We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection.

Wavelets emerged in the 1980s as a powerful tool for signal processing, see [20, 21, 28]. Since then, they have found many applications in this field, such as estimation, detection, classification, compression, filtering and synthesis, see e.g. [5, 8, 12, 19, 29, 36]. In these papers, the coefficients are implicitly assumed to be independent of one another and the exposed methods are based on scalar transformations on each wavelet coefficient of the considered signal. Nonetheless, it was observed that the wavelet coefficients of many real-world signals exhibit some correlations. In particular, the large wavelet coefficients tend to propagate across scales at the same locations, see [32, 33]. In image processing, this phenomenon is related to the fact that the contours of a picture generate discontinuities. Therefore, methods that exploit dependencies between wavelet coefficients should yield better results in the applications.

In order to develop such methods, M. Crouse, R. Nowak and R. Baraniuk [10] introduced a simple probabilistic model allowing to capture correlations between wavelet coefficients : the Hidden Markov Tree (HMT) model, which we now briefly recall. Let us consider a Markov chain X indexed by the binary tree with state space {0, 1}. Basically, the Markov property enjoyed by X means that if the state Xu of a vertex u is x ? {0, 1}, then the states of the two sons of u are chosen independently according to the transition probabilities from x . Conditionally on the Markov chain X , the wavelet coefficients are independent and the wavelet coefficient indexed by a given dyadic interval ? is a centered Gaussian random variable whose variance is large (resp. small) if the state of the vertex of the binary tree that corresponds to ? is 1 (resp. 0). The correlations between the wavelet coefficients are thus given by the underlying Markov chain. Furthermore, the unconditional law of each coefficient is a Gaussian mixture. This property agrees with the observation that the wavelet coefficient histogram of a real-world signal is usually more peaky at zero and heavy-tailed than the Gaussian. Moreover, let us mention that the HMT model was used in image processing, see [9,11] for instance.

In this paper, we investigate the pointwise regularity properties of the sample paths of a model of random wavelet series which is closely related to the HMT model. The regularity of a function at a given point is measured by its Hölder exponent, which is defined as follows, see [25].

Definition (Hölder exponent). Let f be a function defined on R and let x ? R. The Hölder exponent h f (x ) of f at x is the supremum of all h > 0 such that there are two reals c > 0 and ? > 0 and a polynomial P enjoying ?x ? [x ? ?, x + ?]

| f (x ) ? P(x ? x )| ? c |x ? x |h .

Since we are interested in local properties, it is more convenient to work with wavelets on the torus T = R/Z, so that the random wavelet series that we study throughout the paper is in fact a random process on T. This process is defined in Sect. 2 and is denoted by R. Let ? be the canonical surjection from R to T. Note that R ? ? is a one-periodic random function defined on R. The Hölder exponent of R at a point x ? T is then defined by h R (x ) = h R?? (x?), where x? is a real number such that ? (x?) = x . Equivalently, h R (x ) is the supremum of all h > 0 such that d(x , x ) ? ? =?

|R(x ) ? P(x ? x )| ? c d(x , x )h for all x ? T, some positive reals c and ? and some function P on T such that P ? ? coincides with a polynomial in a neighborhood of zero. Here, d denotes the quotient distance on the torus T.

Our main purpose is to perform the multifractal analysis of the sample paths of the random wavelet series R. This amounts to studying the size properties of the iso-Hölder set

Eh = {x ? T | h R (x ) = h}
(

A remarkable property, due to the correlations between wavelet coefficients, is that the spectrum of singularities of the process R is itself a random function. None of the multifractal stochastic processes studied up to now, such as the usual Lévy processes [22], the Lévy processes in multifractal time [4] or the random wavelet series with independent coefficients introduced by J.-M. Aubry and S. Jaffard [2,23], enjoys this property. The random wavelet series based on multifractal measures studied by J. Barral and S. Seuret [3] do not satisfy it either, even though their wavelet coefficients exhibit strong correlations. This remark also holds for the wavelet series based on branching processes introduced by A. Brouste [6]. In this last model, for any dyadic interval ?, the wavelet coefficient indexed by ? is either Gaussian or zero depending on whether or not the vertex of the binary tree that corresponds to ? belongs to a certain Galton-Watson tree. This model can be seen as a particular case of the HMT model by assuming that the underlying Markov chain cannot map to the state 1 a vertex whose father is mapped to the state 0. Let us also mention that A. Brouste, in collaboration with geophysicists, used this model to study the surface roughness of certain rocks, see [7].

For certain values of the parameters of the model, the sample paths of the random wavelet series R enjoy the remarkable property that almost every one of them displays an oscillating singularity at almost every point of the torus. We refer to Subsect. 3.3 for details. Note that this property also holds for the models of random wavelet series studied by J.-M. Aubry and S. Jaffard [2,23]. Conversely, the random wavelet series considered by J. Barral and S. Seuret [3] and those introduced by A. Brouste [6] do not exhibit any oscillating singularity.

