A Dichotomy for the Weierstrass-type functions
aa r X i v : . [ m a t h . D S ] A ug A DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS
HAOJIE REN AND WEIXIAO SHENA bstract . For a real analytic periodic function φ : R → R , an inte-ger b ≥ λ ∈ (1 / b , W ( x ) = P n ≥ λ n φ ( b n x ): Either W ( x ) is real ana-lytic, or the Hausdor ff dimension of its graph is equal to 2 + log b λ . Fur-thermore, given b and φ , the former alternative only happens for finitelymany λ unless φ is constant.
1. I ntroduction
We study the fractal properties of the graphs of Weierstrass type functions(1.1) W ( x ) = W φλ, b ( x ) = ∞ X n = λ n φ ( b n x ) , x ∈ R where b >
1, 1 / b < λ < φ ( x ) : R → R is a non-constant Z -periodicLipschitz function. The most famous example, with φ ( x ) = cos(2 π x ), wasintroduced by Weierstrass, and it is a continuous nowhere di ff erentiablefunction, see [10]. The graphs of Weierstrass-type and related functions areamong the most studied objects in fractal geometry since the birth of thissubject, see [5], [8, Section 8.2] and [6, Chapter 5], among many others.The goal of this paper is to prove the following theorem. Main Theorem.
Let b ≥ be an integer, λ ∈ (1 / b , and let φ be a Z -periodic real analytic function. Then exactly one of the following holds: (i) W is real analytic; (ii) the graph of W has Hausdor ff dimension equal to (1.2) D = + log b λ. Moreover, given b and non-constant φ , the first alternative only holds forfinitely many λ ∈ (1 / b , . Kaplan, Mallet-Paret and Yorke [15] proved that in the case that φ is atrigonometric polynomial, either W is a C curve or the box dimension ofthe graph of W is equal to D , without the assumption that b is an integer. Ourtheorem is a similar dichotomy with box dimension replaced by Hausdor ff Date : August 11, 2020. dimension which is much more di ffi cult to compute. The price we payhere is the assumption that b is an integer which enables us to approach theproblem from dynamical point of view.An immediate consequence is the following corollary which in particularrecovers the main theorem in [25]. Corollary 1.1.
Let b ≥ be an integer, λ ∈ (1 / b , and let φ ( x ) = cos(2 π x + θ ) , where θ ∈ R . Then the Hausdor ff dimension of the graph of W is equalto D. Historical remarks.
A map W as in (1.1) has the following remarkableproperty(1.3) W ( x ) = φ ( x ) + λ W ( bx ) , so the graph of W exhibits approximate self-a ffi nity with scales b and 1 /λ ,and it is natural to conjecture that the Hausdor ff dimension of its graph isequal to D . However, one has to be careful since the function W can besmooth for certain choices of λ, b , φ . (This is easily seen: for any real ana-lytic Z -periodic function W and φ ( x ) = W ( x ) − λ W ( bx ), one has W φλ, b ( x ) = W ( x ).) The pioneering work of Besicovitch and Ursell ([5]) showed thatthe Hausdor ff dimension of a function of the form P ∞ n = b − α n φ ( b n x ) is equalto 2 − α provided that b n + / b n → ∞ and log b n + / log b n →
1. (See [1] forrecent advances for maps of such modified form.) A map as in (1.1) is eas-ily seen to be H¨older continuous of exponent 2 − D which implies that theHausdor ff dimension of its graph is at most D . Many authors have studiedthe anti-H¨older property of these functions [15, 23, 22], with the strongestform given in [13], see Theorem 2.2. This anti-H¨older property implies that W is not di ff erentiable and also that the box and packing dimension of itsgraph are equal to D . Moreover, in [22], it is proved that the Hausdor ff dimension of the graph of such a W is strictly greater one. In [20], it wasshown that the Hausdor ff dimension of W has a lower bound of the form D − O (1 / log b ).The first example of maps in the form (1.1) for which the graph is shownto exactly have Hausdor ff dimension D was given by Ledrappier [17]. Us-ing dimension theory for (non-uniformly) hyperbolic dynamical systemsdeveloped in [18] and a Marstrand type projection argument, Ledrappierproved that the Hausdor ff dimension of the graph of a Takagi function (tak-ing b = φ ( x ) = dist( x , Z ) in (1.1)) is equal to D , provided that theBernoulli convolution P n ± (2 λ ) − n has Hausdor ff dimension one. The lastproperty, studied first by Erd¨os [7], was shown by Solomyak [26, 21] tohold for almost every λ ∈ (1 / , λ outside a set of Hausdor ff dimension zero in Hochman [12]. Man-delbrot [19] conjectured that the Hausdor ff dimension of the graph of W DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 3 is equal to D for φ ( x ) = cos(2 π x ) and all λ ∈ (1 / b , b , firstfor λ close to 1 in [3] and then for all λ ∈ (1 / b ,
1) in [25], in which a re-sult of Tsujii [27] also played an important role. See also [16]. The case φ ( x ) = sin(2 π x ) has also been settled shortly after in [29].It had been known much earlier that the Bernoulli convolution has Haus-dor ff dimension less than one when 2 λ is a Pisot number. So Ledrappier’sapproach has it limitation (as already pointed by himself). Built upon thecelebrated breakthrough [12], it has been shown recently in [4] that theHausdor ff dimension of the graph of Takagi functions equal to D for all λ ,via analysis on entropy of convolutions of measures.Let us mention that the box and Hausdor ff dimensions of Weierstrass-type functions with random phases were obtained in respectively [11] and[14]. See also [24].See [2] and also [6, Chapter 5] for more remarks on Weierstrass-typefunctions. Main findings.
We shall now be more technical and explain the mainfindings in this paper. Let Z + denote the set of positive integers and let N denote the set of nonnegative integers. Let Λ = { , , ..., b − } , Λ = S ∞ n = Λ n and Σ = Λ Z + . For j = j j j · ·· ∈ Σ , define(1.4) Y ( x , j ) = Y φλ, b ( x , j ) = − ∞ X n = γ n φ ′ (cid:18) xb n + j b n + j b n − + · · · + j n b (cid:19) , x ∈ R where(1.5) γ = b λ ∈ b , ! . This quantity appeared in [17] as the slopes of the strong stable manifoldsof a dynamical system which has the graph of W | [0 , as an attractor. Inboth the approaches of [17] and [4], certain separation properties of thesefunctions Y ( x , j ) play an important role.These functions Y ( x , j ) are indeed related to the Weierstrass-type functionin a more direct way. Using the identity (1.3) one can show that if W isLipschitz, then W ′ ( x ) = Y ( x , j ) holds for Lebesgue a.e. x ∈ R and for any j ∈ Σ . In particular, we have Y ( x , i ) ≡ Y ( x , j ) for all i , j ∈ Σ in this case. SeeLemma 2.1. Definition 1.1.
Given an integer b ≥ and λ ∈ (1 / b , , we say that a Z -periodic C function φ ( x ) satisfies • the condition (H) ifY ( x , j ) − Y ( x , i ) . , ∀ j , i ∈ Σ . HAOJIE REN AND WEIXIAO SHEN • the condition (H ∗ ) ifY ( x , j ) − Y ( x , i ) ≡ , ∀ j , i ∈ Σ . Surprisingly, nothing happens between these two extreme cases.
Theorem A.
Fix b ≥ integer and λ ∈ (1 / b , . Assume that φ is Z -periodicand C . Then exactly one of the following holds: (i) W φλ, b is C and φ satisfies the condition (H ∗ ); (ii) W φλ, b is not Lipschitz and φ satisfies the condition (H). To prove Theorem A, we introduce a concept called C k -regulating pe-riod which is a real number t for which W ( x + t ) − W ( x ) is C k . A keyestimate is that a positive C -regulating period t is bounded from below interms of the C -norm of W ( x + t ) − W ( x ), provided that W is not-Lipschitz.This is obtained from the anti-H¨older property established in [13, 15]. SeeLemma 2.2.The proof of the main theorem is then completed by the following theo-rem and a theorem in [25]. Theorem B.
If a real analytic Z -periodic function φ ( x ) satisfies the condi-tion (H) for an integer b ≥ and λ ∈ (1 / b , , thendim H ( { ( x , W φλ, b ( x )) | x ∈ [0 , } ) = D . Theorem B is obtained by modifying the argument of [4] where the di-mension of planar self-a ffi ne measures are studied which in particular showsthat the Hausdor ff dimension of W is equal to D in the case φ ( x ) = dist( x , Z )and b =
2. The strong separation property (H) and the real analytic assump-tion compensate the non-linearity we have to face.Indeed, let µ denote the lift of the standard Lebesgue measure on [0 , W | [0 , . By [17], µ and its projections π j µ along the strongunstable manifold of a dynamical system F (which keeps the graph of W | [0 , invariant) are exact dimensional and that dim( π j µ ) is equal to a constant α for typical j ∈ Σ , see §3.1. We need to show that α =
1. The measure π j µ can be decomposed into measures of similar form in smaller scales,see (3.5). Assuming the contrary, we shall apply Hochman’s criterion onentropy increase ([12]) to obtain a contradiction. An important step is tointroduce a suitable sequence of partitions for the space X of the transfor-mations involved, see (3.8). For the case φ = dist( x , Z ), the set X is a subsetof A , , the space of a ffi ne maps from R to R , and a sequence of suitablepartitions were constructed in [4] using a rescaling-invariant metric in thespace A , . Although we do not have such a metric in our nonlinear case,we deduce a strong separation property of maps in X from the condition(H) under the assumption that φ is real analytic, see §5. With this strong DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 5 separation property, we construct a sequence of partitions of X explicitly,see §6. Proof of the Main Theorem.
By Theorems A and B, we know that either (i)or (ii) holds. To show the last statement, we apply Theorem from [25],which asserts that for λ close to 1 / b (i.e. γ close to 1), the graph of W φλ, b hasHausdor ff dimension D >
1. Assume by contradiction that there are infin-itely many λ k ∈ (1 / b ,
1) such that W φλ k , b satisfies (i). Then λ k are boundedaway from 1 and Y φλ k , b ( x , · · · ) ≡ Y φλ k , b ( x , · · · ) , that is,(1.6) ∞ X n = γ n − (cid:0) φ ′ ( x / b n ) − φ ′ (( x + / b n ) (cid:1) = λ = λ k . For each x ∈ R , the left hand side of (1.6) is a power seriesin γ with radius of convergence at least one. It has infinitely many zeroscompactly contained in the unit disk, so φ ′ ( x / b n ) = φ ′ (( x + / b n ) . It follows that φ ′ is a constant, hence φ is a constant, a contradiction! (cid:3) Proof of Corollary 1.1.
