$L^q$ dimensions and projections of random measures

We prove preservation of $L^q$ dimensions (for $1<q\le 2$) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on $1$-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for $L^q$ dimensions.


Introduction
In recent years there has been great interest in understanding the size of linear (and non-linear) images of sets and measures of dynamical and arithmetic origin. Here 'size' may refer to some fractal dimension, or to Lebesgue measure/absolute continuity.
Even if one is concerned with sets, the proofs usually involve measures supported on them; in this article, we deal primarily with measures (by a measure we always mean a Borel locally finite measure on some Euclidean space). If μ is a measure on a space X and f X Y : → is a map, we denote the push-forward of μ via f by fµ, that is, is a 'nice' Lipschitz map and dim is some notion of dimension for measures, then 'typically' µ Π is 'as large as possible' in the sense that dim dim µ µ Π = if k dim ⩽ µ , and k dim µ Π = if k dimµ > (in the latter case, one expects µ Π to also be absolutely continuous). A precise version of this heuristic is given by Marstrand's projection theorem (and its variants) which, for the case of measures, says that, for any measure μ on d R , there is an equality k dim min dim , for almost all linear maps : d , whenever dim is either Hausdorff or L q dimension ( q 1 2 ⩽ < ); these notions of dimension will be defined later. See e.g. [19, chapter 9] and [16, theorem 1.1]. However, for measures with a dynamical or arithmetic structure, such as self-similar measures or measures invariant under some algebraic dynamical system, one would like to say more, ideally finding the precise set of exceptional linear maps Π.
Early results of this type for sets were obtained in [3,11,22]. Recall that the (lower) Hausdorff dimension of a measure μ is where A dim H is the Hausdorff dimension of A. A general method to bound the Hausdorff dimension of projected measures was developed in [14], with variants and applications given in [1,7,9,10]. Among other things, the equality k dim m in dim , is established for many classes of measures (satisfying certain necessary assumptions), including self-similar measures, more general random cascades on self-similar sets, products of m × -invariant measures on [0, 1], and Bernoulli and Gibbs measures for the natural symbolic coding of the × × ( ) m n , -toral automorphisms, and all linear maps Π (apart from obvious exceptions). A recent breakthrough on the dimensions of self-similar measures [15] also has applications on the dimension of projections, see [25]; this work again deals with Hausdorff dimensions of measures.
Although Hausdorff dimension is no doubt highly relevant, there are many other concepts of dimension of a measure which are also important both mathematically and in applications. Chief among them is the one-dimensional parameter family of dimensions known as L q dimensions: let n I : , where n D is the family of dyadic cubes j j 2 0, 1 : For q q 0, 1 > ≠ , the lower L q dimension D q ( ) µ is the (lower) suitably normalized scaling exponent of n q ( ) µ C as n → ∞: For simplicity we always take logarithms to base 2, unless otherwise noted. The upper L q dimension D q is defined analogously. When the limit in question exists, it is denoted D q ( ) µ ; in this case we say that the L q dimension exists. This family of dimensions measures the degree of singularity of a measure according to its global fluctuations, and are a central ingredient of the multifractal formalism. Of special relevance is the value q = 2; D 2 is also known as the correlation dimension of μ. This is partly because (lower) correlation dimension can also be defined in terms of energies: The map q D q µ is non-increasing, and D D dim (see e.g. [8]). In general, q D q may be strictly decreasing (this is a reflection of the multifractality of μ), but it may also be constant. For example, if μ is Ahlfors-regular with exponent d (that is, if C r Bx r C r , = for all q. For many measures of dynamical origin, such as self-similar measures, the limit in the definition of D q is known to exist, see [23].
The only previous result on L q dimensions of all projected measures (as opposed to almost all) was obtained in [20]. There it is proved that if , µ ν are self-similar measures satisfying certain natural assumptions, then for any q 1, 2 ( ] ∈ , for all orthogonal projections Π onto lines, other than the principal ones (which are clearly exceptional for products).
In this article we prove preservation of L q dimensions for q 1, 2 ( ] ∈ under all projections, for a class of planar measures which include certain self-similar and stochastically self-similar measures, and for certain products of two measures. Precise definitions are given in the next section. Among other applications, we improve upon the main result of [20] in several different directions, and obtain a different (and somewhat more elementary) proof of a projection result from [14] and sharpen it in some special cases.
We follow the general approach of [20], with suitable variants. A central element in the main result of [20] is the existence of a certain subadditive cocycle over an irrational rotation. In the present setting, there is also a subadditive cocycle at the core of the proofs, but the base transformation is now a circle extension of a shift space. Most of the additional work is then concerned with studying this somewhat more complex dynamical object. Nevertheless, we also introduce some generalizations and clarifications that are valid also in the deterministic setting of [20].