Certain random sets related to the iso-Hölder sets Eh enjoy a notable geometric property which was introduced by K. Falconer [17]. To be specific, we establish that, for certain values of h, the sets

Eh = x ? T h R (x ) ? h are almost surely sets with large intersection, see Subsect. 3.4. In particular, this implies that they are locally everywhere of the same size, in the sense that the Hausdorff dimension of the set Eh ? V does not depend on the choice of the nonempty open subset V of the torus. This also implies that the size properties of the sets Eh are not altered by taking countable intersections. In fact, the Hausdorff dimension of the intersection of countably many sets with large intersection is equal to the infimum of their Hausdorff dimensions. This property is somewhat counterintuitive in view of the fact that the intersection of two subsets of the torus with Hausdorff dimensions s1 and s2 respectively is usually expected to be s1 + s2 ? 1 (see [18, Chap. 8] for precise statements). The occurrence of sets with large intersection in the theory of Diophantine approximation and that of dynamical systems was pointed out by several authors, see [14,16,17] and the references therein. Their use in multifractal analysis of stochastic processes is more novel and was introduced by J.-M. Aubry and S. Jaffard. Indeed, they established in [2] that sets with large intersection arise in the study of the Hölder singularity sets of certain random wavelet series. As shown in [15], such sets also appear in the study of the singularity sets of Lévy processes.

The rest of the paper is organized as follows. In Sect. 2 we give a precise definition of the model of random wavelet series that we study. Our main results are stated in Sect. 3 and are proven in Sects. 4 to 7.

In order to define the process that we study, let us introduce some notations. Throughout the paper, N (resp. N0) denotes the set of positive (resp. nonnegative) integers and ? is the collection of the dyadic intervals of the torus T, that is, the sets of the form ? = ? (2? j (k + [0, 1))) with j ? N0 and k ? {0, . . . , 2 j ? 1}. The integer ? = j is called the generation of ?. Furthermore, let us consider a wavelet ? in the Schwartz class (see [30]). For any dyadic interval ? = ? (2? j (k + [0, 1))) ? ?, let ?? denote the function on T which corresponds to the one-periodic function x ? m?Z

? (2 j (x ? m) ? k). Then, the functions 2 ? /2??, together with the constant function equal to one on T, form an orthonormal basis of L2(T), see [34].

Recall that there is a one-to-one correspondence between the set ? of all dyadic intervals of the torus and the set

? U = { } ? The set U is formed of the empty word ? and the words u = u1 . . . u j of finite length j ? 1 in the alphabet {0, 1}. The integer u = j is called the generation of u. In addition, let ? = 0 and U ? = U \ {?}. For every word u ? U ?, the word ?(u) = u1 . . . u u ?1 is called the father of u. Then, the directed graph with vertex set U and with arcs (?(u), u), for u ? U ?, is a binary tree rooted at ?. To be specific, the bijection from U to ? is ? ? u ? ?u = ? ?

u j 2? j + [0, 2? u )? . u j=1 Thus, for every ? ? ?, there is a unique vertex u? ? U such that ?u? = ?.

In the following, X = (Xu )u?U denotes a {0, 1}-valued stochastic process indexed by the binary tree U . The ? -field

Gu = ? (Xv, v ? U \ (uU ?)) can be considered as the past before u, because the set uU ? is composed of all the descendants of the vertex u in the binary tree U , that is, words of the form uv with v ? U ?. Conversely, the future after u begins with its two sons u0 and u1. For any integer j ? 0, let ?0, j and ?1, j denote two probability measures on {0, 1}2. From now on, we assume that the process X is a Markov chain with transition probabilities given by the measures ?0, j and ?1, j . This means that the following Markov condition holds: (M) For any vertex u ? U and any subset A of {0, 1}2,

P((Xu0, Xu1) ? A | Gu ) = ?Xu, u ( A).

Equivalently, the conditional distribution of the vector (Xu0, Xu1) conditionally on the past Gu is the probability measure ?Xu, u . Thus, the Markov condition satisfied by X is conceptually similar to that enjoyed by inhomogeneous discrete time Markov chains.

The random wavelet series R that we study is then defined by

R = = u?U ??? 2?h u 1{Xu=1} + 2?h u 1{Xu=0}

??u 2?h ? 1{Xu? =1} + 2?h ? 1{Xu? =0} ??, where 0 < h < h ? ?. The mean value of every sample path of R vanishes. Moreover, for any ? ? ?, the wavelet coefficient indexed by ? is ? ? C? = 2?h ? if Xu? = 1 ? The coefficient C? should be considered as large in the first case and small in the second one (it even vanishes if h = ? and ? ? 1). The model that we study is thus defined in the same way as the HMT model, except that, conditionally on the underlying Markov chain, the wavelet coefficients are deterministic instead of Gaussian.

Note that the Markov condition (M) implies that the probability measures ?1, j affect
the propagation of the large wavelet coefficients of R across scales, while the probability
measures ?0, j govern their appearance. In what follows, the influence of each of these
two phenomena is reflected by the values of the parameters
and
? j = 2 ?1, j ({(

? j = 1 ? ?0, j ({(0, 0)}),
respectively. Indeed, for any integer j ? 0, ? j is the expected number of sons mapped
to the state 1 of a vertex with generation j that is mapped to the state 1 and ? j is the
probability that a vertex with generation j that is mapped to the state 0 has at least one
son mapped to the state 1.
3. Statement of the Results
3.1. Preliminary remarks. As shown by (

The Hölder exponent of R at any point of the torus is highly related to the size of the
wavelet coefficients which are indexed by the dyadic intervals located around x . More
precisely, it can be computed using the following proposition, which is a straightforward
consequence of [26, Prop. 1.3].
(

|C?| ? ? (2? ? + d(x , x?))h .
(b) If (

By virtue of (

h ? h R (x ) ? h.

As a consequence, for every sample path of the random wavelet series R and every
h ? [0, h) ? (h, ?], the iso-Hölder set Eh defined by (

Before stating these theorems, we need to introduce some further notations related
to the parameters ? j defined by (

j = inf{ j0 ? 0 | ? j ? j0 ? j > 0} ? ? ? ? = lim inf

j?? ? = ?? log ? j +...+log ? j j log 2 if j < ? if j = ?.

