RESONANCE BETWEEN PLANAR SELF-AFFINE MEASURES

. We show that if { φ i } i ∈ Γ and { ψ j } j ∈ Λ are self-aﬃne iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-aﬃne measures µ and ν , the inequality dim H ( µ ∗ ν ) < min { 2 , dim H µ + dim H ν } implies that there is algebraic resonance between the eigenvalues of the linear parts of φ i and ψ j . This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line.


Introduction
In the 1960s, Furstenberg conjectured that if X, Y ⊆ [0, 1] are closed sets invariant under multiplication by integers m and n, respectively, then for any s = 0, the inequality (1. 1) dim(X + sY ) < min{1, dim X + dim Y } implies that log m log n ∈ Q. Here and in the following, dim denotes the (lower) Hausdorff dimension for both sets and measures. This conjecture was one of several that aimed to capture the idea that if log m log n ∈ Q, then expansions in base m and n should have no common structure: Indeed, the right-hand side of (1.1) is always an upper bound for dim(X + sY ), while the strict inequality (1.1) implies that many of the fibers {(x, y) ∈ X × sY : x + y = z} are large, which means that X and Y should have arithmetically similar structure in many places. The phenomenon (1.1) is usually referred to as resonance: X and Y are said to resonate if they satisfy (1.1) for some s, and otherwise they are said to dissonate.
It is also natural to ask if a similar phenomenon holds in a more general setting: For dynamical systems X and Y , does (1.1) imply some kind of algebraic or arithmetic similarity between the sets or the dynamics? The first result in this direction is due to Moreira [10] from 1998, who proved that for two self-conformal sets on the line, (1.1) cannot hold in the presence of an irrationality assumption if one of the sets is totally non-linear. Recall that a set K ⊆ R is called self-conformal if for some C 1+ε -contractions f i . In 2009, Peres and Shmerkin [25] established analogous results for sums of selfsimilar sets on the line: The dimension of the sum is maximal unless the contraction ratios of the defining similarities form an arithmetic set. A set A ⊆ R is called arithmetic if A ⊆ αN for some α ∈ R. Recall that a set is self-similar if it satisfies (1.2) with f i being similarities. An analogous result for convolutions of Cantor measures on the line was obtained shortly after by Nazarov, Peres and Shmerkin [23]: Indeed, by replacing sum with convolution in (1.1), one can formulate the concept of resonance for measures.
A major breakthrough on the topic of resonance between dynamical systems was achieved by Hochman and Shmerkin in 2012 [19], who introduced a powerful method called the local entropy averages to attack problems regarding projections (and therefore sums) of dynamically defined fractals. Hochman and Shmerkin managed to both prove the original conjecture of Furstenberg, and extend to the setting of measures the existing results on the sums of self-similar and self-conformal sets on the line.
Namely, they proved that if {f i } i∈Γ and {g j } j∈Λ are families of C 1+ε -contractions on R that satisfy the open set condition, then for any associated self-conformal measures µ and ν and any t > 0, dim(µ * S t ν) = min{1, dim µ + dim ν} unless the asymptotic contraction ratios of φ i and ψ j form an arithmetic set. Recall that µ is self-conformal if for a probability vector (p i ) i∈Γ and f i as above. Due to well-known variational principles, the result for self-conformal measures implies the result for sets as well.
Recently, this result was generalized not to require any separation conditions in the self-similar setting by Bruce and Jin [9] and in the self-conformal setting by Bárány, Käenmäki, Wu and the author in a work in progress [8].
Perhaps surprisingly, almost nothing seems to be known of this phenomenon in higher dimensions. While there certainly exists literature on bounding the size of sums or convolutions from below, such as the famous inverse theorem of Hochman [17,18], it primarily focuses on showing that, for very general X and Y , the sum (or convolution) X + Y is strictly larger than X, unless X and Y have a very special structure. See also [14,27] and the references therein for progress in related phenomena. However, the existing results do not aim to capture the spirit of the phenomenon predicted by Furstenberg, that "geometric resonance" of dynamically defined sets should imply a kind of "algebraic resonance" between the dynamics.