The model
Our general setup is as follows. A rule is an iterated function system f f , , k 1 ( ) … , where each map f j is a strictly contractive similarity on d R (the ambient dimension d will always be either 1 or 2, later on we will impose an additional homogeneity assumption on the rules). We will work with a finite set of N rules f f , , stands for the closed ball of radius R centered at the origin. Given a sequence Y N : 1, , we define the space of words of length n (possibly with n = ∞) with respect to ω by the formula Note that all n ( ) X ω are subsets of a common tree k : 1 , , We define a compact set B : .
… ω + (where ul denotes the concatenation of u and l ). In other words, these disks are nested. Moreover, their diameters tend to zero uniformly. Alternatively, where u n | is the restriction of the infinite word u to its first n coordinates. Given u n ( ) X ∈ ω , we also define the cylinder u [ ] ω as the set of infinite words in ( ) X ω ∞ that start with u, and note that We remark that we do not assume that B u : ω are disjoint or any other separation condition. Moreover, we do not exclude the possibility that there is a single rule (N = 1), in which case ( ) ω C is a deterministic self-similar set. Even though ( ) ω C is defined for every ω, our results will be probabilistic in nature, and we will be drawing ω according to an invariant ergodic probability measure μ for the left-shift T on Y.
Similar models have been considered in the literature, sometimes under the names 'homogeneous random fractal' or 1-variable fractal, see e.g. [2,13,26].
We will not be interested in the sets ( ) ω C themselves, but rather in measures supported on them. For each i, let p p p , , be a probability vector. On each ( ) X ω ∞ we can then define the product measure p .
The projection of ( ) η ω via the coding map is a Borel probability measure ( ) η ω on ( ) ω C . In the deterministic case N = 1, this is simply a self-similar measure on ( ) ω C . The random case arises naturally, even if a priori one is interested only in deterministic self-similar measures. For example, conditional measures on slices of (deterministic) self-similar and self-affine measures often have this form, and one can decompose an arbitrary self-similar measure as d ( ) for an appropriate choice of weights p i and measure μ; see section 6.4 below. Although the family of product measures just described provides our main class of examples, the proofs extend to more general families of Borel probability measures on X ∞ satisfying the following conditions: η ω is continuous (considering the weak topology on the space of Borel probability measures on X ∞ , where X ∞ is endowed with the product topology). In other words, for any continuous function g on X ∞ , we have When ( ) η ω is a product measure as above, this condition holds with equality and K = 1. This suggests that, in general, the measures ( ) η ω satisfying (2) (or rather their projections ( ) η ω under ∆ ω ) can be thought of as satisfying some kind of 'sub-self-similarity'. In the deterministic case (in which there is a unique measure η ), Gibbs measures for Hölder potentials also satisfy (2). We also note that, since cylinders generate the Borel σ-algebra of each space ( ) X ω ∞ , it follows from (2) that for any u n ( )

L q dimensions of projections
Now we specialize to d = 2 and assume that the rules have the form f f , , , where the function f : . In other words, each rule is a homogeneous IFS (only the translations differ).
We consider the unit circle S 1 endowed with the corresponding normalized Haar measure L. Furthermore, define the continuous map Y S : Recall that ω is said to be μgeneric if converges to μ (here, and throghout the paper, convergence of probability measures is understood to be weak convergence). The orthogonal projection onto the line generated by We can now state our first main theorem. C . Assume furthermore that the product measure µ × L is ergodic for the skew-product S defined in (5).
Then for each q 1, 2 ( ] ∈ there is a number D(q), such that for μ-almost all ω it holds that Furthermore, the convergence of n n q for all μ-generic points ω, or if D q 1 ( ) ⩾ , then the above conclusions hold for all μ-generic ω.
Examples and applications of this result will be discussed in section 6. The assumption that each rule is homogeneous is critical for our method, and it would be interesting to know whether it can be dropped (we recall that for Hausdorff dimension there are similar results which do not require homogeneity, see [7,14]).