By the Main Theorem, it su ffi ces to show that W isnot real analytic. Arguing by contradiction, assume that W is real analytic.Let W ( x ) = P n ∈ Z a n e π inx be the Fourier series expansion of the Z -periodicreal analytic function W . Then | a n | is exponentially small in | n | . However,comparing the Fourier coe ffi cients of both sides of the identity (1.3), weobtain that a b k = ( λ k + e i θ / k ≥
1, absurd! (cid:3)
Problems. (1) Let b > λ ∈ (1 / b ,
1) and φ ( x ) = cos(2 π x ). Does W = W φλ, b have a C k regulating period, 1 ≤ k ≤ ∞ ? If the answer isyes and T > C k regulating period, then we can interpret thegraph of W | [0 , T ) as an invariant repeller of the smooth dynamical sys-tem ( x , y ) ( bx mod T , ( y − cos(2 π x )) /λ + W ( bx mod T ) − W ( bx ))and apply the corresponding dimension theory. If the answer is no,then it would be interesting to study the oscillation of the functions W ( x + T ) − W ( x ) for T > D -dimensional Hausdor ff measure of the graph of W equalto zero, even assuming b is an integer greater than one? It is well-known that the graph of W | J , for any bounded interval J , has finite D -dimensional Hausdor ff measure. HAOJIE REN AND WEIXIAO SHEN
In [22], the case φ ( x ) the Rademacher function and b = φ ( x ) = ( { x } ∈ [0 , / , − { x } ∈ [1 / , , where { x } ∈ [0 ,
1) denote the fractional part of x . In this case, it wasproved that the D -dimensional Hausdor ff measure of the graph of W | [0 , is a positive real number if and only if the Bernoulli convo-lution P n ± λ n is absolutely continuous with respect to the Lebesguemeasure and its density is in the class L ∞ . It is conceivable that forgeneral φ and b , the problem is related to the joint essential bound-edness of the densities of the occupation measures of W ( x ) − Γ u ( x ), u ∈ Σ . See §3 for the definition of Γ u . Organization.
We prove Theorem A in §2. The rest of the paper isdevoted to the proof of Theorem B. In §3, we recall some results from theLedrappier-Young theory and state Theorem B’ which is a reduced form ofTheorem B. The rest of the paper is then devoted to the proof of TheoremB’ and an outline can be found at the end of §3.2.
Acknowledgment.
We would like to thank the participants of the dynam-ical systems seminar in the Shanghai Center for Mathematical Sciences, anin particular, Guohua Zhang for suggesting the name of regulating period.WS is supported by NSFC grant No. 11731003.2. T he conditions (H) and (H ∗ )Throughout we fix an integer b ≥ λ ∈ (1 / b , Z -periodicand continuous function φ : R → R , define W = W φ = W φλ, b as in (1.1). Theorem 2.1.
Assume that φ is Z -periodic and of class C . Then exactlyone of the following holds: (i) W is C and φ satisfies the condition (H ∗ ); (ii) W is not Lipschitz and φ satisfies the condition (H). Remark 2.1.
At the cost of more technicality, the theorem can be provedunder a weaker assumption that φ is C . The main idea of the proof is to analyze the regulating periods of W defined as follows. Definition 2.1.
For each k ∈ Z + , we say that t ∈ R is a C k -regulating period of W = W φ if W ( x + t ) − W ( x ) is a C k function. In this case, we put (2.1) E k ( t ) = sup x ∈ R | ( W ( x + t ) − W ( x )) ( k ) | < ∞ . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 7
It is easy to see that for a given k the set of all C k -regulating periods of W form an additive subgroup of R . If φ is C k , then every number of theform mb − n , where n ∈ Z + and m ∈ Z , is a regulating period of W , which arecalled trivial regulating period. If W is C k , then the subgroup is equal to R .It is fairly easy to show that if W is Lipschitz and φ is C k then W is C k , andthe condition (H ∗ ) holds, see Lemma 2.1. Assuming that W is not Lipschitz,we prove an lower bound of | t | in terms of E ( t ) (Lemma 2.2) and show thatevery C -regulating period is rational (Corollary 2.1).Assuming by contradiction that W is not Lipchitz and φ fails to satisfythe condition (H). We show that violation of the condition (H) yields a non-trivial regulating period of the form 1 / p , where p is an integer greater than 1and co-prime with b . Given such an integer p , we define a renormalization(in §2.2) of φ as follows: R p ( φ ) = X k ∈ Z c kp e π ikx , where c m denotes the m -th Fourier coe ffi cient of φ . The properties that W φ is not Lipschitz and φ does not satisfy the condition (H) are inheritedby the renormalization R p ( φ ) (Proposition 2.1). Hence we can repeat theprocedure infinitely often. However, this would imply that W is Lipschitz,a contradiction!We start with the following easy observation. Lemma 2.1.
If W is Lipschitz and φ is C k for some k ∈ Z + , then W is C k and Y ( x , i ) ≡ Y ( x , j ) for all i , j ∈ Σ .Proof. Assume W is Lipschitz. Then there exists a constant C > x ∈ R , W ′ ( x ) exists and | W ′ ( x ) | ≤ C . From W ( x ) = φ ( x ) + λ W ( bx ), we obtain that W ′ ( x ) = φ ′ ( x ) + γ − W ′ ( bx ) , a . e . It follows that for a.e. x ∈ R , if ( x − n ) ∞ n = is a backward orbit of x then forany n ≥ W ′ ( x − n ) exists, | W ′ ( x − n ) | ≤ C , and W ′ ( x − n − ) = φ ′ ( x − n − ) + γ − W ′ ( x − n ) . Therefore for any i , j ∈ Σ , Y ( x , i ) = Y ( x , j ) = − W ′ ( x )holds for a.e. x ∈ R . Since Y ( x , i ) and Y ( x , j ) are C k − functions, this impliesthat Y ( x , i ) ≡ Y ( x , j ). As W ( x ) = W (0) − R x Y ( t , ) dt is the integral of a C k − function, W is C k . (cid:3) So we need to show that φ satisfies the condition (H) when W is notLipschitz. We shall use the following result due to Hu and Lau, see [13, HAOJIE REN AND WEIXIAO SHEN
Theorem 4.1]. See also Kaplan, Mallet-Parret and York [15] for the casethat φ is a trigonometric polynomial. Theorem 2.2.
Assume that φ is Lipschitz but W is not Lipschitz. Then thereexists c > and κ > such that for any δ ∈ (0 , and any x ∈ R thereexists y ∈ R such that c δ < y − x < δ and | W ( y ) − W ( x ) | ≥ κ | y − x | α , where α = − D. Regulating periods.Lemma 2.2 (Key Estimate) . Suppose that φ is Lipschitz but W is not Lips-chitz. Then there exist constants t > and C > such that if t ∈ R \ { } isa C -regulating period of W, then either | t | > t or E ( t ) | t | D ≥ C . Proof.
Let c , κ be as in Theorem 2.2 and let K > | W ( x ) − W ( y ) | ≤ K | x − y | α for all x , y ∈ R . We may assume that t >
0. Let f ( x ) = W ( x + t ) − W ( x )and choose x such that | W ( x + t ) − W ( x ) | = max x ∈ R | W ( x + t ) − W ( x ) | = : ∆ . Note that f ′ ( x ) =
0. Write x j = x + jt for each j ∈ Z . Claim.
There exist constants t > C > t ≥ t or ∆ ≥ Ct α .To prove this claim, fix a large positive integer m such that(2.2) m α ≥ K / ( κ c α ) . Assume that t < / m . By Theorem 2.2, there exists y such that cmt < y − x < mt and | W ( y ) − W ( x ) | ≥ κ | y − x | α . Let m ′ be minimal such that x m ′ ≥ y . Then cm < m ′ ≤ m , and(2.3) | W ( x m ′ ) − W ( x ) | ≥ | W ( y ) − W ( x ) | − | W ( y ) − W ( x m ′ ) |≥ κ c α m α t α − Kt α ≥ Kt α , where we have used (2.2) for the last inequality. On the other hand, bymaximality of x , we have | W ( x m ′ ) − W ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ′ − X j = ( W ( x j + ) − W ( x j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m ′ ∆ ≤ m ∆ . Together with (2.3), this implies that(2.4) ∆ ≥ Kt α / m . Thus the claim holds with t = / m and C = K / m . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 9
Now let us assume that t < t , so that ∆ ≥ Ct α . Let J = (cid:20) q Ct α ( E + t (cid:21) , where E = E ( t ). For any 0 ≤ j ≤ J , since f ( x j ) − f ( x ) = Z x j x Z xx f ′′ ( y ) dydx , we obtain | f ( x j ) − f ( x ) | ≤ E x j − x ) = E j t ≤ Ct α ≤ ∆ , and hence | f ( x j ) | ≥ ∆ / f ( x j ) f ( x ) >
0. Therefore, for all 0 ≤ k ≤ J , | W ( x k ) − W ( x ) | = k − X j = | f ( x j ) | ≥ k ∆ / . Since | W ( x J ) − W ( x ) | ≤ K | x J − x | α ≤ K J α t α , we obtain(2.5) ∆ ≤ Kt α / J − α . Together with ∆ ≥ Ct α , this implies that J is bounded from above, hence( E + t − α is bounded away from zero. Thus either t or t D E ( t ) is boundedaway from zero. (cid:3) Corollary 2.1. If φ is Lipschitz but W is not Lipschitz, then every C -regulating period of W is rational.Proof. Arguing by contradiction, assume that W has a C -regulating period t ∈ R \ Q . Then for each n ≥ t n : = dist( nt , Z ) is a non-zero C -regulatingperiod, and E ( t n ) = E ( nt ) ≤ nE ( t ) , so by Lemma 2.2, | t n | has a lower bound of the form Cn − / D , where D > C >
0. This contradicts with Dirichlet’s theorem which asserts that foreach irrational real number t and any positive integer Q , there is an integer q with 1 ≤ q ≤ Q such that dist( qt , Z ) < / Q . (cid:3) Lemma 2.3.