The purpose of the present work is to provide an extension of this principle to the planar setting. However, in formulating an extension beyond the line, one has to be careful: The direct extension, that (1.4) dim(X + Y ) < min{2, dim X + Y } unless there is algebraic resonance between the dynamics of X and Y , breaks down easily. Indeed, one can of course isometrically embed any sets X and Y on the line to the plane, and their sum will always have dimension at most 1. It is also not difficult to construct examples of X and Y with dimension strictly greater than one by taking product sets. Thus, in order to expect (1.4) to imply algebraic resonance, one has to assume that X and Y are "spread out" in sufficiently many directions, in some sense.
In this paper, we consider the size of µ * ν when µ and ν are self-affine measures on the plane, that is, they satisfy (1. 3) with f i being invertible affine contractions on R 2 . Let RP 1 denote the collection of one-dimensional subspaces of R 2 . For a 2 × 2-matrix A, let |λ 1 (A)| ≤ |λ 2 (A)| denote its eigenvalues. Let A also denote the action induced by A on RP 1 . We say that a system Φ = {f i (x) = A i x + a i } i∈Γ of affine contractions on R 2 satisfies 1) the strong separation condition if there exists a bounded open set V = ∅ such that for every i = j ∈ Γ, f i (cl(V )) ⊆ V and f i (cl(V )) ∩ f j (cl(V )) = ∅, 2) hyperbolicity if there exists at least one i ∈ Γ such that |λ 1 (A i )| < |λ 2 (A i )|, 3) irreducibility if for every θ ∈ RP 1 , there exists i ∈ Γ such that A i θ = θ, and 4) the domination condition if there exists a multicone C ⊆ RP 1 , i.e. a finite union of closed cones, such that A i C ⊆ int(C) for each i ∈ Γ.
Let us comment on the assumptions of the theorem. The assumption of strong separation is classical in the study of iterated function systems, since it makes it possible to view the attractor as a dynamical system, giving access to a multitude of tools from ergodic theory. However, during recent years, much attention has been directed towards establishing existing results without assuming any separation conditions, and we expect it can be removed from our result as well.
The assumption of hyperbolicity ensures that the systems Φ and Ψ are "strictly self-affine". This is crucial in our approach, since strictly self-affine measures have a very special tangential structure that we will heavily use. This structure is formulated in Proposition 4.1 which is the main technical contribution of this paper, and we believe it can find applications outside this work as well. Without the hyperbolicity assumption, the contractions are similarities up to a change of basis, meaning that the self-affine measures are essentially self-similar. Of course, it would be interesting to find an analogous result for resonance between planar self-similar measures.
The assumption of irreducibility is our way to ensure that the measures µ and ν are "spread out" in sufficiently many directions, which is something one has to assume as explained in the preceding discussion. Without this assumption, it is easy to construct examples for which the conclusion of the theorem does not hold, by e.g. constructing measures on Bedford-McMullen carpets. However, we do not know if it is enough to assume irreducibility for just one of the systems Φ and Ψ.
If it happens that dim µ ≥ dim ν ≥ 1 or 1 ≥ dim µ ≥ dim ν, then the strong separation, hyperbolicity and irreducibility are enough to ensure that µ * ν has the maximal dimension. The case dim µ > 1 > dim ν is more delicate, since now µ might be large enough to "absorb" some of ν in the convolution, and some kind of independence of their local structures is required as in the analogous one-dimensional results. In establishing this independence, we require the domination condition since it gives us a way to connect the eigenvalues of A i and B j to the dynamics of the scenery processes of µ and ν around typical points. However, it is likely that the assumption is just a by-product of our argument.
Finally, we remark that Theorem 1.1 combined with the variational principle of Bárány, Käenmäki and Rossi [5,Proposition 2.4] yields an analogous result for sets: be systems of affine contractions on R 2 that satisfy the strong separation condition, domination, and irreducibility. Let X and Y denote the attractors of Φ and Ψ, and is an arithmetic set.