Note that ? is at most one. Moreover, ? j being the expected number of sons mapped to the state 1 of a vertex with generation j that is mapped to the state 1, the parameter ? expresses the trend with which large wavelet coefficients propagate across scales. For any integer j ? 0, we also need to consider the number ? j = 2 ? ?1,n ({1, 1}) ,

n n= j ?n = j ? which naturally occurs in the study of a specific random fractal set related to the process R, see Lemma 6 below.

Throughout the rest of the paper, we suppose that ? is less than one. This assumption
implies that the large wavelet coefficients of R cannot propagate too much across scales.
It is not very restrictive, in view of the fact that the decomposition of a typical real-world
signal in a wavelet basis has very few large coefficients.
3.2. Law of the spectrum of singularities. Theorems 1 and 2 below give the law of the
spectrum of singularities of the sample paths of the random wavelet series R. Recall
that it is the mapping dR : h ? dim Eh , where Eh is defined by (

With a view to recalling the definition of Hausdorff dimension, let us first define the notion of Hausdorff measure on the torus. To this end, let D denote the set of all nondecreasing functions g defined on a neighborhood of zero and enjoying lim0+ g = g(0) = 0. Any function in D is called a gauge function. For every g ? D, the Hausdorff g-measure of a subset F of T is defined by

Hg(F ) = l?i?m0 ? H?g(F ) with

H?g(F ) = F?inpfUp |Up|<? p=1 ? g(|U p|).

The infimum is taken over all sequences (U p) p?1 of sets with F ? p U p and |U p| < ? for all p ? 1, where | · | denotes diameter (in the sense of the quotient distance on the torus). Note that Hg is a Borel measure on T, see [35]. The Hausdorff dimension of a nonempty set F ? T is then defined by

dim F = sup{s ? (

The pointwise regularity properties of the sample paths of the process R depend
significantly on whether or not the series j 2 j ? j converges, where ? j is given by (

S j = {u ? U | u = j and Xu = 1} .

This means that ? j (z) = E[z#S j ] for any complex number z and any integer j ? 0. The functions ? j may be calculated recursively in terms of the transition probabilities of the Markov chain X . Indeed, the Markov condition (M) implies that for any integer j ? 0, E[z#S j+1 | F j ] =

, where F j is the ? -field generated by the variables Xu , for u ? U with u ? j . Theorem 1. Let us suppose that

j 2 j ? j < ? and ? < 1.
(a) If ? < 0, then with probability one, h R (x ) = h for all x ? T. Therefore, with
probability one, for all h ? [0, ?],
(

Moreover, if there is an integer j? ? 0 such that ?1, j ({(0, 0)}) = 0 for any j ? j?, then

P(dR (h) = ??) = ? j? (0) ·

j (1 ? ? j )2 . ? j= j? If not, P(dR (h) = ??) is positive and it is equal to one if and only if

Remark 1. As ? < 1, Theorem 1 implies that the iso-Hölder set Eh has full Lebesgue measure in the torus with probability one. Thus, the Hölder exponent of almost every sample path of the process R is h almost everywhere. Remark 2. The process R enables to provide a partial answer to a question raised by S. Jaffard in [24]. For many examples of random processes F , such as the usual Lévy processes [22], the Lévy processes in multifractal time [4] or several models of random wavelet series [2,23], even though the function x ? h F (x ) is random, the spectrum of singularities of F is a deterministic function. Of course, this property does not hold in general: consider for instance a fractional Brownian motion whose Hurst parameter follows a Bernoulli law. The random wavelet series R supplies a more elaborate example. Indeed, Theorem 1 shows that its spectrum of singularities may be random. Theorem 2 below indicates that this property still holds if the series j 2 j ? j diverges.

If the series j 2 j ? j diverges, large wavelet coefficients can appear with a relatively large probability at each scale, which makes the sample paths of R very irregular in their irregularity, as shown by the following result. In its statement, h is defined by h = inf h > 0 j 2(1?h/ h) j ? j = ? , which is clearly greater than or equal to h.

Theorem 2. Let us suppose that

j 2 j ? j = ? and ? < 1. (a) If h < h, then with probability one, for all h ? [0, ?], dR (h) = h/h if h < h ? h (b) If h ? h, then with probability one, for all h ? [0, ?], ? ?? dR (h) = h/h if h < h < h (c) If h/h ? ? , then with probability one, (d) If h/h < ? , then with probability one,

Moreover, P(dR (h) = ? ) is positive and it is equal to one if and only if either j 2 j ? j /? j+1 = ? or ?1, j ({(0, 0)}) = 0 for all j large enough.

Remark 3. As ? < 1, Theorem 2 implies that the iso-Hölder set Emin(h,h) has full Lebesgue measure in the torus with probability one. So, the Hölder exponent of almost every sample path of the random wavelet series R is min(h, h) almost everywhere. In addition, if h = h, then the set Eh is almost surely equal to the whole torus, so that the Hölder exponent of R is almost surely h everywhere. Remark 4. For some values of the parameters, the spectrum of singularities of the random wavelet series R need not be concave. Therefore, this spectrum cannot be determined using the multifractal formalisms derived in Besov or oscillation spaces. We refer to [25] for details concerning these multifractal formalisms. Moreover, in general, the spectrum of singularities of R does not coincide with its large deviation spectrum (see e.g. [31] for a fuller exposition), that is, the mapping h ? l?i?m0 ? limj?s?up j 1

log2 #{? ? ? | ? = j and 2?(h+?) j ? |C?| ? 2?(h??) j }.