Assume that φ is C k for some integer k ≥ and does notsatisfy the condition (H). Assume also that W is not Lipschitz. Then there isan integer p > such that ( p , b ) = , and such that / p is a C k -regulatingperiod of W.Proof. Since φ does not satisfies the condition (H), there exist i , j ∈ Σ with i , j and such that Y ( x , i ) ≡ Y ( x , j ). Without loss of generality, we mayassume that i , j . Let r n = ( i + i b + · · · + i n b n − ) / b n , s n = ( j + j b + · · · + j n b n − ) / b n . Then r n , s n for any n ≥
1. For each n , r n − s n and t n : = dist( r n − s n , Z ) are C k -regulating periods of W . We first prove Claim. sup ∞ n = E k ( t n ) < ∞ .Indeed, for each n ≥ − Y ( b n x , i ) = ∞ X m = γ m φ ′ ( b n − m x + r m ) = n X m = γ m φ ′ ( b n − m x + r m ) + γ n ∞ X ℓ = γ ℓ φ ′ ( b − ℓ x + r n + ℓ ) , − Y ( b n x , j ) = ∞ X m = γ m φ ′ ( b n − m x + s m ) = n X m = γ m φ ′ ( b n − m x + s m ) + γ n ∞ X ℓ = γ ℓ φ ′ ( b − ℓ x + s n + ℓ ) , and hence(2.6) n X m = γ m φ ′ ( b n − m x + r m ) − n X m = γ m φ ′ ( b n − m x + s m ) = γ n ∞ X ℓ = γ ℓ φ ′ ( b − ℓ x + s n + ℓ ) − ∞ X ℓ = γ ℓ φ ′ ( b − ℓ x + s n + ℓ ) , where r = s =
0. Let F n ( x ) = n X m = λ m φ ( b m ( x + r n )) − n X m = λ m φ ( b m ( x + s n )) = W ( x + r n ) − W ( x + s n ) . Then, F ′ n ( x ) = n X m = γ − m φ ′ ( b m ( x + r n )) − n X m = γ − m φ ′ ( b m ( x + s n )) = γ − n n X m = γ n − m (cid:0) φ ′ ( b m x + r n − m ) − φ ′ ( b m x + s n − m ) (cid:1) = ∞ X ℓ = γ ℓ (cid:16) φ ′ ( b − ℓ x + s n + ℓ ) − φ ′ ( b − ℓ x + r n + ℓ ) (cid:17) , where the second equality holds because for any 0 ≤ m ≤ n , b m r n ≡ r n − m , b m s n ≡ s n − m mod 1, and the last equality follows from (2.6). As E k ( t n ) = sup x ∈ R | F ( k ) n ( x ) | , it is bounded from above by a constant. The claimis proved.Note that t n ,
0, so by Lemma 2.2, t n is bounded away from zero. Nowtake n i → ∞ so that r n i → r and s n i → s . As the proof of the claim shows, F ′ n lies in a compact family of C k − functions, so W ( x + r ) − W ( x + s ) is C k .Therefore, t = dist( r − s , Z ) = lim n i →∞ t n i is a C k -regulating period of W . ByCorollary 2.1, t ∈ Q . Since t n is bounded away from zero and for n > m , b m ( r n − s n ) = ( r n − m − s n − m ) mod 1 = ± t n − m mod 1 , we obtain that b m ( r − s ) < Z for all integers m ≥
0. Therefore, t does nothave a finite b -adic expansion. So we can write t in the form q / p with p ≥
1, ( q , p ) = p has a prime factor p with p ∤ b . As DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 11 W ( x + / p ) − W ( x ) is a finite sum of translations of W ( x + r ) − W ( x + s ),hence C k , we obtain that 1 / p is a C k -regulating period of W . (cid:3) Renormalization.
For each C function φ : R → R of period 1 andany integer p >
1, let e R p φ ( x ) = X k ∈ Z c kp e π ikpx , and R p φ ( x ) = X k ∈ Z c kp e π ikx = e R p φ ( x / p ) , where c k is the k -th Fourier coe ffi cient of φ . As c k = O ( k − ), e R p φ ( x ) and R p φ ( x ) are C functions.Let P ( φ ) = { p ∈ Z + : p > , ( p , b ) = , / p is a C -regulating period of W φ } . For each p ∈ P ( φ ), we call R p φ (resp. e R p φ ) a renormalization (resp. pre-renormalzation ) of φ .The main properties of the renormalization is stated in the followingproposition. Proposition 2.1.
Assume that φ is C . Let p ∈ P ( φ ) . Then the followinghold: (1) For S p ( φ ) = φ − e R p φ , W S p φ is C and sup x ∈ R | ( W S p φ ) ′ ( x ) | ≤ C , where C > is a constant depending only on φ . (2) W φ is Lipschitz if and only if W R p φ is Lipschitz. (3) φ satisfies the condition (H) if and only if so does R p φ . (4) If q ∈ P ( R p φ ) then pq ∈ P ( φ ) . We need a lemma to prove the proposition.
Lemma 2.4 (Rescaling) . Let τ be a C k function of period for some k ∈ Z + ,and let ˜ τ ( x ) = τ ( px ) , where p ≥ is an integer with ( p , b ) = . Then, (i) t is a C k -regulating period of W τ if and only if tp is a C k -regulatingperiod of W ˜ τ . (ii) τ satisfies the condition (H) if and only if so does ˜ τ .Proof. It is straightforward to check that W ˜ τ ( x ) = W τ ( px ) for all x ∈ R , so W ˜ τ ( x + t / p ) − W ˜ τ ( x ) = W τ ( px + t ) − W τ ( px ) . The statement (i) follows. To prove (ii), we observe that for ˜ u j , ˜ v j ∈ { , , . . . , b − } , j = , , . . . , there exists u j , v j ∈ { , , . . . , b − } , j = , , . . . , such that p ( ˜ u + ˜ u b + · · · ˜ u n b n − ) = u + u b + · · · + u n b n − mod b n , p (˜ v + ˜ v b + · · · ˜ v n b n − ) = v + v b + · · · + v n b n − mod b n . Vice versa, since ( p , b ) =
1, given u , u , · · · , v , v , · · · , we can find ˜ u , ˜ u , · · · , ˜ v , ˜ v , · · · so that the above properties hold. Moreover, u u · · · = v v · · · if and onlyif ˜ u ˜ u · · · = ˜ v ˜ v · · · . Since Y ˜ τ ( x , ˜ u ˜ u · · · ) − Y ˜ τ ( x , ˜ v ˜ v · · · ) = p ( Y τ ( px , u u · · · ) − Y τ ( px , v v · · · )) , the statement follows. (cid:3) Proof of Proposition 2.1. (1) Note W φ ( x ) = W e R p φ ( x ) + W S p φ ( x ). For each m ∈ Z , let a m = R W φ ( x ) e − π imx dx be the m -th Fourier coe ffi cient of W φ ( x ).Note that Z W e R p φ ( x ) e − π imx dx = ( a m if p | m p ∤ m , and Z W S p φ ( x ) e − π imx dx = ( p | m , a m if p ∤ m . Thus for all m ∈ Z ,(1 − e π im / p ) Z W φ ( x ) e − π imx dx = (1 − e π im / p ) Z W S p φ ( x ) e − π imx dx , i.e., Z ( W φ ( x + / p ) − W φ ( x )) e − π imx dx = Z ( W S p φ ( x + / p ) − W S p φ ( x )) e − π imx dx . Therefore W S p φ ( x + / p ) − W S p φ ( x ) = W φ ( x + / p ) − W φ ( x ) is C . Further-more, | a m | , p ∤ m , is of order m − . For p | m , the m -th Fourier coe ffi cient of W S p φ is zero, and for p ∤ m , it is a m . Thus W S p φ ( x ) is C . As in Lemma 2.1,we have ( W S p φ ) ′ ( x ) = − P ∞ n = γ n ( S p φ ) ′ ( x / b n ). So W S p φ is C . Since | ( S p φ ) ′ ( y ) | = | X p ∤ m c m me π imy | ≤ X m ∈ Z | mc m | = : C < ∞ , sup x | ( W S p φ ) ′ ( x ) | ≤ C γ/ (1 − γ ) = : C .(2) Since W e R p φ ( x ) = W R p φ ( px ), W R p φ ( x ) is Lipschitz if and only if so is W e R p φ . By (1), W φ ( x ) − W e R p φ ( x ) is Lipschitz. So the statement holds.(3) Since W S p φ ( x ) = W φ ( x ) − W e R p φ ( x ) is Lipschitz, by Lemma 2.1, Y S p φ ( x , i ) ≡ Y S p φ ( x , j ) for any i , j ∈ Σ . So φ satisfies the condition (H) if and only if sodoes e R p φ . By Lemma 2.4 (ii), e R p φ satisfies the condition (H) if and only ifso does R p φ . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 13 (4) Since R p φ is C , by Lemma 2.4 (i), pq ∈ P ( e R p φ ). Since W S p φ is C ,this implies that pq ∈ P ( φ ). (cid:3) We shall now complete the proof of Theorem 2.1.
Completion of proof of Theorem 2.1. If W is Lipschitz, then (i) holds byLemma 2.1. Assume now that W is not Lipschitz and let us prove that(ii) holds. Arguing by contradiction, assume that φ does not satisfies thecondition (H). By Lemma 2.3, P ( φ ) is not empty.Given p ∈ P ( φ ), by Proposition 2.1, W R p φ is not Lipschitz and R p φ doesnot satisfies the condition (H). So by Lemma 2.3, there is q ∈ P ( R p φ ).By Proposition 2.1 (4), pq ∈ P ( φ ). By definition, p , q ≥
2, so pq > p .Therefore, P ( φ ) is an infinite set.Let p < p < · · · be the elements of P . Then clearly W e R pk φ ( x ) → x ∈ R . By Proposition 2.1 (1), sup | ( W e R pk φ − W ) ′ ( x ) | ≤ C . Itfollows that W is Lipschitz, a contradiction! (cid:3)
3. P reliminaries for the proof of T heorem BIn the remainder of the paper, we shall prove Theorem B. So fix an integer b ≥ λ ∈ (1 / b ,
1) and assume that φ is a Z -periodic analytic functionwhich satisfies the condition (H). We start with recalling some basic factsfrom the Ledrappier-Young theory.A probability measure ω in a metric space X is called exact-dimensional if there exists a constant α ≥ ω -a.e. x ,lim r → log ω ( B ( x , r ))log r = α. In this case, we write dim ω = α . By the mass distribution principle, thisimplies that for any Borel subset E of X with ω ( E ) >
0, we have dim H ( E ) ≥ α. Ledrappier’s Theorem.
