , YAO XIAO for any ( θ , . . . , θ n , . . . ) ∈ ((Σ n ) N , ( µ n ) N ). It is easy to see that (˜ µ n ) ⊗ isergodic measure on (Σ , σ n ). We refer the interesting readers to [13] fora very good introduction of the n -Bernoulli measure. It follows that h ((˜ µ n ) ⊗ , σ n ) = H (˜ µ n ) = s Z Σ n inf x ∈ I n ( ω ) S n g ( x ) dµ n ( ω ) + I n , where I n := log X [ ω ] n ∈ G k ( α,δ ; n,ǫ ) sup x ∈ I n ( ω ) exp( − sS n g ( x )) . Let µ ⊗ n = π ∗ (˜ µ n ) ⊗ and ν n = ( A n ) ∗ µ ⊗ n . We have ν n ∈ M (Λ , T ) and h ( ν n , T ) − s Z log | T ′ ( x ) | dν n ( x ) ≥ − | s | n sup [ ω ] n sup x,y ∈ I n ( ω ) | S n g ( x ) − S n g ( y ) | + I n n . Also, it is not difficult to prove that for n sufficiently large, we have(14) Z gdν n = Z A n gdµ ⊗ n ≥ δ − ǫ and(15) (cid:12)(cid:12)(cid:12) Z F k dν n − α (cid:12)(cid:12)(cid:12) ∞ ≤ ǫ. Then for any ǫ < τ , we have ν n ∈∈ X ( α, δ, k, τ ) andsup { h ( µ, T ) − sλ ( µ, T ) : µ ∈ X ( α, δ, k, τ ) }≥ lim sup n →∞ (cid:18) h ( ν n , T ) − s Z log | T ′ ( x ) | dν n ( x ) (cid:19) ≥ lim sup n →∞ n log X [ ω ] n ∈ G k ( α,δ ; n,ǫ ) sup x ∈ I n ( ω ) exp( − sS n g ( x )) . Taking ǫ →
0, we complete the proof of Proposition 1. (cid:3)
Proof.
Now we are going to prove the upper bounddim H ( ˜Λ m ) ≤ h ( m, T ) λ ( m, T )of the first part of Theorem 1. Here, we need some ideas of thermal-formalism method. We will make some delicate modification of thearguments in [7]. Some key estimates there break down in our cases,and we need to be very careful to deal with the hyperbolicity.We define h k ( α, δ, s, τ ) := sup { h ( µ, T ) − sλ ( µ, T ) : µ ∈ X ( α, δ, k, τ ) } . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS11
It is not difficult to prove that s h k ( α, δ, s, τ )is strictly decreasing in [0 , + ∞ ). Moreover,(16) h k ( α, δ, , τ ) ≥ , and lim s → + ∞ h k ( α, δ, s, τ ) = −∞ . Thus, there exists unique s ∗ = s ∗ ( α, δ ; k, τ ) ∈ [0 , ∞ ) such that(17) h k ( α, δ, s ∗ , τ ) = 0 . Indeed, we have(18) s ∗ = sup (cid:26) h ( µ, T ) λ ( µ, T ) : µ ∈ X ( α, δ, k, τ ) (cid:27) . Since the map s h k ( α, j , s, τ ) is decreasing and has a zero at s = s ∗ , by Proposition 1, we have t = g k (cid:16) α, j , s ∗ + ǫ (cid:17) < ǫ >
0. Thus, there exists L = L ( j ) ≥ j + 1, for any l ≥ L ,we have(19) lim sup n →∞ n log g k (cid:16) α, j ; s ∗ + ǫ, n, l (cid:17) < − t . It follows that there exists U = U ( l ) ∈ N such that for any u ≥ U ,we have(20) X [ ω ] n ∈ G k ( α, j ,u, l ) (diam I u ( ω )) s ∗ + ǫ ≤ exp( − u t . Recalling that˜Λ m = n x ∈ Λ : ( A n k ) ∗ δ x ∗ −−−→ n →∞ m for some { n k } ∞ k =1 such that lim k →∞ n k = ∞ o , it is straightforward to see that˜Λ m ⊂ ∞ [ j =1 ∞ [ l = L ( j ) ∞ \ k =1 ∞ \ n =1 ∞ [ u = n G k (cid:18) α, j , u, l (cid:19) ⊂ ∞ [ j =1 ∞ [ l = L ( j ) ∞ \ k =1 ∞ [ u ≥ U G k (cid:18) α, j , u, l (cid:19) We denote E j,k,l = ∞ [ u ≥ U G k (cid:18) α, j , u, l (cid:19) . ∗ , YAO XIAO Thus, we proved that H s ∗ + ǫ ( E j,k,l ) ≤ X u ≥ U X [ ω ] n ∈ G k ( α, j ,u, l ) (diam I u ( ω )) s ∗ + ǫ ≤ X u ≥ U exp( − u t < + ∞ . It follows that dim H E j,k,l ≤ s ∗ . And this implies that dim H ˜Λ m ≤ s ∗ ,that is dim H ˜Λ m ≤ sup (cid:26) h ( µ, T ) λ ( µ, T ) : µ ∈ X ( α, δ, k, τ ) (cid:27) ≤ sup (cid:26) h ( µ, T ) λ ( µ, T ) : µ ∈ X ( α, k, τ ) (cid:27) where X ( α, k, τ ) := (cid:26) µ ∈ M (Λ , T ) : (cid:13)(cid:13)(cid:13)(cid:13)Z F k dµ − α (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ τ, λ ( µ, T ) > (cid:27) . Noting that λ ( m, T ) >
0, we have there exists k ∈ N such that M m,k \ M p (Λ , T ) = ∅ for any k ≥ k by Lemma 4.Since both M m,k and M p (Λ , T ) are compact, we know that thereexists γ > d ( ν , ν ) = ∞ X n =1 n k f n k (cid:12)(cid:12)(cid:12) Z f n dν − Z f n dν (cid:12)(cid:12)(cid:12) ≥ γ for any ν ∈ M m,k , ν ∈ M p (Λ , T ). Choose k ∗ ∈ N such that k ∗ ≥ k and k ∗− < γ .We claim that for any k ≥ k ∗ , there exists τ = τ ( k ) such that forany τ ∈ (0 , τ ) we have ˜ X ( α, k, τ ) = X ( α, k, τ ), where˜ X ( α, k, τ ) := (cid:26) µ ∈ M (Λ , T ) : (cid:13)(cid:13)(cid:13)(cid:13)Z F k dµ − α (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ τ (cid:27) . Otherwise, we know that X ( α, k, τ ) ( ˜ X ( α, k, τ ). Thus, there existsa sequence { τ n } ∞ n =1 with lim n →∞ τ n = 0 and invariant measures { µ n } in M (Λ , T ) with λ ( µ n , T ) = 0 such that (cid:13)(cid:13)(cid:13)(cid:13)Z F k dµ n − Z F k dm (cid:13)(cid:13)(cid:13)(cid:13) ∞ = (cid:13)(cid:13)(cid:13)(cid:13)Z F k dµ − α (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ τ n . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS13
We could assume that lim n →∞ µ n = ν for some ν ∈ M (Λ , T ) up to asub-sequence. It follows that λ ( ν, T ) = lim n →∞ λ ( µ n , T ) = 0 and Z F k dν − Z F k dm = 0 . Hence, we get d ( ν, m ) = ∞ X n = k +1 n k f n k (cid:12)(cid:12)(cid:12) Z f n dν − Z f n dm (cid:12)(cid:12)(cid:12) ≤ ∞ X n = k +1 n − ≤ k ∗ − < γ. It is contradictory to the fact that d ( ν, m ) ≥ γ by (21) since ν ∈M p (Λ , T ) and m ∈ M m,k . Since τ is arbitrary, for any k ≥ k ∗ , it iseasy to prove thatdim H ˜Λ m ≤ sup (cid:26) h ( µ, T ) λ ( µ, T ) : Z F k dµ = α (cid:27) = sup (cid:26) h ( µ, T ) λ ( µ, T ) : Z F k dµ = α, λ ( µ, T ) > (cid:27) = sup (cid:26) h ( µ, T ) λ ( µ, T ) : Z F k dµ = Z F k dm, λ ( µ, T ) ≥ δ ∗ (cid:27) for some δ ∗ = δ ∗ ( m ). Noting that ∞ \ k = k ∗ (cid:26) µ ∈ M (Λ , T ) : Z F k dµ = Z F k dm (cid:27) = { m } , it is easy to get dim H ( ˜Λ m ) ≤ h ( m, T ) λ ( m, T ) . (cid:3) For any m in M (Λ , T ) and R >
0, we define B d ( m, R ) := { µ ∈ M (Λ , T ) : d ( µ, m ) ≤ R and λ ( µ, T ) > } . We can prove the following result by almost the same method pre-sented here with minor changes. And we omit its proof here. We pointout this result is essential to the proof of Theorem 2.