Indeed, this last function clearly maps any real h ?/ {h, h} to ??.

Remark 5. Recall that, owing to Theorem 1, the spectrum of singularities of R may be random when j 2 j ? j < ?. Theorem 2 shows that this property still holds when j 2 j ? j = ?. Specifically, the spectrum of singularities of R is random if and only if h/h is less than ? , the sum j 2 j ? j /? j+1 is finite and ?1, j ({(0, 0)}) is positive for infinitely many integers j ? 0.

Let us give an explicit example of probability measures ?0, j and ?1, j for which all
these conditions hold. Given a real a ? (

p j ?1 + (1 ? p j )?0 ?2 , where ?0 and ?1 are the point masses at zero and one, respectively. Clearly, ?1, j ({(0, 0)}) is positive for all j , the number j vanishes and ? = 1 ? a < 1. For every integer n ? 0, let jn = bn+1 ? 1. Observe that, for all n greater than some n0, the sum ? jn is at least 2a(1?1/b)( jn+1)?1 and there exists a real q jn?1 such that 2?( jn?1) Furthermore, let q j?1 = 0 for every integer j ? 1 that is not of the form jn with n > n0. Then, let us consider that the probability measures ?0, j are given by

?0, j = q j ?1 + (1 ? q j )?0 ?2 .

It is easy to check that the sum j 2 j ? j diverges and that h is at least h/(a(1 ? 1/b)). Let us suppose that a < 1/(2 ? 1/b). This assumption ensures that h/h is less than ? . Moreover,

? n=n0+1 2 jn?1? jn?1 ? jn ?

2a(1?1/b) jn
? n=n0+1 jn2 2a(1?1/b)( jn+1)?1 ? 21?a(1?1/b)
? 1
j=1
j 2
,
which ensures the finiteness of the sum j 2 j ? j /? j+1. Theorem 2 finally implies that
the spectrum of singularities of the process R is random when the probability measures
?0, j and ?1, j are chosen as above.
(

1 |x ? x |? both have Hölder exponent h at x , in spite of the fact that their oscillatory behavior is completely different, see [26].

The oscillating singularity exponent was introduced in [1] in order to describe the oscillatory behavior of a function near a given point and thus to determine if a function behaves rather like a cusp or like a chirp in a neighborhood of a point. It is defined using primitives of fractional order. To be specific, for any t > 0, any locally bounded function f defined on R and any x ? R with h f (x ) < ?, let htf (x ) denote the Hölder exponent at x of the function (Id ??) ?t/2(? f ), where ? is a compactly supported smooth function which is equal to one in a neighborhood of x and (Id ??) ?t/2 is the operator that corresponds to multiplying by ? ? (1 + ? 2)?t/2 in the Fourier domain. Definition (Oscillating singularity exponent). Let f be a locally bounded function defined on R and let x ? R with h f (x ) < ?. The oscillating singularity exponent of f at x is ? f (x ) = ?htf (x ) ?t t=0+ ? 1 ? [0, ?].

If ? f (x ) > 0, then f is said to display an oscillating singularity at x .

It is proven in [26] that if f is defined by (

The oscillating singularity exponent ?R (x ) of the random wavelet series R at any point x ? T such that h R (x ) < ? is then defined in the natural way, that is, ?R (x ) is set to be equal to ?R?? (x?) for any real number x? enjoying ? (x?) = x , where ? is the canonical surjection from R to T. The following result, which is proven in Sect. 7, gives the value of the oscillating singularity exponent of R at every point of the iso-Hölder set Eh .

Proposition 2. For every h ? [h, h] and every x ? Eh , ?R (x ) = h/h ? 1 if h < h 0 if h = h < ?.

Remark 6. Proposition 2 ensures that the random wavelet series R displays an oscillating
singularity at every point of the set Eh , for any h ? (h, h). Moreover, it is necessary
to assume the finiteness of h for h = h in the statement of Proposition 2 because the
oscillating singularity exponent is not defined for points at which the Hölder exponent
is infinite.
Remark 7. In the case where h < h, Theorem 2 and Proposition 2 ensure that almost
every sample path of the random wavelet series R displays an oscillating singularity at
almost every point of the torus. Note that this remarkable property is also verified by the
models of random wavelet series with independent coefficients which were studied in
[2,23].
3.4. Large intersection properties of the singularity sets. For certain values of h, the
sets Eh defined by (

Let us first recall the definition and the basic properties of the classes Gg(R). To begin with, they are defined for functions g in a set denoted by D1. This is the set of all gauge functions g ? D such that r ? g(r )/r is positive and nonincreasing on a neighborhood of zero. For any g ? D1, let ?g denote the supremum of all ? ? (0, 1] such that g is nondecreasing on [0, ?] and r ? g(r )/r is nonincreasing on (0, ?] and let ?g denote the set of all dyadic intervals of diameter less than ?g, that is, sets of the form ? = 2? j (k + [0, 1)) with j ? N0, k ? Z and |?| < ?g. The outer net measure associated with g is defined by ?F ? R

M?g(F ) =

? inf (?p)p?1 p=1 g(|? p|), where the infimum is taken over all sequences (? p) p?1 in ?g ?{?} such that F ? p ? p. In addition, for g, g ? D1, let us write g ? g if g/g monotonically tends to infinity at zero. We can now give the definition of the classes Gg(R). Recall that a G?-set is one that may be expressed as a countable intersection of open sets.

class Gg(R) of sets with large intersection in R with respect to g is the collection of all G?-subsets F of R such that M?g(F ? U ) = M?g(U ) for every g ? D1 enjoying g ? g and every open set U .