Convolutions of Cantor measures
Recall that the convolution µ ν * of two measures , µ ν on d R is the push-down of the product µ ν × under the addition map x y x y , ( ) + . In this section we will be concerned with convolutions of two measures on R, one of which is a deterministic measure supported on a selfsimilar set, while the other is a random measure satisfying properties analogous to those of the previous section. Recently, there has been much interest in understanding the behavior of various dimensions of measures under convolution, in relation to their algebraic and geometric structure, see e.g. [14,15,20]. Again, most of the known results are for Hausdorff (or entropy) rather than L q dimensions. The exception is [20, theorem 1.1], which we generalize below.
Fix N rules of the form f f , , be a family of measures satisfying assumptions (a)-(c) above (with ( ) ν ω in place of ( ) η ω ). We will assume without loss of generality that ; we can always achieve this via an affine change of coordinates, which will not affect the statement of the theorem. As before, we denote ( ) where ∆ ω is the coding map. The measures ( ) ν ω are then supported on the Cantor sets 0, 1 [ ] We consider yet another rule g g , , k . Again, we assume that g 0, 1 This is a special case of the preceding framework with N = 1 rule, but we repeat some definitions for the sake of fixing notation. We denote the code space by k 1, , ′ (again allowing n = ∞), and the coding map by : The cylinder of infinite words in X′ ∞ starting with u will be denoted simply by [u].
Further, let ϑ be a Borel probability measure satisfying the analogue of (c) above in the random case, that is, we assume that for some constant K 0 > ′ , and set ϑ ϑ = ∆′ . We fix r N ∈ such that b a 1 min and, for this choice of r, we fix l N ∈ large enough so that   and also that β can be made arbitrarily large by choosing l appropriately. Consider the space S 1 β obtained by taking the interval , [ ) β β − and identifying its ends, i.e. β β − = , and endow it with the normalized Lebesgue measure β L . Furthermore, define the continuous map Y : → R α by the formula : where, as before, T denotes the left shift operator on Y, and ( ) + β stands for the natural sum in S 1 β . Also, for each n N ∈ let us define the nth rotation Y S S R : , , n S n r r n r n We can now state our main result on convolutions.

Theorem 2.2. Let μ be an ergodic, invariant measure for
for all μ-generic points ω, or if D q 1 ( ) ⩾ , then the above conclusions hold for all μ-generic ω.
Note that x y x A y , t ( ) + is, up to affine homeomorphism, the orthogonal projection with angle t arctan( ); hence this result can also be interpreted in terms of projections of the product measure ( ) ν ϑ × ω . Again, the homogeneity assumption on the rules is crucial. Also, we do not know if the statement holds if ϑ is also chosen randomly according to a sequence of rules. Although this appears natural, it does not seem possible to build a cocycle like the one at the core of the proof in this setting.

Some auxiliary results from ergodic theory
In this section we collect some ergodic-theoretic facts. The results are rather standard, but we include proofs for completeness, as we have not been able to find exact references. Since the proofs are the same, we state them in greater generality than needed in our later applications.

Compact group extensions and generic points
We begin with some general definitions. Let Y T , , ( ) µ be a measure-preserving dynamical system for a compact metric space Y together with its Borel σ-algebra, G a compact group endowed with the Haar measure m G and Y G : → α a continuous map. Define the skew-product map S on X Y G = × by the formula The study of compact group extensions of this form is classical, going back at least to [18]. The application of this theory to the study of projections was highlighted by Falconer and Jin in [7], who used compact group extensions to study the Hausdorff dimension of projections of a class of random measures (different from ours). A measure ϑ on X is said to project over μ if Y π ϑ µ = , where Y π stands for the projection on Y. Furthermore, ϑ is said to be uniquely ergodic over μ if it is the unique ergodic measure which projects over μ. Notice that the measure m G µ × clearly projects over μ and is also S-invariant by the Fubini-Tonelli theorem, since μ is T-invariant and m G is the Haar measure on G.