Let µ denote the pushforward of the Lebesguemeasure in [0 ,
1) to the graph of W by x ( x , W ( x )). To complete the proofof Theorem B, it su ffi ces to show that dim( µ ) ≥ D , since it is well-knownthat W ( x ) is a C − D function and hence the Hausdor ff dimension of its graphis at most D .The graph of W | [0 , is invariant under the dynamical system F : [0 , × R → [0 , × R , ( x , y ) bx mod 1 , y − φ ( x ) λ ! and µ is an invariant probability measure. The Ledrappier-Young’s dimen-sion theory of dynamical systems applies in this setting, which relates the dimension of µ with its projection along some dynamical defined flows. Weshall now recall the results obtained in Ledrappier [17].As before let Λ = { , , . . . , b − } and let Σ = Λ Z + . Let σ : Σ → Σ denote the shift map ( i i · · · ) ( i i · · · ). Let ν denote the even distributedprobability measure on Λ and let ν Z + denote the product (Bernoulli) measureon Σ . For each i ∈ Λ , define(3.1) g i ( x , y ) = (cid:18) x + ib , λ y + φ (cid:18) x + ib (cid:19) (cid:19) . Define the ‘inverse’ of F as G : [0 , × R × Σ → [0 , × R × Σ , ( x , y , i ) ( g i ( x , y ) , σ ( i )) . Then(3.2) µ = b b − X i = g i µ. Direct computation shows that Dg i ( x , y ) Y ( x , i ) ! = b Y (( x + i ) / b , σ ( i )) ! . So Dg i n g i n − · · · g i contracts the vector (1 , Y ( x , i )) at the exponential rate − log b . Let Γ i ( x ) = Z x Y ( t , i ) dt . So for each y , x y +Γ i ( x ) is the integral curve of the vector filed (1 , Y ( x , i ))which passes through (0 , y ). For each i ∈ Σ , this defines a foliation in[0 , × R whose leaves are “parallel” to each other. For i ∈ Σ , define(3.3) π i ( x , y ) = y − Γ i ( x ) , ( x , y ) ∈ [0 , × R . So π i is the projection of ( x , y ) into the line x = { y +Γ i ( x ) } y ∈ R . We call π i the flow projection function with respect to i .The following result is a part of [17, Proposition 2] which serves as ourstarting point to calculate the Hausdor ff dimension of the graph of W . Theorem 3.1. If φ : R → R is a Z -periodic continuous piecewise C func-tion, then (1) µ is exact dimensional; (2) there is a constant α ∈ [0 , such that for ν Z + -a.e. i ∈ Σ , π j µ is exactdimensional and dim( π j µ ) = α . (3)(3.4) dim( µ ) = + ( D − α. Therefore, Theorem B is reduced to the following
DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 15
Theorem B’.
Fix an integer b ≥ and λ ∈ (1 / b , . Assume that φ is a realanalytic Z -periodic function which satisfies the condition (H). Then α = ,where α is the constant in Theorem 3.1. A transition formula.
We shall follow the strategy in [4], built on [12],to prove Theorem B’. For i = i i · · · i n ∈ Λ n , write g i = g i ◦ g i ◦ · · · g i n . Byiterating the formula (3.2), we obtain µ = b n X i ∈ Λ n g i µ and hence for each j ∈ Σ , π j µ decomposes into measures on small scales as(3.5) π j µ = b n X i ∈ Λ n π j ◦ g i µ. This resembles the case of self-similar / self-a ffi ne measures, as the maps π j g i satisfies the following transition rule, which implies that each of themeasure in the right hand side of (3.5) is a translated rescaling of a measureof the form π i µ .Recall that Λ = S ∞ n = Λ n . For each i = i i · · · i n ∈ Λ , set | i | = n and(3.6) i ∗ = i n i n − · · · i . Lemma 3.1.
For any j ∈ Σ and i ∈ Λ , (3.7) π j g i ( x , y ) = λ | i | π i ∗ j ( x , y ) + π j g i (0) Proof.
By induction it su ffi ces to consider the case i = i ∈ Λ . According todefinition, we have π j g i ( x , y ) = π j (cid:18) x + ib , λ y + φ (cid:18) x + ib (cid:19)(cid:19) = λ y + φ (cid:18) x + ib (cid:19) + Z x + ib ∞ X n = γ n φ ′ (cid:18) sb n + j b n + · · · + j n b (cid:19) ds = λ y + λ Z x γφ ′ ( u + ib ) du + λ Z x ∞ X n = γ n + φ ′ (cid:18) ub n + + ib n + + · · · + j n b (cid:19) du + π j g i (0) . (cid:3) To apply the argument in [4], we need to show the following:(i) Most of the measures in the right hand side of (3.5) has certain en-tropy porous property. This will be done in §4 and is similar to thecorresponding part of [4].(ii) Maps in the space(3.8) X = { π j ◦ g i (cid:12)(cid:12)(cid:12) j ∈ Σ , i ∈ Λ } , satisfy a suitable separation condition. This will be done in §5 andour argument uses essentially the real analytic assumption on φ .This separation property enables us to define a sequence of suitablepartitions of X in §6.After these preparations, the proof of Theorem B’ will be given in §7.3.3. Entropy of measures.
We shall recall definition and basic propertiesof entropy of measures which is a basic tool for the proof of Theorem B’.Consider a probability space ( Ω , B , ω ). A (countable) partition Q is acountable collection of pairwise disjoint measurable subsets of Ω whoseunion is equal to Ω . We use Q ( x ) to denote the member of Q which contains x . If ω ( Q ( x )) >
0, then we call the conditional measure ω Q ( x ) ( A ) = ω x , Q ( A ) = ω ( A ∩ Q ( x )) ω ( Q ( x ))a Q -component of ω . We define the entropyH ( ω, Q ) = X Q ∈Q − ω ( Q ) log b ω ( Q ) , where the common convention 0 log 0 = P , we define the condition entropy as H ( ω, Q|P ) = X P ∈P ,ω ( P ) > ω ( P ) H ( ω P , Q ) . When Q is a refinement of P , i.e., Q ( x ) ⊂ P ( x ) for each x ∈ Ω , we have H ( ω, Q|P ) = H ( ω, Q ) − H ( ω, P ) . We shall consider the case where there is a sequence of partitions Q i , i = , , · · · , such that Q i + is a refinement of Q i . In this situation, we shallwrite ω x , i = ω x , Q i , and call it a i-th component measure of ω . For a finite set I of positive integers, suppose that for each i ∈ I , there is a random variable f i defined over ( Ω , B ( Q i ) , ω ), where B ( Q i ) is the sub- σ -algebra of B whichis generated by Q i . Then we shall use the following notation P i ∈ I ( B i ) = P ω i ∈ I ( B i ) : = I X i ∈ I ω ( B i ) , where B i is an event for f i . If f i ’s are R -valued random variable, we shallalso use the notation E i ∈ I ( f i ) = E ω i ∈ I ( f i ) : = I X i ∈ I E ( f i ) . For example, we have H ( ω, Q m + n |Q n ) = E ( H ( ω x , n , Q m + n )) = E i = n ( H ( ω x , i , Q i + m )) . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 17
These notations were used extensively in [12] and [4].In particular, we shall often consider the case
Ω = R and B the Borel σ -algebra. Let L n denote the partition of R into b -adic intervals of level n ,i.e., the intervals [ j / b n , ( j + / b n ), j ∈ Z . Let P ( R ) denote the collectionof all Borel probability measures in R . For an exact dimensional probabilitymeasure ω ∈ P ( R ), its dimension is closely related to the entropy, as shownin the following fact which is [28, Theorem 4.4]. See also [9, Theorem 1.3]. Proposition 3.1. If ω ∈ P ( R ) is exact dimensional, then dim( ω ) = lim n →∞ n H ( ω, L n ) . These notations P i ∈ I ( B i ), E i ∈ I ( f i ) will also apply to the case where Ω = X , B is the collection of all subsets of X , and ω is a discrete measure.In the following, we collect a few well-known facts about entropy andconditional entropy. Lemma 3.2 (Concavity) . Consider a measurable space ( Ω , B ) which is en-dowed with partitions Q and P such that P is a refinement of Q . Let ω, ω ′ be probability measures in ( Ω , B ) . The for any t ∈ (0 , ,tH ( ω, Q ) + (1 − t ) H ( ω ′ , Q ) ≤ H ( t ω + (1 − t ) ω ′ , Q ) , tH ( ω, P|Q ) + (1 − t ) H ( ω ′ , P|Q ) ≤ H ( t ω + (1 − t ) ω ′ , P|Q ) . Lemma 3.3.
Let ω ∈ P ( R ) . There is a constant C > such that for anya ffi ne map f ( x ) = ax + c, a , c ∈ R , a , and for any n ∈ N we have (cid:12)(cid:12)(cid:12) H ( f ω, L n + [log b | a | ] ) − H ( ω, L n ) (cid:12)(cid:12)(cid:12) ≤ C . Lemma 3.4.
Given a probability space ( Ω , B , ω ) , if f , g : Ω → R aremeasurable and sup x | f ( x ) − g ( x ) | ≤ b − n then | H ( f ω, L n ) − H ( g ω, L n ) | ≤ C , where C is an absolute constant.
4. E ntropy porosity
This section is devoted to analysis of entropy porosity of the projectedmeasures π j µ . This property will be used in applying Hochman’s criterionto obtain entropy growth under convolution. Definition 4.1 (Entropy porous) . Let ω ∈ P ( R ) . We say that ω is ( h , δ, m ) -entropy porous from scale n to n if P ω n ≤ i ≤ n m H ( ω x , i , L i + m ) < h + δ ! > − δ. The main result of this section is the following Theorem 4.1. Before thestatement of the theorem, we need to introduce a notation.
Notation.
For each integer n ≥
0, let ˆ n be the unique integer such that(4.1) λ ˆ n ≤ b − n < λ ˆ n − . In particular, ˆ0 =
0. With this notation, there is a constant C > j ∈ Σ , i ∈ Λ ˆ n and any m ∈ N ,(4.2) (cid:12)(cid:12)(cid:12) H ( π j g i µ, L n + m ) − H ( π i ∗ j µ, L m ) (cid:12)(cid:12)(cid:12) ≤ C . Indeed, by Lemma 3.1, π j g i µ is equal to the pushforward of π i ∗ j µ by a map λ | i | x + c , for some c ∈ R . So the statement follows from Lemma 3.3. Theorem 4.1.
Fix an integer b ≥ and λ ∈ (1 / b , . Assume that φ : R → R is a Z -periodic piecewise C function such that W = W φλ, b is nota Lipschitz function. Then for any ε > , m ≥ M ( ε ) , k ≥ K ( ε, m ) andn ≥ N ( ε, m , k ) , the following holds: For any j ∈ Σ and u ∈ Λ ˆ t , t ∈ N , ν ( i = ( i i · · · ) ∈ Σ : π j g u g i i ··· i ˆ n µ is ( α, ǫ, m ) − entropyporous from scale t + n + to t + n + k )! > − ε. We shall follow the argument in [4, Section 3] to prove this theorem.In particular, we shall use (3.5) to decompose a measure π j µ as a convexcombination of measures of the form π j g i µ .4.1. Uniform continuity across scales.
Following [4], we say that a mea-sure ω ∈ P ( R ) is uniformly continuous across scales if for every ε > δ > x ∈ R and r ∈ (0 , ω ( B ( x , δ r )) ≤ εω ( B ( x , r )) . A family M of measures in P ( R ) is called jointly uniformly continuousacross scales if for every ε > δ > ω ∈ M , any x ∈ R and any r ∈ (0 , Lemma 4.1.