1.1. On the proof of Theorem 1.1. For simplicity, we suppose that the contractions in Φ and Ψ map the unit ball into disjoint ellipses. While stronger than the separation condition that we assume, the same argument works for the classical strong separation condition up to minor technical additions.
We will prove the statement by contradiction: We show that if any of the condi- is not an arithmetic set holds, then we will have dim(µ * ν) = min{2, dim µ + dim ν}. Our proof is based on the local entropy averages of Hochman-Shmerkin [19]: instead of proving directly that dim(µ * ν) is large, we will show that the entropy of µ ′ * ν ′ is large on many scales, where µ ′ and ν ′ are magnifications of µ and ν along properly chosen filtrations of their supports.
The machinery of Hochman-Shmerkin [19] asserts that dim(µ * ν) is at least the average of finite-scale entropies of magnifications of µ and ν. With this machinery, the cases i) and ii) above are very simple in principle. Indeed, magnifying both µ and ν along the "cylinder ellipses" ϕ i 1 • · · · • ϕ in (B(0, 1)) and ψ j 1 • · · · • ψ jn (B(0, 1)), on the limit they both resemble orthogonal projections of the original measures, by the hyperbolicity assumption. As such, they will have entropy close to either 1 in the case i), or dim µ and dim ν in the case ii), by the strong projection theorem for selfaffine measures [2, Theorem 7.1], due to Bárány, Hochman and Rapaport. Since the ellipses on which the magnifications are supported on have major axes pointing in different directions by the irreducibility assumption, their convolution has a productlike structure on the plane and thus has entropy close to min{2, dim µ + dim ν}.
The situation dim µ > 1 > dim ν is more delicate, since the magnifications of µ along the cylinder ellipses are no longer enough to store a sufficient amount of the dimension of µ. Instead, we will apply the local entropy averages with µ magnified along dyadic squares and ν along the cylinder ellipses.
For any such magnifications µ ′ and ν ′ of µ and ν, and any orthogonal projection π, applying the chain rule of entropy yields that the entropy of µ ′ * ν ′ is equal to The key geometric ingredient in our proof is the observation that each µ ′ has a fiber structure in the sense that πµ ′ is (close to) a slice measure of µ, for a properly chosen π. The precise form of this structure is stated in Proposition 4.1. Similar fiber structures have been previously observed for self-affine sets by Käenmäki, Koivusalo and Rossi [21], for self-affine measures on Bedford-McMullen carpets by Ferguson, Fraser and Sahlsten [13] and for self-affine measures with an additional projection condition by Kempton [22]. Combining this with the dimension conservation phenomenon that follows for planar self-affine measures from the Ledrappier-Young formula of Bárány [1] and the general fact that the average entropy of µ ′ over many scales is close to the dimension of µ, we see that upon averaging, (1.5) completing the proof by the local entropy averages and another application of the dimension conservation of [1] and the projection theorem of [2]. Proving (1.7) is the part where the assumption iii) on the eigenvalues of A i and B j steps in. We need to inspect the dynamics of the sequences (πµ ′ ) and (πν ′ ) obtained by continuously magnifying µ and ν, and show that they are independent of each other in a sense. Since πµ ′ is (close to) a slice of µ and πν ′ is an orthogonal projection of ν, they both have "dynamically self-similar" structure, which allows us to study the independence of (πµ ′ ) and (πν ′ ) via mixing properties of certain underlying dynamical systems. Although this part bears some resemblance to the works on convolutions of self-similar measures on the line, new ideas are required due to the underlying dynamics being more complicated.