Proposition 3.1. A measure ϑ on X is uniquely ergodic over an ergodic measure μ if and only if it is the unique S-invariant measure which projects over μ.
Proof. For the ⇐ implication, we note that ϑ must be ergodic: showing that ϑ is ergodic as claimed. The ⇐ implication is now clear.
Thus, suppose that ϑ is uniquely ergodic and let ζ be an S-invariant measure that projects over μ. We must show that ζ ϑ Since μ is ergodic, it is an extreme point of the set D whence, by Bauer's characterization of extreme points [4, chapter IX, theorem 3], we have that Y π ρ δ = µ . But then, since ρ is supported on the set of ergodic measures and ϑ is uniquely ergodic, we obtain that 1 : π ρ µ ρ σ π σ µ ρ σ σ π σ µ ρ ϑ which implies that ρ δ = ϑ and concludes the proof. □ Recall that given a measure-preserving system Z R , , It follows from the ergodic theorem that if σ is ergodic then σ-almost every z Z ∈ is generic for σ.
The special case of the next lemma in which the base system (Y, T) is uniquely ergodic is a classical result of Furstenberg, see e.g. [6, theorem 4.21]. The general case goes along the same lines and is surely known, but we give the proof for completeness.
is ergodic then it is uniquely ergodic over μ.
Proof. Clearly we have Y π ϑ µ = , so that it remains to check that ϑ is the unique ergodic measure with this property. Now, since ϑ is ergodic we have that ϑ-almost every g X , (˜˜) ω ∈ is generic for ϑ. Furthermore, we have that Indeed, observe that for any i , Then for any continuous function f we have that ) ω ′ is generic for ϑ. Now, let ρ be an ergodic measure on X which projects over μ. For any such ρ the set has full μ-measure on Y. In particular, the set Λ ∩ Λ ϑ ρ is nonempty, where Λ ϑ is defined by analogy with (13). Notice that by (12) we have that for any ω ∈ Λ ∩ Λ ϑ ρ there exists g G ∈ such that g , ( ) ω is generic for both ϑ and ρ. But by definition of generic point this implies that for any continuous function, which shows that ϑ ρ = . □ We finish this section with the following uniform convergence result, which again is classical in the uniquely ergodic case.
is ergodic, then for every continuous function f X : → R and every μ-generic point ω, the ergodic averages Proof. Suppose the statement does not hold for some μ-generic ω and continuous f. Then we can find 0 ε > , a sequence n j → ∞ and points g G j ∈ such that After passing to a further subsequence, we may assume that : , which equals μ thanks to our assumption that ω is generic.
It now follows from proposition 3.1 and lemma 3.2 that . This contradicts (14), as desired. □

Subadditive cocycles and generic points
It is well known that if (X, S) is uniquely ergodic, then ergodic averages of continuous functions converge uniformly. This fails for cocycles over uniquely ergodic systems, but one side of the inequality holds: this was observed by Furman [12, theorem 1]. An inspection of the proof of the subadditive ergodic theorem given by Katznleson and Weiss [17] yields a more general result. First, we introduce a definition.

Definition 3.4. A function
Theorem 3.5. Let X S , , ( ) µ be an ergodic measure-preserving system with X compact and S continuous. Let n n be an upper C-approximable subadditive cocycle on X, that is, Then for any μ-generic x X ∈ , it holds that We start by noting that upper C-approximable functions are bounded from above and integrable by definition. Fix N N ∈ , 0 ε > and let By assumption, there exists a continuous function X : Note also that (16) and use the last inequality, we obtain for all sufficiently large n N ∈ , where we have written n as (m + 1)N + r for some m N ∈ and r N 1 ⩽ ⩽ . Dividing the above inequality by Nn yields Observe that m n N 1 → as n → ∞, so that by taking lim sup n→∞ in both sides and using the fact that x is generic, we may conclude that Since the bound in (19) holds for arbitrary N N ∈ and 0 ε > , we obtain the result. □ In the special case of a compact group skew-product Y G S m , , , we can apply lemma 3.3 to get the following improvement.
) µ be a measure-preserving system with Y compact and T continuous, let G be a compact group with Haar measure m G , and let Y G : → α be continuous function. Denote the associated skew-product measure-preserving system by X S m , , be a an upper C-approximable subadditive cocycle over X.
Proof. This follows from lemma 3.3 by letting n → ∞ in the pointwise bound (18).