Under the assumption of Theorem 4.1, for any ε > thereexists δ = δ ( ε ) > such that for any j ∈ Σ and any y ∈ R , π j µ (cid:0) B ( y , δ ) (cid:1) < ε. Proof.
Arguing by contradiction, assume that this is false. Since the familyof probability measures π j µ is compact in the weak star topology, it followsthat there exists j ∈ Σ and y ∈ R such that π j µ has an atom at y . Thismeans that the set X = { x ∈ [0 ,
1) : W ( x ) = Γ j ( x ) + y } has positive Lebesgue measure. Let x be a Lebesgue density point of X andlet J n be the b -adic interval of level n which contains x . Then | J n ∩ X | / | J n | → n → ∞ . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 19
Let i n ∈ Λ , n = , , . . . , be such that b n x ∈ [ i n / b , ( i n + / b ) mod 1.Let i n = i i · · · i n . Then for each n , g i n maps [0 , × R onto J n × R . ByLemma 3.1, π j g i n ( x , y ) = λ n π j n ( x , y ) + π j g i n (0 , , where j n = i ∗ n j . Note that g i n ( x , W ( x )) = ( S n ( x ) , W ( S n ( x )), where S n ( x ) = i n + i n − b + · · · + i b n − b n + xb n . Thus for x ∈ S − n ( X ∩ J n ) ⊂ [0 , W ( x ) − Γ j n ( x ) = y n : = ( y − π j g i n (0 , /λ n . Thus |{ x ∈ [0 ,
1) : W ( x ) = Γ j n ( x ) + y n }| = | X ∩ J n | / || J n | → , as n → ∞ . In particular, this implies that the sequence y n is bounded. Let n k be a subsequence such that j n k → j ∞ and y n k → y ∞ in respectively Σ and R . Then for Lebesgue a.e. x ∈ [0 , W ( x ) ∈ Γ j ∞ ( x ) + y ∞ . By continuity, itfollows that W ( x ) = Γ j ∞ ( x ) + y ∞ is a C function, a contradiction! (cid:3) Proposition 4.1.
Under the assumption of Theorem 4.1, the family of mea-sures { π j µ } j ∈ Σ is jointly uniformly continuous across scales.Proof. It su ffi ces to prove that there is κ > j ∈ Σ , any x ∈ R and any r ∈ (0 , π j µ ( B ( x , κ r ) ≤ π j µ ( B ( x , r )) . To this end, let δ = δ (1 / > M > δ be a constant such that π j µ is supported in [ − M , M ] for each j ∈ Σ .Put κ = λδ/ (3 M ). Given r ∈ (0 , n = n ( r ) ∈ N such that3 M ≤ λ − n r < λ − M . Note that λ − n κ r < δ < M . We shall show that for each i = i i · · · i n ∈ Λ n ,(4.5) π j g i µ ( B ( x , κ r )) ≤ π j g i µ ( B ( x , r )) . Once this is proved, (4.4) follows from (3.5).To prove (4.5), we first apply Lemma 3.1 and obtain x ( i ) ∈ R , such thatfor any R > π j g i µ ( B ( x , R )) = π i ∗ j ( B ( x ( i ) , λ − n R )) . If | x ( i ) | ≥ M , then B ( x ( i ) , λ − n κ r ) is disjoint from [ − M , M ] since λ − n κ r ≤ M . Thus the left hand side of (4.5) is zero and hence the inequality holds.Assume now that | x ( i ) | < M . Then B ( x ( i ) , λ − n r ) ⊃ [ − M , M ] , so the right hand side of (4.5) is equal to 1 /
2. On the other hand, B ( x ( i ) , λ − n κ r ) ⊂ B ( x ( i ) , δ ) . Thus the left hand side of (4.5) is at most 1 / (cid:3) Corollary 4.1. α > .Proof. By Proposition 4.1, there is δ > π j µ ( B ( y , δ n )) ≤ − n π j µ ( B ( y , j ∈ Σ , n ∈ Z + and y ∈ R . It follows thatlim sup r → log π j µ ( B ( y , r ))log r ≥ log δ − > . Thus α > (cid:3) Entropy porosity of π j µ . In this subsection we complete the proof ofTheorem 4.1.
Lemma 4.2.
For any ε > , m ≥ M ( ε ) , n ≥ N ( ε, m ) , inf j ∈ Σ ν n ( i ∈ Λ n : α − ε < m H ( π ij µ, L m ) < α + ε )! > − ε. Proof.
Denote h m ( j ) = m H ( π j µ, L m ). Let us first show that h m is continuousin j ∈ Σ . Indeed, the supports of supp( π j µ ) are uniformly bounded and j π j µ is continuous in the weak star topology. Since π j µ has no atom, forany I ∈ L m , j π j µ ( I ) is continuous. Thus1 m H ( π j µ, L m ) = m X I ∈L m , I ⊆ [0 , h (cid:0) π j µ ( I ) (cid:1) is continuous in j , where h ( t ) = t log b t is a continuous function in [0 , ∞ ).Since h m converges to α ν Z + -a.e., the sequence { h m } also converges to α in measure, i.e. Ω m : = ( j ∈ Σ : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m H ( π j µ, L m ) − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε ) satisfies ν Z + ( Ω m ) → m → ∞ . So there exists M ( ε ) such that when m ≥ M ( ε ), ν Z + ( Ω m ) > − ε/ m ≥ M ( ε ). As Ω m is an open subset of Σ , there exists N : = N ( m , ε ) such that the union X N of the N -th cylinders completely containedin Ω m has ν Z + -measure greater than 1 − ε . For each n ≥ N , X n ⊃ X N . Thelemma follows. (cid:3) We shall need the following two lemmas which are respectively Lemma3.7 and Lemma 3.10 in [4].
DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 21
Lemma 4.3.
For any ε > , there exists δ > such that the followingholds. Let m , ℓ ∈ N and k > k ( m , ℓ ) be given, and suppose that τ ∈ P ( R ) is a measure and β > is a constant such that for a (1 − δ ) -fraction of ≤ t ≤ k, we can write τ as a convex combination τ = p τ + P i ≥ p i τ i , τ i ∈ P ( R ) , p < δ so as to satisfy the following three conditions (1) m H ( τ i , L t + m ) ≥ β, i ≥ . (2) diam ( supp ( τ i )) ≤ b − ( t + ℓ ) , i ≥ . (3) τ ( I ) < δτ ( J ) whenever I ⊆ J are concentric intervals, | I | = b − ℓ | J | = b − ( t + ℓ ) .Assume further that (cid:12)(cid:12)(cid:12) k H ( τ, L k ) − β (cid:12)(cid:12)(cid:12) < δ . Then τ is ( β, ε, m ) -entropy porousfrom scale to k. Lemma 4.4.
For every ε > there exists δ > with the following property.Let ℓ ∈ N and m > m ( ε, ℓ ) , and let τ ∈ P ( R ) be a measure such that τ ( I ) < δ τ ( J ) whenever I ⊆ J are concentric intervals, | I | = b − ℓ | J | = b − ( k + ℓ ) for every k ∈ N . Let n > n ( m , ℓ ) and suppose that τ is ( α, δ, m ) -entropyporous from scales n to n = n + n. Then for any f ( x ) = ax + c, a ∈ R \ { } and c ∈ R , f τ is ( α, ε, m ) -entropy porous from scales n − [log b | a | ] to n − [log b | a | ] . Lemma 4.5.
Under the assumption of Theorem 4.1, for any ε > , there ex-ists δ > such that if m ≥ M ( ε ) and k ≥ K ( ε, m ) and if (cid:12)(cid:12)(cid:12) k H ( π j µ, L k ) − α (cid:12)(cid:12)(cid:12) < δ , then π j µ is ( α, ε, m ) -entropy porous from scale to k.Proof. (1) Assume without loss of generality that π j µ is supported in [0 , j ∈ Σ . Fix ε >
0. Let δ > δ < α . Let β = α − δ/ > . By Proposition 4.1,there exists ℓ ∈ N , such that for any j ∈ Σ , we have(4.6) π j µ ( I ) < δ π j µ ( J )whenever I ⊆ J are concentric intervals with 1 ≥ | I | = b − ℓ | J | .By Lemma 4.2, when m ≥ M ( ε ) and n ≥ N ( ε, m ), we have(4.7) ν ˆ n ( i ∈ Λ ˆ n : α − δ < m H ( π ij µ, L m ) < α + δ )! > − δ Increasing M ( ε ) if necessary, we may assume that M ( ε ) > C , ℓ ) /δ ,where C is as in (4.2).Fix m > M ( ε ) and assume k > K ( ε, m ) : = N ( ε, m ) /δ . Let us show thatfor any N ( ε, m ) < n ≤ k , and for t = n − ℓ , the measure τ = π j µ can bewritten in the form P p i τ i with the properties (1)-(3) in Lemma 4.3. Indeed, since m > C /δ , by (4.2), for any i ∈ Λ ˆ n , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m H ( π j g i µ, L n + m ) − m H ( π i ∗ j µ, L m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C m < δ . So (4.7) implies that the set I n = ( i ∈ Λ ˆ n : 1 m H ( π j g i µ, L m + n ) > α − δ ) has cardinality greater than (1 − δ ) b ˆ n . We define τ , τ , . . . to be equal to π j g i µ with i ∈ I n , p = p = · · · = b − ˆ n and define p = − I n b − ˆ n and τ to be the average of π j g i µ for those i ∈ Λ ˆ n \ I n . Then τ = p τ + p τ + · · · and p < δ . Moreover,(1) For each i = , , . . . ,1 m H ( τ i , L t + m ) ≥ m ( H ( τ i , L n + m ) − ℓ ) > α − δ = β. (2) Since we assume that all the π j µ are supported in [0 ,
1] and λ ˆ n ≤ b − n by definition of ˆ n , by Lemma 3.1, each of τ , τ , · · · is supported inan interval of length b − n ≤ b − ( t + ℓ ) .(3) The property (3) follows from (4.6).Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k H ( τ, L k ) − β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k H ( τ, L k ) − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | α − β | < δ, by Lemma 4.3, we obtain that τ is ( β, ε, m )-entropy porous from scale 1 to k , hence it is ( α, ε, m )- entropy porous from scale 1 to k . (cid:3) Proof of Theorem 4.1.