1.2. Structure. In Section 2 we introduce our setting more rigorously, and collect some general known results and short lemmas on self-affine measures, dynamical systems and linear algebra. Section 3 is devoted to translating the local entropy averages machinery of [19] to our setting, while in Section 4 we state our main geometric and dynamical results, and explain how the proof of Theorem 1.1 is concluded using these. Section 5 is perhaps the most technical one, devoted to the proof of the main geometric result, Proposition 4.1. The arguments here were inspired by the work of Kempton [22]. Finally, in Section 6 we investigate the dynamics of the sequences of magnifications of µ and ν, and prove the required lower bounds for their average entropies. Orthogonal projection onto the line θ π i Projection onto the ith coordinate µ i,θ A "slice measure" of µ; see (4.1) S t , T x Scaling by 2 t and translation by −x, respectively R θ A rotation taking θ onto the y-axis The singular value decomposition The largest ellipse contained in Q i,θ,k θ(A) The direction of the longer axis of A(B(0, 1)) θ(i), θ − (i), θ * (i) Limit orientations; see Lemma 2.1 A Operator norm of A A| θ The restriction of A onto the line θ The singular values of A An increasing sequence; see (5.1) Magnifications of µ and ν; see Notation 3.1 ρ(n, (i, θ)) The reflection done by A −1 i|n on the line θ Coding of π 2 ν j| i k via Z Ψ ; see (6.7) F ′ , G ′ Functions F and G without reflection

Preliminaries
In this paper, a measure refers to a Radon measure on a metrizable topological space. The notation P(X) stands for probability measures on the space X. For a measure µ on X and a subset Y ⊆ X, µ| Y denotes the restriction of µ onto Y , µ Y := µ(Y ) −1 µ| Y the normalized restriction when µ(Y ) > 0. For a measurable function f , let f µ := µ • f −1 denote the push-forward. The space of probability measures is always equipped with the weak- * topology which we metrize using the Lévy-Prokhorov metric d LP , where A ε denotes the open ε-neighbourhood of A. We measure the size of measures with the lower Hausdorff dimension, and occasionally with upper and lower local dimensions, where B(x, r) denotes the closed ball centered at x and of radius r. The measure µ is called exact dimensional if dim loc µ(x) = dim loc µ(x) almost everywhere. The following connection between Hausdorff and lower local dimension is well-known: For any measure µ, dim µ = ess inf x∼µ dim loc µ(x).
2.1. Symbolic dynamics. Let Γ be a finite set with #Γ ≥ 2, and let Φ = {ϕ i } i∈Γ be an iterated function system of contractions on R d . It is well-known that there exists a unique compact set K, called the attractor of Φ, such that We refer to Falconer's book [11] for standard properties of iterated function systems. If the functions ϕ i are of the form linear maps with A i < 1 and a i ∈ R d , then the IFS Φ is called self-affine and its attractor a self-affine set. Write Γ * = n Γ n for the set of finite words composed of elements of Γ. For a finite word a = i 0 i 1 . . . i n ∈ Γ * we write ϕ a = ϕ i 0 • · · · • ϕ in . Let |a| denote the number of elements in a. For finite words a and b, let ab ∈ Γ |a|+|b| denote their concatenation.
For a word i ∈ Γ N ∪ Γ * and and integer k ≤ |i|, let i| k ∈ Γ k denote its projection to the first k coordinates. When k = 0, set i| k = ∅. For i, j ∈ Γ N , let i ∧ j := i| k , where k is the largest integer for which i| k = j| k . Define a distance d on Γ N by d(i, j) = 2 −|i∧j| for every i, j ∈ Γ N . For a finite word a ∈ Γ * , write [a] for the cylinder set {i ∈ Γ N : i| |a| = a}. It is not difficult to see that the cylinder sets are closed and open in the topology generated by d.
It is sometimes convenient to consider the two-sided sequence space Γ Z . For i = . . . i −2 i −1 ; i 0 i 1 i 2 . . . ∈ Γ Z and m ≤ n ∈ Z, write i| n m = i m i m+1 . . . i n . The metric d extends to Γ Z by replacing i| k by i| k −k in the definition of i ∧ j. The cylinder sets of Γ Z are given by [i] n m := {j ∈ Γ Z : j| n m = i| n m }. There is a natural surjection Γ Z → Γ N given by the restriction to the "positive coordinates", . . . i −1 ; i 0 i 1 . . . =: i → i + := i 0 i 1 . . .. Similarly, we define the projection to the "negative coordinates" by i − : We let σ denote the left-shift on both Γ N and Γ Z , given by σ(i 0 i 1 . . .) = i 1 i 2 . . . and σ(i −1 ; i 0 i 1 . . .) = . . . i 0 ; i 1 i 2 . . .. The tuples (Γ N , σ) and (Γ Z , σ) are referred to as the one-sided and two-sided shift spaces, respectively. For any σ-invariant probability measure ν on Γ Z , there is a unique σ-invariant probability measure on Γ Z which we also denote by ν, given by ν( This is referred to as the natural extension of ν.