Notation and preliminaries
Recall that Y N 1, , For each n N ∈ let us define the nth rotation Y S S R : for a certain constant d u Also, for each n N ∈ we define L n ( ) ω as the unique nonnegative integer such that and consider the family of intervals n D ω . With this, for q > 1 we define the functions Y S :

A submultiplicative cocycle
Our aim is to show that given q > 1 there exists a continuous subadditive cocycle q q n n , for every v Y S , 1 ( ) ω ∈ × . We do this in two steps. We first show that the family log q n n , ( ) N τ ∈ for q n , τ defined in (22) constitutes, up to additive constants, a subadditive cocycle. Then, we prove that there exists a 'smooth' analogue q n , τ of q n , τ which is continuous. From these facts it will follow that the cocycle log q q n , τ = F enjoys the desired properties. In this section we establish the core of this program, by showing that there exists K 1 > 1 such that for any n m , where we have suppressed q from the notation for simplicity. This implies that the family K log n n 1 ( ) N τ ∈ is a subadditive cocycle. We begin by introducing a definition.
) ω ∈ × and proceed to show (24). Recall that : : , Therefore, using (27), (28) and Hölder we obtain that Summing over all I n m ( ) D ∈ ω + such that I J ⊂ , we get Now, using (20) it is not hard to see that for any such interval I and u v J ( ) . Write n n m 1 λ λ = ω ω + + , and note that the family is composed of consecutive intervals of equal length between 1 2 and . Since the same is true for the family m , .

A continuous analogue of τ n
We now construct for each n N ∈ a continuous function n τ which is comparable up to multiplicative constants to n τ . To this end, we consider For any fixed y R ∈ the function x xy : , and is equal to 1 on the interval y y 2 , 2 We claim that n τ is continuous. Indeed, this is a consequence of the following fact. ρ ω is continuous. Then, for any continuous function h X Z : Proof. By uniform continuity, given 0 Note that L n ( ) ω is continuous, since it depends only on the first n coordinates of ω, and hence x n ( ) ( ) ψ ω is jointly continuous. We can then apply lemma 4.4 with X Y   D ω and j j j 2 2 , 3 2 : On the other hand, to establish the leftmost inequality we observe that for any Y ω ∈ and j Z ∈ we have the inclusion Notice that log n τ is well defined by (34), since n τ is strictly positive by its mere definition. Furthermore, each log n τ is continuous, since n τ is. Thus, we conclude that the sequence log n n ( ) N τ ∈ is a continuous subadditive cocycle on Y S 1 × .

The proof of equation (23)
Write log n n φ τ = for simplicity. We can now show that The '⩾' inequality is clear, since the limit in the left is taken along a subsequence. To see the other inequality, fix k and choose n such that L k L n n 1 On the other hand, the sequence k k q ( ) ν C is always decreasing for q > 1, since for a dyadic interval J I I 1 2 = ∪ one has J I I . Hence provided n is large enough.
The claim for D q follows by taking a limit along an appropriate subsequence of k, and the case of D q is analogous. ( ) η ω

A subadditive cocycle for the measures
The foregoing analysis of the measures v ( ) η ω has a corresponding, but simpler, correlate for the measures ( ) η ω . Since the proofs are very similar, we only state the results, leaving the details to the interested reader. Let n Q denote the family of dyadic squares j j j j j j 2 , 1 2 2 , 1 2 : , .
Then one can check, as before, that there exists a sequence of continuous functions n ξ , such that