Proposition 2.
Let m be an invariant measure in M (Λ , T ) such that λ ( m, T ) > and B d ( m, R ) as above. Furthermore let A r := { x ∈ Λ , Asym ( { ( A n ) ∗ δ x } ∞ n =1 ) ∩ B d ( m, r ) = ∅} ∗ , YAO XIAO for any r ∈ (0 , R ) . Then we have dim H A r ≤ sup (cid:26) h ( µ, T ) λ ( µ, T ) : µ ∈ B d ( m, r ) (cid:27) . Moran constructions
Since the proof of the lower bound of Theorem 1 relies heavily onthe constructions of Moran sets, which can be seen as the generalizedcantor sets with product structures.We first establish a framework of abstract construction of the Moranset, which may be of independent interest. Indeed this kind of tech-niques have been greatly used in a certain amount of literature, forexample see [7, 6, 1], however, it seems that there is no unified frame-work.4.1.
Basic settings in Moran Construction.
Let I = { I n,j : 1 ≤ j ≤ m n , n ∈ N } be a sequence of the intervals in [0 ,
1] such that(1) intI n,i T intI n,j = ∅ for any n ∈ N and 1 ≤ i = j ≤ m n ;(2) I n,i contains at least one element I n +1 ,j for any n ∈ N , and1 ≤ i ≤ m n ;(3) for each n ∈ N , I n +1 ,j is contained in one of the elements in I n,i for 1 ≤ j ≤ m n +1 .We denote r n = min ≤ i ≤ m n diam( I n,i ) and R n = max ≤ i ≤ m n diam( I n,i ). Weassume that lim n →∞ R n = 0. Let Y := ∞ T n =1 m n S i =1 I n,i . Here, I n,i is said to bethe i -th fundamental interval in the n -th level of Y . We will call Y to be Moran set. For any x ∈ Y , there exists a sequence of nestedintervals { I n,l n ( x ) } ∞ n =1 such that x = ∞ T n =1 I n,l n ( x ) .The following observation is crucial in the estimates of the lowerbound of the Hausdroff dimension of the Moran set. Lemma 6.
Assume that the fundamental intervals satisfy the followingconditions lim n →∞ log R n log r n = 1 and lim n →∞ log r n +1 log r n = 1 . (22) Let η be a measure supported on Y which satisfies the balanced property (23) lim n →∞ min ≤ i ≤ m n log η ( I n,i )max ≤ i ≤ m n log η ( I n,i ) = 1 . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS15
Then we have lim sup r → log η ( B ( x, r ))log r = lim sup n →∞ log η ( I n,l n ( x ) )log r n lim inf r → log η ( B ( x, r ))log r = lim inf n →∞ log η ( I n,l n ( x ) )log r n for any x ∈ Y .Proof. For any x ∈ Y , r >
0, there exists a nested sequence of intervals { I k,l k ( x ) } ∞ k =1 , such that x ∈ I k,l k ( x ) for any k ∈ N . And there alwaysexits a unique integer n such that R n +1 < r < R n .We assume that there exists N = N ( x, n, r ) intervals in { I n,j } m n j =1 which intersect with B ( x, r ). It follows that N r n ≤ X B ( x,r ) ∩ I n,j = ∅ | I n,j | ≤ R n . Thus, we get N ≤ R n r n . Combining the fact that B ( x, r ) contains atleast one element in { I n +1 ,j : 1 ≤ j ≤ m n +1 } , we havelog η ( B ( x, r ))log r ≤ min ≤ j ≤ m n +1 log η ( I n +1 ,j )log R n and log η ( B ( x, r ))log r ≥ log R n − log r n log R n +1 + max ≤ j ≤ m n log η ( I n,j )log R n +1 . By (22) and (23), we proved the desired conclusion. (cid:3)
It seems that the conditions in Lemma 6 are too strong and awkwardat the first glance. Indeed they are very useful especially in estimatingthe lower bound of the Hausdroff dimension in the Moran constructions.4.2.
Moran construction driven by dynamical system.