By Lemma 3.1, π j g w g i i ··· i ˆ n = λ ˆ n + | w | π i ˆ n ··· i w ∗ j µ + Constant . So by Lemma 4.4, it su ffi ces to prove that when m > M ( ε ), k ≥ K ( ε, m ) and n ≥ N ( ε, m , k ), for any h ∈ Σ ,(4.8) ν (cid:0)(cid:8) i ∈ Σ : π i i ··· i ˆ n h µ is ( α, ε, m ) − entropy porous from scale 1 to k (cid:9)(cid:1) > − ε. Given ε >
0, let δ , M ( ε ) and K ( ε, m ) be given by Lemma 4.5. For this δ > k ≥ K ( δ ) and n ≥ N ( δ, k ), ν ( i ∈ Σ : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k H ( π i ˆ n i ˆ n − ··· i h µ, L k ) − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ )! > − δ. Therefore, when m ≥ M ( ε ), k ≥ max( K ( ε, m ) , K ( δ )) and n ≥ N ( δ, k ), (4.8)holds. (cid:3) DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 23
5. T ransversality
In this section, we deduce from the condition (H) some quantified esti-mates. These estimates will be used to construct a sequence of partitions L n of the space X in the next section which in turn is used in the last sectionto prove Theorem B. The main result of this section is summarized in thefollowing theorem. Theorem 5.1.
Suppose that a real analytic Z -periodic function φ ( x ) sat-isfies the condition (H) for some integer b ≥ and λ ∈ (1 / b , . Thenthere exist positive integers ℓ , Q and a constant ρ > with the followingproperty. For any u , v ∈ Σ with u n , v n , (5.1) sup x ∈ [0 , | Γ ′ u ( x ) − Γ ′ v ( x ) | ≥ ρ b − Q n , and (5.2) X I ∈L ℓ I ⊂ [0 , inf x ∈ I | Γ ′ u ( x ) − Γ ′ v ( x ) | ≥ ρ sup x ∈ [0 , | Γ ′ u ( x ) − Γ ′ v ( x ) | . For the proof, we observe that for any integer k ≥
0, the family Γ ( k ) u , u ∈ Σ , is compact with respect to the topology of uniform convergence in R . Together with the condition (H), this implies the maps in(5.3) F n : = { Γ u − Γ v : u n , v n and u j = v j for 1 ≤ j < n } are uniformly separated with constants depending on n . In order to quan-tify the dependence of the constants in n , we shall use the following factfrequently, which can be checked directly by definition of Γ : If u = ( u m ) ∞ m = , v = ( v m ) ∞ m = ∈ Σ and u = v , u = v , . . . , u n − = v n − butu n , v n , where n ∈ Z + , then for any k ≥ , (5.4) Γ ( k ) u , v ( x ) = (cid:18) γ b k − (cid:19) n − Γ ( k ) σ n − ( u ) ,σ n − ( v ) x + u + · · · + u n − b n − b n − ! , where Γ u , v = Γ u − Γ v . Definition 5.1.
For an integer k ≥ , we say that a map ψ : [ a , b ) → R isk -regular if ψ is C k and sup x ∈ [ a , b ) | ψ ( k ) ( x ) | ≤ x ∈ [ a , b ) | ψ ( k ) ( x ) | . Lemma 5.1.
There exists a constant ε > and a positive integer Q suchthat for any f ∈ F , (5.5) sup x ∈ [0 , | f ′ ( x ) | ≥ ε , and for any x ∈ [0 , , there exists k ∈ { , , . . . , Q } such that (5.6) | f ( k ) ( x ) | ≥ ε . Proof.
This follows from the fact that F is compact with respect to thetopology of uniform convergence in the C k sense. More precisely, if (5.5)fails, then there exists f m ∈ F such that sup x ∈ [0 , | f ′ m ( x ) | < / m . Pass-ing to a subsequence we may assume that there exists f ∈ F such thatsup x ∈ R | f ′ m ( x ) − f ′ ( x ) | →
0. Then f ′ ( x ) = x ∈ [0 , f isreal analytic and f (0) =
0, this implies that f ≡
0. However, F does notcontain the zero function by the condition (H), a contradiction.Similarly, if (5.6) fails, then there exists f m ∈ F and x m ∈ [0 ,
1] suchthat | f ( k ) m ( x m ) | < / m , for each m = , , . . . and k = , , . . . , m . Passingto a subsequence, there exists x ∈ [0 ,
1] and f ∈ F such that x m → x and max x ∈ [0 , | f ( k ) m ( x ) − f ( k ) ( x ) | → m → ∞ , for each k = , , . . . . Itfollows that f ( k ) ( x ) = k ≥
1. As f is real analytic and f (0) = f ≡
0, a contradiction. (cid:3)
Lemma 5.2.
Let ε , Q be as in Lemma 5.1. There exist ℓ ∈ N such thatfor any f ∈ F n , n = , , . . . , and any I ∈ L ℓ with I ⊂ [0 , , f : I → R isk-regular and sup x ∈ I | f ( k ) ( x ) | ≥ ε (cid:16) γ b − k (cid:17) n − for some k ∈ { , , . . . , Q } .Proof. For n =
1, there is a constant C > f ∈ F , | f ( k ) ( x ) | ≤ C for any k = , , . . . , Q + x ∈ [0 , ℓ ≥ f ∈ F and each I ∈ L ℓ with I ⊂ [0 , f : I → R is k -regular and sup x ∈ I | f ( k ) ( x ) | ≥ ε for some k ∈ { , , . . . , Q } .For general n , this follows from (5.4). Indeed, there is u , u , . . . , u n − anda map f ∈ F such that f ′ ( x ) = γ n − f ′ x + u + · · · + u n − b n − b n − ! . For any I ∈ L ℓ , there is J ∈ L ℓ such that x ∈ I ⇒ x + u + · · · + u n − b n − b n − ∈ J . Since f is k -regular in J for some k ∈ { , , . . . , Q } , f is k -regular in I forthe same k . (cid:3) Lemma 5.3.
For any integer k ≥ , there exist δ k > and τ k > suchthat the following holds. Let ψ : [0 , → R be a C k function such that | ψ ( k ) ( x ) | ≥ for all x ∈ [0 , . Then there exists a subinterval J of [0 , suchthat | J | ≥ δ k and such that | ψ ′ ( x ) | > τ k for all x ∈ J. DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 25
Proof.
We prove by induction on k . The starting step k = k < m , m ≥
2. Let us prove it for the case k = m . Assume without loss of generality that ψ ( m ) ( x ) ≥ x ∈ [0 , ψ ( m − (1 / ≥
0. Then ψ ( m − ( x ) ≥ for all x ∈ [3 / , ϕ ( x ) = m ψ (( x + / ϕ ( m − ( x ) ≥ x ∈ [0 , J m − of [0 ,
1) such that | J m − | ≥ δ m − and | ϕ ′ ( x ) | ≥ τ m − for all x ∈ J m − . Put J m = { ( x + / x ∈ J m − } , δ m = δ m − / τ m = (1 / m − τ m − . Then | J m | ≥ δ m and | ψ ′ ( x ) | ≥ τ m for all x ∈ J m . (cid:3) Lemma 5.4.
Assume that f : [ a , b ) → R is k-regular for some positiveinteger k. Then there exists δ k > , ρ k > depending only on k and aninterval J ⊂ [ a , b ) with | J | > δ k ( b − a ) such that inf x ∈ J | f ′ ( x ) | ≥ ρ k sup x ∈ [ a , b ) | f ′ ( x ) | . Proof.
Without loss of generality, we may assume that [ a , b ) = [0 ,
1) andsup ≤ x < | f ( k ) ( x ) | = . (Otherwise, we consider λ f ( λ x + c ) instead of f for suitable choices of λ , λ > c ∈ R .) We may assume that for each 1 ≤ k ′ < k , f : [0 , → R is not k ′ -regular, i.e.(5.7) sup x ∈ [0 , | f ( k ′ ) ( x ) | > x ∈ [0 , | f ( k ′ ) ( x ) | , for otherwise we may work on k ′ instead of k . By the mean value theorem, | f ( k − ( x ) − f ( k − ( y ) | ≤ | x − y | ≤ x , y ∈ [0 , x ∈ [0 , | f ( k − ( x ) | < . But then bythe mean value theorem again | f ( k − ( x ) − f ( k − ( y ) | ≤ | x − y | ≤ x , y ∈ [0 , x ∈ [0 , | f ( k − ( x ) | ≤ . Repeating the process,sup x , y ∈ [0 , | f ′ ( x ) − f ′ ( y ) | ≤ k − . On the other hand, by Lemma 5.3, there exists δ k > τ k > J with | J | ≥ δ k such that | f ′ ( x ) | > τ k for all x ∈ J . The lemma follows bytaking ρ k = τ k / (2 k − + τ k ). (cid:3) Proof of Theorem 5.1.
By Lemma 5.2 and Lemma 5.3, we obtain the firstinequality. By Lemma 5.2 and Lemma 5.4, we obtain the second inequality. (cid:3)
6. T he partitions L X i of the space X In this section, we construct a nested sequence of partitions L X i of thespace X in (3.8) and prove a few key properties of these partitions. Theseparation properties given in Theorem 5.1 play a central role in the proofs.Throughout we fix an integer b ≥ λ ∈ (1 / b ,
1) and we assume that φ : R → R is a real analytic Z -periodic function that satisfies the condition(H).Recall that by Lemma 3.1, for any j ∈ Σ and i ∈ Λ , π j g i ( x , y ) = λ | i | ( y − Γ i ∗ j ( x )) + π j g i (0) . So each member of X can be written in the form λ t ( y − ψ ( x )) + c , where t ∈ N , c ∈ R and ψ ( x ) ∈ C ω ( R ) with ψ (0) =
0. We shall call | i | the height ofthe map π j g i . Define π : X → N × R M + by λ t ( y + ψ ( x )) + c → (cid:18) t , ψ ( 1 M ) , ψ ( 2 M ) , . . . , ψ (1) , c (cid:19) , where M = b ℓ and ℓ comes from Theorem 5.1. Definition 6.1.
For each integer n ≥ , L X n consists of non-empty subsets of X of the following form π − ( { t } × I × I × . . . × I M × J ) , where t ∈ N , I , I , · · · , I M ∈ L n , J ∈ L n + [ t log b /λ ] . The partition L X consistsof non-empty subsets of X of the following form π − ( { t } × R × . . . × R × J ) , where t ∈ N , J ∈ L [ t log b /λ ] . Lemma 6.1.
There exists A > such that any i ≥ , each element of L X i contains at most A elements of L X i + .Proof. When i ≥
1, the statement holds with A = b M + . Since Γ j ( x ) isuniformly bounded in [0 , j ∈ Σ , for each t ∈ N , there are only finitelymany members of L X whose elements have height t . So enlarging A , wecan guarantee that the statement holds also for the case i = (cid:3) Lemma 6.2.
There exists R > such that if π j g u and π j g v belong to the sameelement of L X i , where j ∈ Σ , u , v ∈ Λ ˆ n , and i ≥ , then for any x ∈ [0 , andy ∈ R , | π j g u ( x , y ) − π j g v ( x , y ) | ≤ Rb − ( n + i ) . Proof.