Conclusion of the proof
We start by applying (36) to show that D q ( ) ( ) η ω exists and is constant μ-almost everywhere. However, before we can do so it is clear that we must first understand the behavior of the quotient L n n ( ) ω as n tends to infinity. This is the purpose of the following lemma.
But this follows at once from the fact that ω is μ-generic, since the application log 1ω λ ω is continuous on Y. □ Now it follows from the subadditive ergodic theorem that for μ-almost all ω it holds that η ω exists and equals D(q) for μ-almost all ω. Furthermore, if ω is μ-generic, then it follows from theorem 3.5 that η ω . Now we move onto the projections v ( ) η ω . Let us begin by observing that for all v Y S , Indeed, this follows from the well-known facts that D q does not increase under Lipschitz maps, and can never exceed the dimension of the ambient space. Now, since µ × β L is ergodic by assumption, corollary 3.6 combined with lemma 4.5 imply that any μ-generic ω satisfies, for each v S 1 ∈ , Hence, given any Y ω ∈ , if we wish to prove (6), then it suffices to show that Since n n ( ) N φ ∈ is a bounded subadditive cocycle, the subadditive ergodic theorem yields upon an application of the Fubini theorem that μ -almost every for L-almost every v S 1 ∈ . In light of (23), µ and from theorem 4.1 and Fubini, we deduce that this equals D q min ,1 ( ( ) ) (this is the point of the proof where we use that q 2 ⩽ ). Hence, if we let then (38) holds for all ω ∈ E, and if D q 1 ( ) ⩾ , also for ω in the set G of μ-generic points.
We conclude that for every ω in the full μ-measure set ∩ G E, all the inequalities in (37) are equalities, and hence (6) is satisfied. If ⊂ G E, or if D q 1 ( ) ⩾ , then (6) holds for any μ-generic ω. The claim concerning uniform convergence over v S 1 ∈ follows from the above analysis, and the uniformity in corollary 3.6 (which implies that the rightmost inequality in (37) holds uniformly in v, for any fixed ω ∈ G ). This finishes the proof of theorem 2.1.

Preliminaries
The proof of theorem 2.2 follows the same general outline as the proof of theorem 2.1. We will therefore indicate where the main differences lie, and sketch or omit the parts of the proof that closely follow the arguments from theorem 2.1.
we consider the orthogonal projection s Π onto the linear space generated by the vector (1, e s ), i.e.
In the course of the proof it will be important to work with families of pairs (u, v) such that the eccentricity of the rectangle h 0, 1 ω is bounded, and behaves like a rotation under the action of the skew-product S. In order to do this, we need to introduce a number of families of pairs of words. Similar families appear in [20], although here we will need an extra family due to the somewhat more complicated setting.
Hence, let us consider the families ii. An easy calculation using (8) and (9) shows that the size of all rectangles in n ( ) In particular, the eccentricity of rectangles in each family, i.e. their height-width ratio, always stays bounded in between e 3β − and e 3β .
iii. Under the convention From the above considerations it is easy to see that for u v , , e , We note that the family Y ( ) ( ) η ω ω∈ satisfies the following conditions, closely related to (a)-(c) above.

D Galicer et al Nonlinearity 29 (2016) 2609
As a matter of fact, we will prove the result for projections of families of measures ( ) η ω satisfying these conditions (i.e. it will not matter that ( ) η ω is a product measure for each ω).

A submultiplicative cocycle
For each n N ∈ we define L n ( ) ω as the unique nonnegative integer such that a a 2 2 , and, as in the proof of theorem 2.1, consider the nested families of intervals n ( ) D ω given by With this, for q > 1 we define the functions Y S : .
Similarly to the proof of theorem 2.1, we will show that q n , τ is a submultiplicative cocycle (up to a multiplicative constant), and then we will construct a 'nicer' cocylce q n , τ which is comparable to q n , τ . Unlike the situation in theorem 2.1, the functions q n , τ will not be continuous, but will nevertheless be approximable by continuous functions in a suitable way. Since q will remain fixed, we drop it from the notation.
Hence, the first step is to show that there exists K 1 > 1 such that for any n m , N ∈ and s Y S , To see this, let us fix n m , N ∈ , s Y S , 1 ( ) ω ∈ × β , and proceed to show (43). We will consider three separate cases, depending on whether s R , 0 Y ω for the proof of the second case and with ( ) Z ω for the proof of the third case. Except for this difference, the proof of all three cases is completely analogous so we will only comment on the first case only.
The proof is a minor variant of the proof of (24). Given J n ( ) , : and consider the interval J which has the same center as J but of length This can be established in a very similar manner to (27); we omit the details. If we continue to argue as in the proof of theorem 2.1, we further obtain recall (29) and (30). Now, using (39) it is not hard to see that for any such interval I and u v , Combining this with (44), and reasoning exactly as in the end of section 4.2, we finally deduce that the cocycle relation (43) holds for some K 1 > 0 depending on q.