The classes Gg(T) are then defined in the natural way using the classes Gg(R) and the canonical surjection ? from R to T.

class Gg(T) of sets with large intersection in T with respect to g is the collection of all subsets F of T such that ??1(F ) ? Gg(R).

The results of [14] show that the classes Gg(T) of sets with large intersection in the torus enjoy the following remarkable properties.

Theorem 3. For any gauge function g ? D1, (a) the class Gg(T) is closed under countable intersections; (b) every set F ? Gg(T) enjoys Hg(F ? V ) = ? for every g ? D1 with g ? g and every nonempty open set V , and in particular

dim F ? sg = sup{s ? (

As previously announced, the sets Eh defined by (

Proposition 3. Let us assume that h is finite. Then, with probability one, for all h ? [h, min(h, h)), the set Eh belongs to the class GIdh/h (T).

Together with Theorem 3, this result implies that with probability one, for every h ? [h, min(h, h)), the set Eh has infinite Hausdorff measure for every gauge function g ? D1 with g ? Idh/h . This property comes into play in the proof of Theorem 2, because it enables to obtain a sharp lower bound on the Hausdorff dimension of the corresponding iso-Hölder set Eh , see Sect. 6.

In this section, we establish several results that are called upon at various points of the proofs of Theorems 1 and 2.

The Hölder exponent of the random wavelet series R at a given point x of the torus depends on the way large wavelet coefficients are located around x . To be specific, let

S = {u ? U | Xu = 1} =
where the sets S j are defined by (

L? = {x ? T | d(x , xu ) < 2?h u /? for infinitely many u ? S},
where d is the quotient distance on the torus. It is straightforward to check that ? ? L?
is nondecreasing. The following lemma establishes a connection between the sets L?
and the sets Eh and Eh defined by (

Lemma 4. (a) For every h ? [0, h) ? (h, ?], the set Eh is empty.
(b) For every h ? [h, h],
(

L? and Eh = Eh \

L?.
h<??h
h<?<h
Proof. Assertion a is due to the fact that the Hölder exponent of the process R is
everywhere between h and h, as shown by (

? j=0

S j , u j=1 Indeed, let us assume that x ? L?. Then, there are infinitely many dyadic intervals ? = ? (2? j (k + [0, 1))) ? ?, with j ? N0 and k ? {0, . . . , 2 j ? 1}, such that where x? = ? (k2? j ). Proposition 1 ensures that h R (x ) ? ?. Conversely, let us assume that x ? L?. Then, for every dyadic interval ? ? ? such that ? is large enough, if C? = 2?h ? , then d(x , x?) ? 2?h ? /?. Thus, whether C? = 2?h ? or C? = 2?h ? , we have

|C?| ? (2? ? + d(x , x?))?, so that h R (x ) ? ? thanks to Proposition 1.

The proof of the following lemma is modeled on that of Proposition 1 in [27, Chap. 11].

Lemma 5. With probability one, for every ? ? (h, ?), we have L? = T. Proof. Let us consider a real number ? > h. Moreover, for every u ? U , let Bu be the open ball of T with center xu and radius 2?h u /?. It is straightforward to establish the following inclusion of events:

Then, let j0 and j be two integers such that {T = L?} ? ? T = ? ? ? ? and let us assume that T cannot be written as the union over u ? S j?1 ? S j of the balls Bu . So, there is an integer k ? {0, . . . , 2 j ? 1} such that the closed ball with center ? (k2? j ) and radius 2? j?1 is not included in this last union of balls. As a result, the point ? (k2? j ) cannot belong to the union over u ? S j?1 ? S j of the open balls with center xu and radius 2?h u /? ? 2? j?1. Therefore, ? ? ? ? ? T = where A jj,k denotes, for each j ? { j ? 1, j }, the event corresponding to the fact that the Markov chain X maps to 0 all the vertices of the set

A jj,k = {u ? {0, 1} j | d(xu , ? (k2? j )) ? 2?1?h j /?}.

Owing to (

P(A j?1 ? A jj,k ) ? P(?u ? A j,k

Xu = Xu0 = Xu1 = 0) ? ?0, j?1({(0, 0)})# A .
(

? ? ? ? ?

? T =
has probability at most lim inf j v j , which vanishes because there are infinitely many
integers j ? j0 + 1 such that ? j?1 ? 2(?1+2h/(?+h)) j , owing to the fact that ? > h. Then,
(

Let ? ? (h, ?). Lemma 7 below gives a useful decomposition of the set L? defined
by (

L? = {x ? T | d(x , xu ) < 2?h u /? for infinitely many u ? S},

S = {u ? U ? | Xu = 1 and X?(u) = 0}.
(

The second set coming into play in the decomposition of L? is a set denoted by ? and defined as follows. For every vertex u of the binary tree U , let uU denote the set of all words of the form uw with w ? U and let

?u = {v ? uU | ? j ? { u , . . . , v } Xv1...v j = 1}.

The set ?u is empty if Xu = 0. Otherwise, ?u is the largest subtree of U rooted at u and formed of vertices which are mapped to 1 by the Markov chain X . The boundary of ?u is the set ??u = {? = (? j ) j?1 ? {0, 1}N | ? j ? u ?1 . . . ? j ? ?u }.