In manyapplications, the fundamental intervals in the different levels in Moranconstruction was obtained by dynamical system. In this section, we tryto introduce a framework for Moran construction driven by dynamicalsystem.For each n ∈ N ∪ { } , let F n := { ˆ I n,j : 1 ≤ j ≤ s n } be a family of s n disjoint closed intervals in [0 ,
1] such that for each n ∈ N there exists l n ∈ N such that T l n ( ˆ I n,j ) ⊃ s n +1 S j =1 ˆ I n +1 ,j for any j ≤ s n .Set n i = i P j =1 l j for i >
0. We denote I k be the family of sub-intervalssuch that for each J ∈ I k , T n k is bijective from J to ˆ I k,j for some j ≤ ∗ , YAO XIAO s k , and T n l ( J ) ∈ F l for 0 ≤ l ≤ k . Indeed, I k is exactly the set of allfundamental intervals of level k and I = S ∞ k =1 I k is the set of allfundamental intervals of different levels. It is easy to see that {I l } ∞ l =0 is a nested sequence of compact set.Denote Y := T ∞ n =1 I n . Let µ i be a sequence of measures on F i and J n be a interval in I n . For any 0 ≤ k ≤ n , there exits t k = t k ( J n ) ≤ s k such that T n k ( J n ) ⊂ ˆ I k,t k . In other words, we have J n = n \ k =0 T − n k ( ˆ I k,t k ) . We can introduce a family of measures η n on I n such that(24) η n ( J n ) = n Y k =0 µ k ( ˆ I k,t k ) . It is easy to see the family of measures η n are compatible to eachother in the following sense, for any J n ∈ I n ,(25) η m ( J n ) = η n ( J n )for any m ≥ n . It is standard to get a measure η on Y by just takingthe weak- ∗ limit of η n . It follows that(26) η ( J n ) = n Y k =0 µ k ( ˆ I k,t k ) . for any J n ∈ I n .For any x ∈ Y , any k ∈ N S { } , there exists J k = J k ( x ) in I k andˆ I k,t k = ˆ I k,t k ( x ) in F k such that x ∈ ∞ \ k =0 J k and T n k ( x ) ∈ ˆ I k,t k .We can transfer Lemma 6 to the following result, which will be veryhelpful in the construction of Moran set. Lemma 7.
Let η satisfies the balanced property defined in Lemma 6.If the following asymptotic additive property (27) lim n →∞ log diam J n ( x ) n P i =0 log diam I i,t i ( x ) = 1 ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS17 and the tempered growth property (28) lim n →∞ log diam I n +1 ,t n +1 ( x ) n +1 P i =0 log diam I i,t i ( x ) = 1 hold, we have lim sup r → log η ( B ( x, r ))log r = lim sup i →∞ log µ i ( I i,t i ( x ))log diam I i,t i ( x )lim inf r → log η ( B ( x, r ))log r = lim inf i →∞ log µ i ( I i,t i ( x ))log diam I i,t i ( x ) for any x ∈ Y . The asymptotic additive property and tempered growth property arecrucial in the construction of Moran set and Moran measure which isa Bernoulli-like measure on the Moran set. By Lemma 7, we only needto gluing a sequence of subsystems carefully in the construction.4.3.
Preparation of geometric Moran construction.