By definition of the partition L X i , we have | π j g u ( ) − π j g v ( ) | = O ( b − ( n + i ) ) DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 27 and | Γ u ∗ j ( k / M ) − Γ v ∗ j ( k / M ) | ≤ b − i for each 1 ≤ k ≤ M . Note that the last inequality also holds for k = I ∈ L ℓ with I ⊂ [0 ,
1) thereexists 0 ≤ k < M such that I = [ k / M , ( k + / M ). Thusinf x ∈ I | Γ ′ u ∗ j ( x ) − Γ ′ v ∗ j ( x ) | ≤ b − i M . By Theorem 5.1, it follows that(6.1) sup x ∈ [0 , | Γ ′ u ∗ j ( x ) − Γ ′ v ∗ j ( x ) | ≤ ρ − Mb − i , hence(6.2) sup x ∈ [0 , | Γ u ∗ j ( x ) − Γ v ∗ j ( x ) | ≤ ρ − Mb − i . Since π j g u ( x , y ) − π j g v ( x , y ) = − λ ˆ n ( Γ u ∗ j ( x ) − Γ v ∗ j ( x )) + π j g u ( ) − π j g v ( )the lemma follows. (cid:3) Lemma 6.3.
There exists a constant C ∈ Z + such that for any u , v ∈ Λ n ,n ≥ , and j ∈ Σ , L X Cn ( π j g u ) , L X Cn ( π j g v ) .Proof. Choose C ∈ Z + such that ρ b − Q n > ρ − Mb − Cn holds for all n ≥
1. Since u and v are distinct elements of Λ n , by Theo-rem 5.1, sup x ∈ [0 , | Γ ′ u ∗ j ( x ) − Γ ′ v ∗ j ( x ) | ≥ ρ b − Q n > ρ − Mb − Cn . As in the proof of (6.1), we see that π j g u and π j g v cannot belong to the sameelement of L X Cn . (cid:3) For a discrete probability measure η in the space X and a Borel probabil-ity measure µ in R , let η.µ denote the Borel probability measure in R suchthat for any Borel subset of R , η.µ ( A ) = η × µ ( { ( Ψ , x ) ∈ X × R : Ψ ( x ) ∈ A } ) . Lemma 6.4.
For any ε > , there exists p > and δ ∗ > such that thefollowing holds if i and k are su ffi ciently large. If η is a probability measuresupported in an element of L X i such that each element in the support of η has height ˆ n and such that k H (cid:0) η, L X i + k (cid:1) > ε, then ν ˆ i ( u ∈ Σ ˆ i : 1 k H (cid:0) η. (cid:0) δ g u ( ) (cid:1) , L i + k + n (cid:1) ≥ δ ∗ )! > p . Proof.
Let M = b ℓ + , where ℓ is as in Theorem 5.1 and assume ˆ i > ℓ . Itsu ffi ces to prove that for each integer 0 ≤ T < b ˆ i − ℓ − , there exists at leastone element x of X T = ( Tb ˆ i + jM : 0 ≤ j < M , j ∈ Z ) such that(6.3) 1 k H ( η.δ ( x , W ( x )) , L i + k + n ) > ε M , Indeed, once this proved, the desired estimate holds with δ ∗ = ε/ (2 M ) and p = / M .So let us fix T . Write e x j = Tb ˆ i + jM , 0 ≤ j < M and let e z j = (cid:0)e x j , W ( e x j ) (cid:1) .Define F : supp( η ) → R M , by F ( Ψ ) = (cid:0) Ψ ( e z ) , Ψ ( e z ) , . . . , Ψ ( e z M − ) (cid:1) . Claim.
There exists a constant e C such that H (cid:0) η, L X i + k (cid:1) ≤ H (cid:0) F η, L R M i + k + n (cid:1) + e C . To prove this claim, take I ∈ L R M i + k + n . It su ffi ces to show that the cardi-nality of the set { J ∈ L X i + k (cid:12)(cid:12)(cid:12) J ∩ F − ( I ) , ∅ and J ∩ supp( η ) , ∅} is uni-formly bounded. For any Ψ ( m ) ∈ supp( η ) with F ( Ψ ( m ) ) ∈ I , m = ,
2, write Ψ ( m ) ( x , y ) = λ ˆ n ( y − Γ u ( m ) ( x )) + c ( m ) . For each 1 ≤ j < M , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) Ψ (2) ( e z j ) − Ψ (1) ( e z j ) (cid:17) − (cid:16) Ψ (2) ( e z j − ) − Ψ (1) ( e z j − ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = O ( b − ( i + k + n ) )which means that λ ˆ n (cid:12)(cid:12)(cid:12) ( Γ u (2) − Γ u (1) ) ( e x j ) − ( Γ u (2) − Γ u (1) ) ( e x j − ) (cid:12)(cid:12)(cid:12) = O ( b − ( i + k + n ) ) , i.e. (cid:12)(cid:12)(cid:12) ( Γ u (2) − Γ u (1) ) ( e x j ) − ( Γ u (2) − Γ u (1) ) ( e x j − ) (cid:12)(cid:12)(cid:12) = O ( b − ( i + k ) ) . Therefore, inf x ∈ [ e x j − , e x j ) | Γ ′ u (2) ( x ) − Γ ′ u (1) ( x ) | = O ( b − ( i + k ) ) . For each element L of L ℓ which is contained in [0 ,
1) there exists 1 ≤ j < M such that [ e x j − , e x j ) ⊂ L . Soinf x ∈ L | Γ ′ u (2) ( x ) − Γ ′ u (1) ( x ) | = O ( b − ( i + k ) ) . DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 29
By Theorem 5.1, it follows thatsup x ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) Γ u (2) − Γ u (1) (cid:1) ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( b − ( i + k ) ) . Since Γ j (0) = j ∈ Σ , we obtain thatsup x ∈ [0 , | Γ u (2) ( x ) − Γ u (1) ( x ) | = O ( b − ( i + k ) ) . In particular, λ ˆ n (cid:12)(cid:12)(cid:12) ( Γ u (2) − Γ u (1) ) ( e x j ) (cid:12)(cid:12)(cid:12) = O ( b − ( i + k + n ) ) . Since Ψ (2) ( e z j ) − Ψ (1) ( e z j ) = − λ ˆ n ( Γ u (2) − Γ u (1) ) ( e x j ) + c (2) − c (1) , we also obtain that (cid:12)(cid:12)(cid:12) c (2) − c (1) (cid:12)(cid:12)(cid:12) = O ( b − ( i + k + n ) ) . By definition of L X i + k , we conclude the proof of the claim.Since H (cid:0) F η, L R M i + k + n (cid:1) ≤ M − X j = H (cid:0) η.δ e z j , L i + k + n (cid:1) , the claim implies that for at least one e z j we have1 k H (cid:0) η.δ e z j , L i + k + n (cid:1) ≥ ε M − e CkM . So (6.3) follows provided that k is su ffi ciently large. (cid:3)
7. P roof of T heorem B’In this section, we shall apply Hochman’s criterion on entropy increas-ing to complete the proof of Theorem B’. The basic idea is to introduce adiscrete measure θ j n = b ˆ n X i ∈ Λ ˆ n δ π j g i ∈ P ( X )for each n ∈ Z + and analyze the entropy of θ j n with respect to the partitions L X i and also the entropy of π j µ = θ j n .µ with respect to the partitions L i . The entropy of θ j n . We start with analyzing the entropy of θ j n withrespect to the partitions L X i . Lemma 7.1.
For ν Z + -a.e. j ∈ Σ , lim n →∞ n H (cid:16) θ j n , L X (cid:17) = lim n →∞ n H ( π j µ, L n ) = α. Proof.
Define π n , π : Σ → R , by π n ( i ) = g i i ··· i ˆ n (0 ,
0) and π ( i ) = lim n →∞ π n ( i ).Then π n − π = O ( b − n ), and hence π j π n − π j π = O ( b − n ). Therefore, H ( π j µ, L n ) = H (cid:16) π j πν Z + , L n (cid:17) = H ( π j π n ν Z + , L n ) + O (1) . For ν -a.e. j ∈ Σ , lim n →∞ n H ( π j µ, L n ) = α , solim n →∞ n H ( π j π n ν, L n ) = α. Since H ( θ j n , L ) = H ( π j π n ν, L n ), the lemma follows. (cid:3) Lemma 7.2.
There exists C ∈ Z + such that for each j ∈ Σ , we have lim n →∞ n H (cid:16) θ j n , L X Cn (cid:17) = log b log(1 /λ ) . Proof.
By Lemma 6.3, there exists C ∈ Z + such that for all n ≥ i , k ∈ Λ ˆ n , π j g i and π j g k lie in distinct elements of L Cn .Therefore H ( θ j n , L Cn ) = ˆ n log b . Since lim n →∞ n / ˆ n = log b /λ , the lemmafollows. (cid:3) From now on, we fix j ∈ Σ so that the conclusion of Lemma 7.1 holds.We shall write θ n = θ j n . Let(7.1) ε = C log b log λ − α > . Decomposition of entropy.
In the following lemma, we decomposethe entropy of θ n and π j µ into small scales. Lemma 7.3.
For any τ > , there exists C ( τ ) > such that if k , n arepositive integers with n > C ( τ ) k, then (7.2) 1 Cn H ( θ n , L X Cn |L X ) ≤ E θ n ≤ i < Cn " k H (( θ n ) Ψ , i , L X i + k ) + τ, (7.3) 1 Cn H ( π j µ, L ( C + n |L n ) ≥ E θ n ≤ i < Cn " k H (( θ n ) Ψ , i .µ, L i + k + n |L i + n ) − τ. DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 31
Proof.
Using Lemma 6.1 and arguing in the same way of [12, Lemma 3.4],we have 1
Cn H ( θ n , L X Cn |L X ) = E θ n ≤ i < Cn " k H (( θ n ) Ψ , i , L X i + k ) + O kn ! . Therefore, when k and n / k are large enough, (7.2) holds. Similarly, we alsohave 1 Cn H ( π j µ, L ( C + n |L n ) ≥ Cn X ≤ i < Cn " k H ( π j µ, L i + k + n |L i + n ) − τ. Note that π j µ = ( θ n ) .µ . By concavity of conditional entropy, we have H ( π j µ, L i + k + n |L i + n ) = H (( θ n ) .µ, L i + k + n |L i + n ) ≥ E θ n ( H (( θ n ) Ψ , i .µ, L i + k + n |L i + n )) . Thus (7.3) holds. (cid:3)
Proof of Theorem B’.
To conclude the proof of Theorem B’, we shallfurther decompose the entropy Q Ψ , i , n , k : = k H ([( θ n ) Ψ , i ] .µ, L i + k + n |L i + n )into smaller scales and compare it with e Q Ψ , i , n , k = b ˆ i X u ∈ Λ ˆ i Z X k H ( Ψ g u µ, L i + k + n ) d ( θ n ) Ψ , i ( Ψ ) , for each Ψ in the support of θ n . Lemma 7.4.