For every sequence ? = (? j ) j?1 in {0, 1}, let where x ?? = where ? is the canonical surjection from R to T. Essentially, a point of the torus belongs to ? if it can be obtained as the intersection of a sequence of nested dyadic intervals indexing large wavelet coefficients of the process R that propagate across scales. The following lemma provides the law of the Hausdorff dimension of ?. We refer to Sect. 3 for the definitions of the parameters appearing in its statement. Lemma 6. If ? < 0, then ? is empty with probability one. If not, then with probability one, ? is empty or has Hausdorff dimension ? and, in addition, ? if there is a j? ? 0 such that ?1, j ({(0, 0)}) = 0 for any j ? j?, then P(? = ?) = ? j? (0) · ? j= j? ? if ?1, j ({(0, 0)}) > 0 for infinitely many integers j ? 0, if j 2 j ? j = ? and if ? > 0, then P(? = ?) is less than one and it is equal to zero if and only if j 2 j ? j /? j+1 = ?.

Proof. The lemma is a straightforward consequence of Proposition 4 in [13], which provides the law of the Hausdorff dimension of the set ?? , and the observation that the sets ?? and ? have the same Hausdorff dimension.

The following lemma supplies a precise statement of the aforementioned decomposition of the set L? in terms of the sets L? and ?.

Lemma 7. For every ? ? (h, ?), we have L? = L? ? ?.

Proof. To begin with, it is easy to check that L? ? L?. Next, let us consider a point x in ?. Then, there are a vertex u ? U and a sequence ? = (? j ) j?1 ? ??u such that x = ? (x?? ). For every integer j ? u , we have ?1 . . . ? j ? S and

d(x , x?1...? j ) ? 2? j < 2?h j/?.

The point x thus belongs to L?. It follows that ? is included in L?. Hence, L? contains both L? and ?.

Conversely, let us consider a point x in L? which does not belong to L?. Then, there is an integer j0 ? 0 such that d(x , xu ) ? 2?h u /? for every vertex u ? S with generation at least j0. We may assume that j0 ? (log2(2h/? ? 1))/(h/? ? 1). Let

S = {u ? S | d(x , xu ) < 2?h u /?} and observe that the set S cannot contain any vertex of S with generation at least j0. Since x ? L?, there exists a sequence (vn)n?1 in S such that vn is increasing. A standard diagonal argument leads to a sequence ? = (? j ) j?1 in {0, 1} such that for every j ? 1, there are infinitely many integers n ? 1 enjoying ?1 . . . ? j = v1n . . . vnj . Let u = ?1 . . . ? j0 and let us consider two integers j ? j0 and n satisfying vn > j and ?1 . . . ? j = v1n . . . vnj . The vertex vn belongs to S and its generation is at least j0, so that vn ? S \ S. Hence, ?(vn) belongs to S. Moreover, d(x , x?(vn)) ? d(x , xvn ) + d(xvn , x?(vn)) < 2?h vn /? + 2? vn ? 2?h ?(vn) /?, pwrhoivcehthenatsu?r1e.s. t.h?ajt =?(vv1nn.). ?.vnS . By repeating this procedure vn ? j times, one can j ? S . In particular, X?1...? j = 1 for every integer j ? j0, so that ? ? ??u . Furthermore, for any j ? j0, d(x , ? (x?? )) ? d(x , x?1...? j ) + d(x?1...? j , ? (x?? )) ? 2?h j/? +

Letting j ? ?, we obtain x = x? . The point x thus belongs to ?.
where F j?1 is the ? -field generated by the variables Xu , for u ? U such that u ? j ?1.
It follows that the set S j is nonempty with probability at most 2 j ? j?1. As the sum
j 2 j ? j converges, the Borel-Cantelli lemma ensures that, with probability one, there
are at most finitely many integers j ? 1 such that S j = ?. Consequently, the set S is
almost surely finite. So, with probability one, for every real ? > h, the set L? given by
(

Eh = ? ? Eh = T \ ? if h = h ??

Theorem 1 is then a direct consequence of Lemma 6.

In order to prove Theorem 2, let us assume that with, observe that a.s. ?h ? [0, h) ? (min(h, h), ?]

Eh = ?,
(

The next result gives an upper bound of the Hausdorff dimension of the iso-Hölder set Eh for any h ? [h, min(h, h)). j 2 j ? j = ? and ? < 1. To begin

dim Eh ? max(h/h, dim ?) and ?h ? (h, min(h, h)) dim Eh ? h/h.
Proof. Owing to (

(

H?Ids (L?) ?

j?N 2?h j/???/2 #S j · (21?h j/?)s ? ?

2 j ? j?1 j 2(21?h j/?)s .

j?N 2?h j/???/2 Since ?/s < h, this last series converges so that the right-hand side tends to zero as ? ? 0. Hence, the Hausdorff Ids -measure of the set L? vanishes. It fol#lows that with probability one, for all h ? [h, min(h, h)), the Hausdorff dimension of h<??h L? is at most h/h. The result then follows from the fact that

Eh = ? ?

L? and ?h ? (h, min(h, h)) Eh ? L?, h<??h h<??h owing to Lemmas 4 and 7.

Lemma 6 and Proposition 8, together with the assumption that ? is less than one,
imply that with probability one, for any h ? [h, min(h, h)), the iso-Hölder set Eh has
Lebesgue measure zero. Moreover, with probability one, this set is empty for every
h ? [0, h) ? (min(h, h), ?], owing to (

In order to give a lower bound on the Hausdorff dimension of the iso-Hölder set Eh for every real h ? [h, min(h, h)), we shall treat two cases separately: h < ? and h = ?. Let us first consider the case in which h is finite. The lower bound then follows from the fact that Eh is a set with large intersection, as shown by the following result. Recall that the classes Gg(T) of sets with large intersection in the torus are defined in Subsect. 3.4.