In order toprove Theorem 1, we need to introduce two Moran sets, while theconstructions in the both of the Moran sets share some similar steps inthe initial constructions.We first try to deal with the common steps in the two constructionsa hyperbolic measure µ . In the first case, we will set µ = m in Section5.1, where m is the given hyperbolic measure in Theorem 1. In thesecond case, µ is an arbitrary given hyperbolic measure in Section 5.2.It is very convenient to transfer most of our discussion in the symbolicspace. Let { ǫ i } ∞ i =1 be a decreasing sequence such that lim i →∞ ǫ i = 0.Recall that f and g = log | T ′ | are uniformly continuous on Λ, thereexists k i ∈ N such that(29) var n ≤ j ≤ i A n f j ◦ π < ǫ i , var n A n g ◦ π < ǫ i , | − log D n ( ω ) n − A n g ( π ( ω )) | < ǫ i (for any ω ∈ Σ)for any n ≥ k i . Noting that µ is an ergodic measure, we have for µ a.e ω ,(30) A n f j ( π ( ω )) → R f j ◦ πdµ for 1 ≤ j ≤ i,A n g ( π ( ω )) → λ ( µ, σ ) , − log µ [ ω | n ] n → h ( µ, σ ) . ∗ , YAO XIAO by Birkhoff’s ergodic theorem and Shannon-Mcmillan-Breiman’s theo-rem. For any δ >
0, there exists a compact set Ω ′ ( i ) ⊂ Σ such that(31) µ (Ω ′ ( i )) > − δ and (30) holds uniformly on Ω ′ ( i ) by Egorov’s theorem. Then thereexists m i ≥ k i such that for any n ≥ m i and any ω ∈ Ω ′ ( i ), we have(32) sup ≤ j ≤ i | A n f j ( π ( ω )) − R f j dµ | < ǫ i , | A n g ( π ( ω )) − λ ( µ, σ ) | < ǫ i , | − log µ [ ω | n ] n − h ( µ, σ ) | < ǫ i , In this way, we get a good block of length at least m i with nicestatistical behaviour. A very naive idea is to product the blocks we havegot in each step to construct the Moran set and introduce a productmeasure on itself. While, this construction is not really good since itis possible that lim sup i →∞ m i +1 m i = ∞ , and this makes it difficult to study the statistical analysis of the pointsin the Moran set. To overcome this difficulty, we introduce a newsequence with very slow growth and the corresponding sequence ofblocks still have well controlled statistical behaviors.We can assume that m i +1 ≥ m i . The key point in the constructionof Moran set is to construct the a suitable sequence { l i,j : 0 ≤ j ≤ p i , i ∈ N } such that(1) p i = m i +1 − m i − l i,j = m i + j .It is a simple observation that l i,p i = m i +1 − i →∞ sup ≤ j
In this section, we will prove the lower bound for Theorem 1. Andwe will separate the proof into two cases.5.1.
The construction of the first Moran set.
Let m be a hyper-bolic measure. LetΣ( i, j ) = { ω ω . . . ω l i,j | ω = ( ω n ) ∞ n =1 ∈ Ω ′ ( i ) } and Ω( i, j ) = { ω ∈ Σ : ω ω . . . ω l i,j ∈ Σ( i, j ) } . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS19 . . . Σ( l,
0) Σ(1 ,
1) Σ(1 , p ) . . . Σ(2 ,
0) Σ(2 ,
1) Σ(2 , p ) . . . Σ( k,
0) Σ( k,
1) Σ( k, p k ) ... . . . l l l p l l l p l k l k l kp k Figure 2.
Construction of the Moran setNow, we will use the framework discussed in
Section µ = m .By the construction in Section 4.3 , we have µ (Ω( i, j )) ≥ µ (Ω ′ ( i )) ≥ − δ for 1 ≤ j ≤ p i .We define the concatenation of Σ( i, , Σ( i, . . . , and Σ( i, p i ) asfollows. p i Y j =0 Σ( i, j ) := (cid:8) ω ω . . . ω l i, ω ω . . . ω l i, , . . . , ω ω . . . ω l i,pi (cid:9) . Similarly, we define the geometric Moran construction in the following, M := ∞ Y i =1 p i Y j =0 Σ( i, j ) . (See figure 2 for the illustration of the construction of the Moran set.)We relabel the sequences { l i,j : 0 ≤ j ≤ p i , i ∈ N } and { Σ( i, j ) : 0 ≤ j ≤ p i , i ∈ N } by { l ∗ i } ∞ i =1 and { Σ ∗ ( i ) } ∞ i =1 for convenience. Correspondingly,we also use the notations { Ω ∗ i } and { ǫ ∗ i } ∞ i =1 . It follows from (33) thatlim i →∞ l ∗ i +1 l ∗ i = 1 . ∗ , YAO XIAO By the Stolz’s theorem, we have(34) lim n →∞ l ∗ + l ∗ + . . . l ∗ n +1 l ∗ + l ∗ + . . . l ∗ n = 1 . Lemma 8.