For any τ > , the following holds provided that k ≥ K ( τ ) :For any Ψ in the support of θ n , (7.4) Q Ψ , i , n , k ≥ e Q Ψ , i , n , k − τ. Proof.
By concavity of conditional entropy, the left hand side of (7.4) is atleast 1 b ˆ i X u ∈ Λ ˆ i Z X k H ( Ψ g u µ, L i + k + n |L i + n ) ! d η ( Ψ ) , where η = ( θ n ) Ψ , i . For each Ψ in the support of η and each u ∈ Λ ˆ i ,the measure Ψ g u µ is supported in an interval of length O ( b − ( i + n ) ), hence H ( Ψ g u µ, L i + n ) is uniformly bounded. The lemma follows. (cid:3) The following lemma will be proved in the next section, using Hochman’scriterion on entropy increase.
Lemma 7.5 (Entropy Increasing) . Assume α < . For every ε > , thereexist δ ∗ ( ε ) > and K ( ε ) > such that for each k ≥ K ( ε ) there existsI ( k , ε ) with the following property. Assume i ≥ I ( k , ε ) . If Ψ is in thesupport of θ n and k H (( θ n ) Ψ , i , L X i + k ) ≥ ε, then Q Ψ , i , n , k ≥ e Q Ψ , i , n , k + δ ∗ ( ε ) . Lemma 7.6.
For any τ > , k ≥ K ( τ ) and n ≥ N ( τ, k ) , the followingholds: E θ n ≤ i < Cn ( e Q Ψ , i , n , k ) > ( α − τ )(1 − τ ) . Proof.
First, we notice that E θ j n ≤ i < Cn ( e Q Ψ , i , n , k ) = Cn X ≤ i < Cn b ˆ i X u ∈ Λ ˆ i Z X k H ( Ψ g u µ, L i + n + k ) d θ j n ( Ψ ) = Cn X ≤ i < Cn b ˆ i X u ∈ Λ ˆ i b ˆ n X v ∈ Λ ˆ n k H ( π j g v g u µ, L i + n + k ) = Cn X ≤ i < Cn b ˆ i + ˆ n X w ∈ Λ ˆ i + ˆ n k H ( π j g w µ, L i + n + k ) . By Lemma 4.2, for each k ≥ M ( τ/ n largeenough: inf j ∈ Σ ν ˆ i + ˆ n ( w ∈ Λ ˆ i + ˆ n : 1 k H ( π w ∗ j µ, L k ) > α − τ/ )! > − τ. By Lemma 3.7 and Lemma 3.3, for w ∈ Λ ˆ i + ˆ n , | H ( π w ∗ j µ, L k ) − H ( π j g w µ, L i + n + k ) | is uniformly bounded. So when k is large enough, the above displayed in-equality implies thatinf j ∈ Σ ν ˆ i + ˆ n ( w ∈ Λ ˆ i + ˆ n : 1 k H ( π j g w µ, L i + n + k ) > α − τ )! > − τ. The lemma follows. (cid:3)
Proof of Theorem B’.
Arguing by contradiction, assume that α <
1. Let ε be given by (7.1) and ε = ε /
2. Let δ ∗ = δ ∗ ( ε /
2) be given by Lemma 7.5and let τ ∈ (0 , δ ∗ ) be a small constant to be determined. Fix k ≥ max( K ( τ ) , K ( ε ) , K ( τ )) , where K ( τ ) is given by Lemma 7.4, K ( ε ) is given by Lemma 7.5 and K ( τ ) is given by Lemma 7.6. Assume that n is large enough. Then theleft hand side of (7.2) tends to ε >
0. By Lemma 6.1, for any i ≥
0, any
DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 33 L X i -component η of θ n , k H ( η, L X i + k ) is bounded from above by a constant.Thus ξ : = P θ n ≤ i < Cn k H (( θ n ) Ψ , i , L X i + k ) > ε ! is bounded from below by a positive constant 2 p . By Lemma 7.5, ξ : = P θ n ≤ i < Cn (cid:16) Q Ψ , i , n , k > e Q Ψ , i , n , k + δ ∗ (cid:17) ≥ ξ − I ( k , ε ) Cn ≥ p . Therefore, by Lemmas 7.6 and 7.4, E θ n ≤ i < Cn (cid:0) Q Ψ , i , n , k (cid:1) ≥ E θ n ≤ i < Cn (cid:16) e Q Ψ , i , n , k (cid:17) + ξδ ∗ − (1 − ξ ) τ ≥ ( α − τ )(1 − τ ) + ξδ ∗ − (1 − ξ ) τ. Choosing τ > E θ n ≤ i < Cn (cid:0) Q Ψ , i , n , k (cid:1) ≥ α + p δ ∗ / . However, as n → ∞ , the left hand side of (7.3) converges to α , a contradic-tion! (cid:3) Proof of the Entropy Increasing Lemma.
In the rest of this section,we shall prove Lemma 7.5. The following is a version of Hochman’s en-tropy increasing criterion, see [12, Theorem 2.8] and [4, Theorem 4.1].
Theorem 7.1 (Hochman) . For any ε > and m ∈ Z + there exists δ = δ ( ε, m ) > such that for k > K ( ε, δ, m ) , n ∈ N , and τ, θ ∈ P ( R ) , if (1) diam ( supp ( τ )) , diam ( supp ( θ )) ≤ b − n , (2) τ is (1 − ε, ε , m ) -entropy porous from scales n to n + k, (3) k H ( θ, L n + k ) > ε ,then k H ( θ ∗ τ, L n + k ) ≥ k H ( τ, L n + k ) + δ, where ∗ denotes the convolution. For η : = ( θ n ) Ψ , i as in Lemma 7.5, we decompose η.µ as follows: η.µ = b ˆ i X u ∈ Λ ˆ i η. g u µ. We first show that the entropy of each term in the right hand side can berepresented by entropy of convolutions of line measures.
Lemma 7.7.
There is a constant C > and for each τ > there existsK ( τ ) such that when i ≥ C k, k ≥ K ( τ ) the following holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k H ( η. g u µ, L i + k + n |L i + n ) − k H (( η.δ g u (0) ) ∗ ( Ψ g u µ ) , L i + k + n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < τ. Proof.
Write z : = g u (0) = ( x , y ). Define F , G : supp( η ) × supp( g u µ ) → R by F ( Ψ , z ) = Ψ ( z ) , G ( Ψ , z ) = Ψ ( z ) + Ψ ( z ) − Ψ ( z ) . Note that F ( η × g u µ ) = η. g u µ and G ( η × g u µ ) is a translation of the convo-lution of η.δ z and Ψ . g u µ . By Lemma 6.2, η.δ z is supported in an intervalof length O ( b − ( i + n ) ). The same is also true for Ψ . g u µ , and hence for themeasure G ( η × g u µ ). It follows that H ( G ( η × g u µ ) , L i + n ) is bounded fromabove by a constant. Thus it is enough to show that F ( Ψ , z ) − G ( Ψ , z ) = O ( b − ( i + k + n ) )under the assumption that i / k is large enough.To this end, write Ψ ( x , y ) = λ ˆ n ( y − Γ v ( x )) + c and Ψ ( x , y ) = λ ˆ n ( y − Γ v ( x )) + c . Then for z = ( x , y ), we have (cid:12)(cid:12)(cid:12) F ( Ψ , z ) − G ( Ψ , z ) (cid:12)(cid:12)(cid:12) = λ ˆ n (cid:12)(cid:12)(cid:12) Z x x ( Y v − Y v )( s ) ds (cid:12)(cid:12)(cid:12) = b − n · O ( | x − x | ) . Note that | x − x | ≤ b − ˆ i = O ( b − log b log / λ i ). So when i / k is su ffi ciently large, | x − x | = O ( b − ( i + k ) ), and hence | F ( ψ, z ) − G ( ψ, z ) | = O ( b − ( i + k + n ) ). (cid:3) The measure η.δ g u ( ) plays the role of θ , and Ψ g u µ plays the role of τ in Hochman’s theorem. Lemma 6.4 shows that for a definite amount of u , η.δ g u ( ) has definite entropy. Proof of Lemma 7.5.
First, by concavity of conditional entropy,1 k H ( η.µ, L i + k + n |L i + n ) ≥ b − ˆ i X u ∈ Λ ˆ i k H ( η. g u µ, L i + k + n |L i + n ) . By Lemma 7.7, for any τ > k H ( η.µ, L i + k + n |L i + n ) ≥ b ˆ i X u ∈ Λ ˆ i k H (( η.δ g u ( ) ) ∗ ( Ψ g u µ ) , L i + k + n ) − τ holds for each Ψ in the support of η , provided that k is large enough and i ≥ C k . By [12, Corollary 4.10], increasing K ( τ ) if necessary, we have(7.6) 1 k H (( η.δ g u ( ) ) ∗ ( Ψ g u µ ) , L i + k + n ) ≥ k H ( Ψ g u µ, L i + k + n ) − τ, for any Ψ and u .Next, let us prove the following Claim.
There exist p , δ o > k large enough, there exists I ( ε, k ) such that the following holds when i ≥ I ( ε, k ). For each Ψ ∈ supp( η ), DICHOTOMY FOR THE WEIERSTRASS-TYPE FUNCTIONS 35 there is a subset Ω Ψ of Λ ˆ i with ν ˆ i ( Ω Ψ ) > p such that for u ∈ Ω Ψ , we have anentropy growth:(7.7) 1 k H (( η.δ g u ( ) ) ∗ ( Ψ g u µ ) , L i + k + n ) ≥ k H ( Ψ g u µ, L i + k + n ) + δ o . Take ξ = min(1 − α, δ ∗ , p ), where δ ∗ = δ ∗ ( ε ) and p = p ( ε ) are as inLemma 6.4. So the set Ω = ( u ∈ Λ ˆ i : 1 k H ( η.δ g u ( ) , L i + k + n ) > ξ ) satisfies ν ˆ i ( Ω ) > p , provided that i , k are large enough. By Theorem 4.1,there exists m , and for each k large enough there exists I k such that when i ≥ I k , for any Ψ in the support of η , we have ν ˆ i ( Ω Ψ ) > − ξ , where Ω Ψ = { u ∈ Λ ˆ i : Ψ g u µ is ( α, ξ/ , m ) − entropy porous from scale n + i to n + k + i } . Thus ν ˆ i ( Ω Ψ ) ≥ p /
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220 H andan R oad , S hanghai ,C hina hanghai C enter for M athematical S ciences , J iangwan C ampus , F udan U niversity , N o onghu R oad , S hanghai , C hina E-mail address ::