Proposition 9. Let us assume that h < ?. Then, with probability one, for every real h ? [h, min(h, h)),

Eh ? GIdh/h (T) and dim Eh ? h/h.
Proof. Lemma 5 shows that, with probability one, for every real ? > h, the set L?
defined by (

for infinitely many (u, p) ? S × Z belongs to the class GIdh/h (R), so that the set L? belongs to the class GIdh/h (T). Furthermore, owing to Lemma 4 and the fact that ? ? L? is nondecreasing, the set Eh is equal to the intersection over the integers n > 1/(h ? h) of the sets Lh+1/n. Each of these sets belongs to the class GIdh/h (T), which is closed under countable intersections thanks to Theorem 3. Hence, with probability one, ?h ? [h, min(h, h))

Eh ? GIdh/h (T).

(

In order to establish the remainder of the proposition, let us begin by observing that
Eh = Eh ? GIdh/h (T) with probability one, by virtue of (

This ensures that the set ?? defined by (

Eh = (Eh \ ?) \

L?.
h<?<h
In addition, (

j ? 1 log 2. h ? Since ?h/ h < h, this last series converges so that the right-hand side tends to zero as ? ? 0. It follows that H?Idh/h·log(L?) = 0. The fact that ? ? L? is nondecreasing implies that because the union can actually be taken on a countable subset of (h, h). Therefore, H?Idh/h·log(Eh ) = ?, so that the Hausdorff dimension of Eh is at least h/h.

Let us now consider the case in which h is infinite. The following result provides a lower bound on the dimension of the iso-Hölder set Eh , for every h ? [h, h). Proposition 10. Let us assume that h = ?. Then, with probability one, for every real h ? [h, h),

dim Eh ? 0.

To prove Proposition 10, we do not use the theory of sets with large intersection. Instead, we follow the main ideas of the proof of Lemma 9 in [22]. Specifically, for any h ? [h, h), we obtain a point yh in the set Eh as the intersection of a sequence (Inh )n?1 of nested closed sets and we show that, with probability one, the construction of this sequence is possible for all h ? [h, h). To this end, let us establish the following preparatory result.

Lemma 11. With probability one: (a) For every u ? U , the set uU ? ? S is nonempty. (b) There is real ? ? 1 such that

? j ? 1 #S j ? ? 2 j ? j?1 j 2. (c) If ? < 0, then the set ? is empty. Conversely, if ? ? 0, then there is a real ? ? 1 such that ? j ? 0 #S j ? ? 2(1??) j/2 j?1 = j ? .

Proof. To begin with, observe that b directly follows from (

P(?v ? uU v ? j =? Xv = 1) ?

?1, ({(

B j?1 (1 ? ? j?1)2 j? u ?1 times that of Buj?1. Arguing by induction, one then readily verifies that

P(Buj ) = P(Buu )

(1 ? ? ?1)2 ? u ?1 .

j
= u +1
Since 2 ? = ?, the preceding product tends to zero as j ? ?. By (

Xu = 0 =? ?u ? u U ? Xu = 1.

Thanks to (

Let us prove c. If ? < 0, then Lemma 6 ensures that the set ? is almost surely empty. Conversely, let us assume that ? ? 0. Owing to the Markov condition (M), for any integer j ? 1, the conditional expectation of #S j conditionally on the ? -field generated by the variables Xu for u ? U with u < j is at most 2 j ? j?1 + ? j?1 #S j?1. Arguing by induction on j , one can establish that with the convention that ??1 = 1. As ? j?1 vanishes, it follows that

? ? If u ? j , the right-hand side vanishes for j large enough. Otherwise, since ? is assumed to be less than one, its limit inferior as j ? ? vanishes. Thus, Furthermore, for any vertex u ? U and any integer j > u , observe that a.s. ?u ? uU

Xu = 0. ?u ? u U

Xu = 0

? Buj , ? j ? 0 E[#S j ] ? ? j ? j E[#S j ] ? ? j?1 k=?1 2k+1?k j?1 =k+1

? , j?1 = j ? ?

j?1 k= j?1 2k+1?k k = j ?

. k = j As a result, for some real c > 0 and every integer j ? j , Markov?s inequality yields = j ? ? ? ?

! E #S j

" 2(1??) j/2 j?1 ? = j ? c 2?(1??) j/4.

We conclude using the Borel-Cantelli lemma.

From now on, we assume that the event on which the statement of Lemma 11 holds occurs. For any h ? [h, h), let us build recursively a sequence (Inh )n?1 of nested closed subsets of the torus which lead to a point of the set Eh . For this purpose, let for any integer j ? 0. Since j 2 j ? j = ? and h = ?, it is easy to check that (? hj ) j?0 is a sequence of positive reals which enjoys

?? > 0 ? hj = o(2?j ) as j ? ?.

Moreover, for any vertex u ? U , let Buh be the open ball with center xu and radius h 2?h u / h and let 23 Bu be the open ball with center xu and radius 3 · 2? u ?1.

Together with the sequence (Inh )n?1 of nested closed sets, we build a nondecreasing sequence ( jnh )n?0 of nonnegative integers. The construction of the set I1h and the integers j0h and j1h depends on whether or not ? is negative.

? Step 1, if ? < 0. Let us build the set I1h and the integers j0h and j1h . As h = ?, the series j 2(1?h/ h) j ? j?1 j 2 converges, so there is an integer j0h ? 4? such that j where ? is given by Lemma 11b. Furthermore, Lemma 11a shows that, for j ? j0h large enough, the set S j0h ? . . . ? S j is nonempty. In addition, there is at least one connected component, denoted by I j , of the complement in the torus of the balls Buh , for u ? S j0h ? . . . ? S j , which has Lebesgue measure at least j = j0h #S j ? 4?

j j = j0h

3 .