For any i ∈ N and ω ∈ M , we have lim n →∞ A n f i ( π ( ω )) = Z f i dµ. Proof of Lemma 8
We denote n q = P qj =1 l ∗ j for any q ∈ N . By (34),we have lim q →∞ n q +1 n q = 1 . It is easy to see that lim q →∞ A n q f i ( π ( ω )) = Z f i dµ for each i ∈ N . For n q ≤ n < n q +1 , we have1 n (cid:12)(cid:12)(cid:12)(cid:12) S n f i ( π ( ω )) − n Z f i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:12)(cid:12)(cid:12)(cid:12) S n q f i ( π ( ω )) − n q Z f i dµ (cid:12)(cid:12)(cid:12)(cid:12) + 1 n (cid:12)(cid:12)(cid:12)(cid:12) S n − n q f i ( σ n − n q π ( ω )) − ( n − n q ) Z f i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:12)(cid:12)(cid:12)(cid:12) S n q f i ( π ( ω )) − n q Z f i dµ (cid:12)(cid:12)(cid:12)(cid:12) + 2( n − n q ) k f i k n ≤ n q (cid:12)(cid:12)(cid:12)(cid:12) S n q f i ( π ( ω )) − n q Z f i dµ (cid:12)(cid:12)(cid:12)(cid:12) + 2( n q +1 − n q ) k f i k n q . Taking n goes to infinity, we complete the proof of Lemma 8. (cid:3) By Lemma 8, it is easy to check that π ( M ) ⊂ Λ µ . Now, we willconstruct a probability measure η on M , and we call it Moran measure.For each w ∈ Σ ∗ ( i ), we define(35) ρ iw := µ ([ w ]) µ (Ω ∗ ( i )) . It is seen that P w ∈ Σ ∗ ( i ) ρ iw = 1 . Let C n := { [ w ] : w ∈ Q ni =1 Σ ∗ ( i ) } . Foreach w = w · · · w n ∈ C n , we define(36) ν ([ w ]) := n Y i =1 ρ iw i . We still denote the extension to Borel σ -algebra σ ( C n : n ≥
1) of M by ν . Let η = π ∗ ν and { I n,j : 1 ≤ j ≤ k n } = π ( n Q i =1 Σ ∗ ( i )). ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS21
We will split the remaining part of the proof into two steps. In thisfirst step, we are going to prove the balanced property for η . For each I n,j , there exists ω = ω ω . . . ω n ∈ C n such that π ( ω ) = I n,j . By thedefinition of η , we know that η ( I n,j ) = n Q i =1 µ ([ w i ]) µ (Ω ∗ ( i )) .It follows from (32), we have − n X i =1 l i ( h ( µ, σ ) + ǫ i ) ≤ log η ( I n,j ) ≤ − n log(1 − δ ) − n X i =1 l i ( h ( µ, σ ) − ǫ i ) . Noting that the both sides of the above inequality are independent ofthe index j , we obtainmin ≤ i ≤ k n log η ( I n,i )max ≤ i ≤ k n log η ( I n,i ) ≥ n log(1 − δ ) + P ni =1 l i ( h ( µ, σ ) − ǫ ∗ i ) P ni =1 l i ( h ( µ, σ ) + ǫ ∗ i )and min ≤ i ≤ k n log η ( I n,i )max ≤ i ≤ k n log η ( I n,i ) ≤ P ni =1 l i ( h ( µ, σ ) + ǫ ∗ i ) n log(1 − δ ) − P ni =1 l i ( h ( µ, σ ) − ǫ ∗ i ) . This proves the balanced property for η as n goes to infinity.In the second step, we will show that the fundamental intervals { I n,k :1 ≤ k ≤ k n , n ∈ N } satisfy the tempered growth property. By Lemma1 we can getlog diam I n,k = log diam D P nj =1 l ∗ j ( ω ) ≤ − ( S P nj =1 l ∗ j g ( π ( ω )) − n X j =1 l ∗ j ǫ ∗ n )= − n X j =1 S l ∗ j g ( π ( σ l ∗ + ...l ∗ j − ω )) + n X j =1 l ∗ j ǫ ∗ n ≤ − n X j =1 l ∗ j λ ( µ, σ ) + n X j =1 l ∗ j ( ǫ ∗ n + ǫ ∗ j ) . Thus, we have(37) log R n ≤ − n X j =1 l ∗ j λ ( µ, σ ) + n X j =1 l ∗ j ( ǫ ∗ n + ǫ ∗ j )and(38) log r n ≥ − n X j =1 l ∗ j λ ( µ, σ ) − n X j =1 l ∗ j ( ǫ ∗ n + ǫ ∗ j ) . ∗ , YAO XIAO By the Stolz’s theorem, it is not difficult to see thatlim n →∞ log R n log r n = 1 and lim n →∞ log r n +1 log r n = 1 . (39)It follows from (5.1), (37) and (38) thatlim n →∞ max j (cid:12)(cid:12)(cid:12) log η ( I n,j )log diam( I n,j ) − h ( µ, σ ) λ ( µ, σ ) (cid:12)(cid:12)(cid:12) = 0 . By Lemma 6, we havelim r → log η ( B ( x, r ))log r = h ( µ, σ ) λ ( µ, σ )for any x ∈ π ( M ). Recall that µ = m , we obtain(40) dim H Λ m ≥ dim H π ( M ) ≥ h ( m, σ ) λ ( m, σ ) . Noting that dim H Λ m ≤ dim H ˜Λ m , we havedim H Λ m = dim H ˜Λ m = h ( m, σ ) λ ( m, σ ) . This also recover first part of Theorem B in our setting.5.2.