We define a distance d on RP 1 , given by the smaller angle between lines. The following lemma is, for us, the key technical consequence of the domination assumption in Theorem 1.1.
exist for every i ∈ Γ N , the convergences are uniform, and the functions i → θ(i), i → θ − (i) and i → θ * (i) are Hölder continuous.
In particular, θ(Γ N ) ⊆ C, where C is the strongly invariant multicone of {A i } i∈Γ .
Proof. Let A i|n = U i|n D i|n V −1 i|n denote the singular value decomposition. By Lemma 2.1, U i|n takes the x-axis onto a line which tends to θ(i) as n → ∞. In particular, U −1 i|n θ remains uniformly bounded away from the x-axis. By [6, Theorem B], i|n θ is pulled very close to the y-axis by D −1 i|n and finally taken close to θ(A −1 i|n ) by V −1 i|n . For the second statement, note that A i|n C ⊆ C and d(A i|n C, θ(A i|n )) → 0 as n → ∞ by the above, since C contains more than one point. Since C is closed, the claim follows.
We require the following technical observation, that the directions θ(i) ⊥ and θ * (i) are bounded away from each other, uniformly in i.
By the assumption, {A i } i∈Γ has a strongly invariant multicone C. By Lemma 2.2, θ(i) ∈ int(C) for every i ∈ Γ N . On the other hand, In particular, θ(i) = θ − (i) and the result follows.
For a matrix A, let v i (A) denote the eigenspace associated to λ i (A) for i = 1, 2. When A is hyperbolic, v i (A) ∈ RP 1 for i = 1, 2. The following observation follows immediately from the eigenvalue decomposition.
2.3. The Furstenberg measures. For any probability vector (p i ) i∈Γ and Bernoulli measureμ := p N on Γ N , there are associated measures µ F and µ * F on RP 1 , called the Furstenberg measures, which are given by forμ-almost every i. The measures µ F and µ * F are supported on the sets θ − (Γ N ) and θ * (Γ N ) = θ(Γ N ) ⊥ , respectively. The product measuresμ × µ F andμ × µ * F are invariant and ergodic under the maps , respectively. When Φ satisfies the domination condition, the measuresμ × µ F and µ × µ * F are images of the Bernoulli measure p Z on Γ Z through the factor maps i → (i + , θ − (i − )) and i → (i + , θ * (i − )), which easily implies the existence, invariance and ergodicity ofμ × µ F andμ × µ * F . These measures do also exist without the domination assumption, with the prescribed properties, which is a classical result of Furstenberg. When Φ is irreducible, it is known that the measures µ F and µ * F are non-atomic.
2.4. Self-affine measures. Recall that K denotes the attractor of the system of affine contractions Φ. Let Π : which we call the natural projection. Fix a probability vector p = (p i ) i∈Γ and letμ = p N denote the Bernoulli measure on Γ N with marginal p. The measure µ := Πμ =μ • Π −1 is called the self-affine measure associated to p, and is wellknown to be a Radon measure supported on K that satisfies Throughout the paper, we will use dim to denote the Hausdorff dimension of both sets and measures. The following strong projection theorem for self-affine measures is due to by Bárány, Hochman and Rapaport [2]. For θ ∈ RP 1 , let π θ : R 2 → θ denote the orthogonal projection onto the line θ.
We remark that the result was proved earlier for non-hyperbolic IFSs by Hochman and Shmerkin in [19].