(

1
? Step 1, if ? ? 0. Since ? ? [0, 1), there is an infinite subset J of N such that
? j . . . ? j?2 ? 2(5?+1)( j?1)/6 for any integer j ? J . Together with Lemma 11c, this
ensures that #S j?1 is bounded by ? 2(2+?)( j?1)/3 for any j ? J . Let j0h denote an
integer in J which is greater than 4? and is large enough to ensure both (

j
This last inequality follows from Lemma 11b?c, along with (

j j = j0h ? 2(2+?)( j0h?1)/3 + ? ? ? 2 2+3? ( j0h?1) + ? 1/2

j
.
This inequality holds for j large enough, because of (

The inequality follows from Lemma 11b and (

j
j = jnh+1
% ,
which holds for j large enough because of (

jn+1

The sets I0h , I1h , . . . given by the preceding procedure form a decreasing sequence of
closed subsets of the torus. Moreover, the diameter of each set Inh is at most ? hjh 2?h jnh/ h ,
n
which tends to zero as n ? ? by virtue of (

Lemma 12. The point yh belongs to the iso-Hölder set Eh .

Proof. Let ? ? (h, h]. Owing to (

n d(yh , xun ) ? 2?h un / h + ? hjh 2?h jnh/ h < 2?h un /?.

n
The last inequality follows from the fact that n ? n0 and un ? jnh . As a result, the
point yh belongs to the set L? defined by (

This proves the lemma in the case where h = h. We may therefore assume that h > h. Lemma 4, together with the fact that ? ? L? is nondecreasing, shows that it suffices to establish that yh ? Lh .

Let us assume that ? < 0. The point yh belongs to I1h , so it cannot belong to any ball
Buh for u ? S j0h ? . . . ? S jh . Moreover, for any integer n ? 1, the point yh belongs to
1
Inh+1, so it cannot belong to any ball Buh for u ? S jnh+1 ? . . . ? S h . It follows that yh
jn+1
does not belong to any ball Buh with u ? S and u ? j0h . Hence, yh does not belong to
the set Lh defined by (

Let us assume that ? ? 0. In this case, the point yh does not belong to any ball Bh u with u ? S and u ? j0h and does not belong to any closed dyadic interval ?u with u ? S j0h?1. Therefore, the point yh cannot belong to the set Lh . It cannot belong to the set ? either. Otherwise, there would exist a vertex u ? S and a sequence ? = (? j ) j?1 ? ??u h enjoying yh = ? (x?? ). If u ? j0 ?1, then the vertex ?1 . . . ? j0h?1 would belong to S j0h?1 and index a closed dyadic interval containing yh . If u ? j0h , then the point yh would belong to the set ?u and thus to the ball Buh , since h > h. In both cases, we would end up with a contradiction. Hence, yh does not belong to ?. Lemma 7 finally ensures that yh ? Lh .

We have established that, with probability one, for any h ? [h, h), it is possible to build a point yh in the set Eh . Proposition 10 is thus proven.

Using Propositions 8, 9 and 10 together with the fact that Eh ? ? by Lemmas 4 and 7, we finally obtain the following result.

dim Eh = max(h/h, dim ?) and ?h ? (h, min(h, h)) dim Eh = h/h.

Theorem 2 is then an immediate consequence of this result and Lemma 6.

In order to prove Proposition 2, let h ? [h, h] and let x ? Eh . Let us first assume that h < h and let us consider a real number ? > h/h ?1. Owing to Lemma 4, for any integer n ? 1 such that h + 1/n < (? + 1)h, there exists a dyadic interval ?n ? ? enjoying ?n ? n, C?n = 2?h ?n and d(x , x?n ) < 2?h ?n /(h+1/n). Note that these intervals ?n are such that d(x , x?n )1+? ? 2? ?n . Proposition 3 in [1] then shows that ?R (x ) ? ?. This inequality holds for any ? > h/h ? 1. Thus, ?R (x ) ? h/h ? 1.

Conversely, let us consider a real number ? > ?R (x ). Owing to Proposition 3 in [1], there exists a sequence (?n)n?1 of dyadic intervals of the torus such that d(x , x?n )1+? ? 2? ?n for all n ? 1, 2? ?n + d(x , x?n ) ?n????? 0 and

log |C?n | log(2? ?n + d(x , x?n )) ?n????? h.

As h < h, for infinitely many integers n ? 1, we have log2 |C?n | ? ?n

log |C?n | ? log(2? ?n + d(x , x?n )) < h, so that Xu?n = 1. Thanks to Lemma 4, it follows that h ? (1 + ?)h. Letting ? ? ?R (x ), we obtain ?R (x ) ? h/h ? 1.

Let us now suppose that h = h < ? and consider a real ? > 0. As h > h, the point x does not belong to Lh by virtue of Lemma 4. Hence, there exists an integer j0 ? 0 such that Xu = 0 for any vertex u ? U enjoying u ? j0 and d(x , xu ) < 2? u . For any integer j ? j0, there is a vertex u j ? U satisfying u j = j and d(x , xu j ) < 2? j . Observe that 2? ?u j + d(x , x?u j ) ?j????? 0 and log |C?u j | log(2? ?u j + d(x , x?u j )) ?j????? h.

Proposition 3 in [1] then ensures that ? ? ?R (x ). We conclude by letting ? ? 0.

Acknowledgement. The author is grateful to Stéphane Jaffard for many useful comments. Communicated by M.B. Ruskai