The construction of second Moran set.
In order to provethe lower bound of the second part of Theorem 1, we need to makesome modifications of the Moran set M constructed above. Let µ bearbitrary hyperbolic measure. Since m is a parabolic ergodic measure,we assume that m supports on π (1 ∞ ) without loss of generality. Thereexists a sequence { k i } ∞ i =1 such thatlim i →∞ k i = ∞ , lim i →∞ k i +1 k i = 1 , lim i →∞ k i ǫ i = 0 . Let k i,j = k i for 1 ≤ j ≤ p i . DenoteΣ( i, j ) = { ω ω . . . ω l i,j k i,j l i,j | ω ∈ Ω ′ ( i ) } and Ω( i, j ) = { w : ω ω . . . ω l i,j k i,j l i,j ∈ Σ( i, j ) } . Next, we still use the framework discussed in
Section
Section 4.3 , we have µ (Ω( i, j )) ≥ µ (Ω ′ ( i ) ≥ − δ. We define a new Moran set in the following,ˆ M := ∞ Y i =1 p i Y j =0 Σ( i, j ) ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS23 and relabel the sequences { l i,j : 0 ≤ j ≤ p i , i ∈ N } and { Σ( i, j ) : 0 ≤ j ≤ p i , i ∈ N } by { l ∗∗ i } ∞ i =1 and { Σ ∗∗ ( i ) } ∞ i =1 . Analogously, we also takethe relabelling sequences { Ω ∗∗ ( i ) } ∞ i =1 , { k ∗∗ i } ∞ i =1 and { ǫ ∗∗ i } ∞ i =1 . For eachˆ w = w k ∗∗ i l ∗∗ i ∈ Σ ∗∗ ( i ), we define a probability measure on Σ ∗∗ ( i ) by(41) ρ i ˆ w := µ ([ w k ∗∗ i l ∗∗ i ]) µ (Ω ∗∗ ( i )) . Denote C n := { [ w ] : w ∈ Q ni =1 Σ ∗∗ ( i ) } . For each w = w · · · w n ∈ C n , wedefine(42) ˆ ν ([ w ]) := n Y i =1 ρ iw i . This measure can be uniquely extended to ˆ M and we still denote it byˆ ν . Let { ˆ I n,j : 1 ≤ j ≤ k n } = π ( n Q i =1 Σ ∗∗ ( i )) and denote ˆ η = π ∗ ˆ ν . We canget the following estimates, − n X i =1 l ∗∗ i ( h ( µ, σ ) + ǫ ∗∗ i ) ≤ log ˆ η ( ˆ I n,k ) ≤ − n X i =1 l ∗∗ i ( h ( µ, σ ) − ǫ ∗∗ i ) , log diam( ˆ I n,k ) ≤ − n X j =1 l ∗∗ j λ ( µ, σ ) + n X j =1 l ∗∗ j (1 + k ∗∗ j ) ǫ ∗∗ j , log diam( ˆ I n,k ) ≥ − n X j =1 l ∗∗ j λ ( µ, σ ) − n X j =1 l ∗∗ j (1 + k ∗∗ j ) ǫ ∗∗ j for any 1 ≤ k ≤ k n . It is almost the same to prove that π ( ˆ M ) ⊂ Λ m and dim H π ( ˆ M ) ≥ h ( µ,T ) λ ( µ,T ) . Noting that µ is arbitrary, we have(43) dim H Λ m ≥ sup µ (cid:26) h ( µ, T ) λ ( µ, T ) : λ ( µ, T ) > , µ is ergodic (cid:27) . We need the following lemma about the hyperbolic dimension of Λ.