An upper C-approximable analogue of τ n
In order to apply corollary 3.6, we need a C-approximable cocycle (recall definition 3.4). Unlike the situation in theorem 2.1, there is now an inherent discontinuity at the end point of the interval , ; note that although the identification of the extreme points is required for applying ergodic-theoretic tools, as far as the geometric definition of n τ is concerned, there is no such identification. This issue arises already in [20, p 107], where (in the course of proving what effectively is a special case of theorem 2.2) it is incorrectly claimed that the functions n φ (corresponding to our n τ ) are continuous. In fact, there is continuity up to the endpoint of the interval. Fortunately, this turns out to be a minor issue: because the discontinuity set is small, the new cocycle is still upper C-approximable.
We proceed to the details. Firstly, in close analogy to section 4.3, we define x y x y , 2 .
and n τ is continuous on Y , . Since we clearly have 0 1 n ⩽ ⩽ τ , the fact that n τ is C-approximable is now a consequence of the following lemma.

Conclusion of the proof
The remaining of the proof of theorem 2.2 follows exactly the same lines as the proof of theorem 2.1. In particular, (23) holds in the current setting. Details are left to the interested reader.

The deterministic case
When there is just N = 1 rule, we obtain the following immediate consequence of theorem 2.1 Proof. The dynamical system Y S S , degenerates to rotation by α on the circle, for which Lebesgue measure is certainly ergodic. This is then just a special case of theorem 2.1. □ Measures η satisfying the assumptions include product (Bernoulli) measures on X ∞ , as well as Gibbs measures for Hölder potentials and for almost-additive sequences of potentials. When η is Bernoulli, then η is a self-similar measure on the corresponding self-similar set, so in particular we obtain existence and preservation of L q dimensions of projections of selfsimilar measures for homogeneous planar iterated function systems (for which the linear part contains an irrational rotation), regardless of overlaps. For Hausdorff dimension this is known to hold also for non-homogeneous systems [7,14]; below we will recover this as another consequence of theorem 2.1.
In a similar way, we have the following consequence of (the proof of) theorem 2.2.
where A t x = tx, and moreover n q n D D log Proof. Since there is no code space, the ergodicity assumption in theorem 2.2 reduces to b L being ergodic for the map s s b a ln( / ) (8) and (9); we take r = 1 since a < b). Since b a log log / is irrational, these systems are isomorphic to irrational rotations for any value of , so the claim follows from (the proof of) theorem 2.

□
This extends [20, theorem 1.1], and most of the generalizations outlined in [20, section 5]. More precisely, we allow overlapping in the construction, our measures on the Cantor sets are more general (including Gibbs measures), and we obtain uniform convergence over compact sets of scalings t.

Random self-similar measures
Next, we go back to the setting of theorem 2.1 with N different rules, but assume that the measures ( ) η ω have the following product structure. For each i N 1, , { } ∈ … , let p p p , , … be a probability vector, and set p . . We want to obtain an explicit formula for the L q dimensions of the projections ( ) η ω ; for this, we need to assume some separation assumption. For simplicity we assume the following very strong separation condition: Similarly, it was shown in the proof of lemma 4.5 that for any ω ∈ G, n log log d . By applying this to the case in which μ is a Bernoulli measure, we obtain the following consequence of theorem 2.1. according to ν in the following way: choose σ ∈ Σ according to the probability vector r; then choose u according to the probability vector p σ . Thanks to the product structure of ν, this extends to infinite sequences as follows.
For each N ω ∈ Σ , let ( ) η ω be the product measure p i