Lemma 9 ([24]) . Under the setting above, we have (44) dim hypH
Λ = sup µ (cid:26) h ( µ, T ) λ ( µ, T ) : λ ( µ, T ) > , µ is ergodic (cid:27) . By Lemma 9, we obtain dim H Λ m ≥ dim hypH Λ. It is proved in [10] if T is a C map, we havedim H Λ = sup µ (cid:26) h ( µ, T ) λ ( µ, T ) : λ ( µ, T ) > , µ is ergodic (cid:27) . ∗ , YAO XIAO Hence, we get dim H Λ m = dim H Λif T is a C map. This recovers the second part of Theorem B. Notingthat dim H ˜Λ m ≥ dim H Λ, we complete the second part of Theorem 1.6.
Proof of Theorem 2
In this section, we will prove Theorem 2. Recall that g ( x ) = log | T ′ ( x ) | for any x ∈ Λ. By Theorem 1, we only need to prove that dim H Λ δ p =dim H ˜Λ δ p with the assumption(45) dim hypH Λ < dim H Λ . Let Λ ∗ := { x ∈ Λ : lim inf n →∞ A n g ( x ) > } and Λ ∗ := { x ∈ Λ : lim sup A n g ( x ) > } . It is proved in [13] that(46) dim hypH
Λ = dim H Λ ∗ . By (46) and (45), we know thatdim H Λ = dim H Λ \ Λ ∗ . We write Λ ∗ = S ∞ k =1 Λ ∗ k , where Λ ∗ k = { x ∈ Λ ∗ : lim sup n →∞ A n g ( x ) > k } for any k ∈ N . For any n ∈ N , we denote C n := (cid:26) µ ∈ M (Λ , T ) : d ( µ, δ p ) ≥ n (cid:27) . Since T has only one fixed point, we know that λ ( µ, T ) > µ ∈ C n . Noting that C n is compact, then for any positive real num-ber ρ n < n , there exists finitely many invariant measures { µ i } l n i =1 ⊂ C n for some l n ∈ N such that C n ⊂ S l n i =1 intB d ( µ i , ρ n ) and δ p / ∈ S l n i =1 intB d ( µ i , ρ n ), where intB d ( µ i , ρ n ) is the interior of the closed ball B d ( µ i , ρ n ) = { µ : d ( µ, µ i ) ≤ ρ n } .Let B n := { x ∈ Λ :
Asym ( { ( A m ) ∗ δ x } ∞ m =1 ) ∩ C n = ∅} and B in := { x ∈ Λ :
Asym ( { ( A m ) ∗ δ x } ∞ m =1 ) ∩ B d ( µ i , ρ n ) = ∅} for 1 ≤ i ≤ l n . It is easy to see that(47) Λ ∗ = ∞ [ n =1 B n = Λ \ Λ δ p . ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS25
By Proposition 2, we havedim H B in ≤ sup (cid:26) h ( µ, T ) λ ( µ, T ) : µ ∈ B d ( µ i , ρ n ) (cid:27) ≤ dim hypH Λ . It follows that dim H B n ≤ max ≤ i ≤ l n dim H B in ≤ dim hypH Λ . Now, we get dim H Λ ∗ ≤ dim hypH Λ . Together with (46), we havedim H Λ ∗ = dim hypH Λ . This implies that dim H Λ δ p = dim H Λ. Sincedim H ˜Λ δ p ≥ dim H Λ δ p , we get(48) dim H ˜Λ δ p = dim H Λ δ p . Acknowledgement
We would like to thank Professor Shen Weixiao for drawing our at-tention to recovering the hyperbolicity in the non-uniformly hyperbolicinterval maps along the subsequence. The research was partially sup-ported by National Key R&D Program of China (2020YFA0713300)and NSFC of China (No.11771233, No.11901311).
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Ergodic Theory Dynam.Systems , 36(7):2334–2350, 2016. School of Mathematics and Statistics, Anyang Normal University,Anyang, 455000, P. R. China.
ULTIFRACTAL ANALYSIS IN NON-UNIFORMLY HYPERBOLIC INTERVAL MAPS27 School of mathematics and statistics, Northwestern Polytechni-cal University, Xi’an, 710000, P. R. China.3