It follows from the Ledrappier-Young formula due to Bárány [3] that planar selfaffine measures satisfy dimension conservation in directions typical for the Furstenberg measure µ F . The Ledrappier-Young formula was shortly after generalized to higher dimensions by Bárány and Käenmäki [3]. The application to dimension conservation was recently generalized by Feng [12] to higher dimensions and for more general self-affine measures. Let µ = µ θ x dπ θ µ(x) denote the disintegration of µ with respect to the orthogonal projection π θ . Theorem 2.6 (Corollary of [1], Theorem 2.7). Let µ be a self-affine measure associated to an affine system of contractions Φ with the strong separation and domination conditions, and µ F the associated Furstenberg measure. Then for µ F -a.e. θ and π θ µa.e. x, the measure µ θ x is exact dimensional and dim µ = dim π θ ⊥ µ + dim µ θ x . 2.5. Flows and eigenvalues. Let I be either R or [0, +∞), and let {T s } s∈I be a family of measurable functions on a measure space (X, µ) with the property that T s • T t = T s+t for each s, t ∈ I. Recall that a real number c ∈ R is an eigenvalue of the flow (X, T s , µ) if there exists a measurable function f : X → C such that f •T s = e(cs)f almost everywhere, for every s ∈ I, where we write e(x) := exp(2πix). If the flow is measure-preserving, i.e. T s µ = µ for every s, then it is not difficult to see that any eigenfunction f satisfies |f | ≡ 1. We now record some standard properties of eigenvalues.
Lemma 2.7 (Special case of Lemma 3.11, [16]). Let (X, T s , µ) be an ergodic measurepreserving flow, and let c ∈ R. Then the discrete-time dynamical system (X, T c , µ) is ergodic if and only if c −1 is not an eigenvalue of (X, T s , µ).
Lemma 2.8. A measure-preserving flow (X, T s , µ) on a standard Borel space X has at most countably many eigenvalues.
Proof. It is not difficult to see that eigenfunctions for different eigenvalues are orthogonal. Since the space L 2 (X) is separable, any collection of orthogonal functions has to be countable.
The proof of the following result for the product of discrete-time dynamical systems can be found in many textbooks. Most proofs utilize the spectral theorem, and applying the spectral theorem for one-parameter families of unitary operators, the same proofs go through for flows as well. Let (Γ N , σ) be a shift space. Given a function f : Γ N → (0, +∞), we may build a flow from (Γ N , σ) by "flowing up" from a point i ∈ Γ N until we reach the time f (i), then switch to the point σi and continue flowing until the time f (σi), and so on. Formally, we let Z = (Γ N × R)/ ∼ eqipped with the quotinent topology, where ∼ denotes the equivalence relation generated by A special property of regular enough suspension flows is that to any eigenvalue corresponds a continuous eigenfunction. Proposition 2.10 ( [24], Proposition 6.2). Let (Z, T s , λ) be the suspension of a shift space (Γ N , σ, µ) under a locally Hölder continuous roof function, where µ is the equilibrium state for a locally Hölder continuous potential on Γ N . Then a number α ∈ R is an eigenvalue of (Z, T s , λ) for a continuous eigenfunction if and only if it is an eigenvalue for a measurable eigenfunction.
It is well-known that Bernoulli measures on Γ N are equilibrium states for locally constant potentials.
The flows Z and Z + have the same eigenvalues.
, it is not difficult to see that the eigenvalues of Z + are also eigenvalues of Z.
For the other direction, suppose that α is not an eigenvalue of Z + . Then by Lemma 2.7, the discrete and an interval I ⊆ R. Since any continuous function on Z can be approximated arbitrarily well (in L 1 ) by simple functions on this kind of sets, in order to show that λ is ergodic under T 1/α , it suffices to show that for a.e. (i, t).
Let A + be the set of full λ + -measure such that for each (j, t) ∈ A + , we have by Birkhoff's ergodic theorem. The second-to-last equality follows from the choice of ℓ, and the last follows from T t -invariance of λ. Now, if A is the embedding of A + to Z, then T ℓ/α A has full λ-measure, and for each (i, t) ∈ T ℓ/α A, Therefore λ is ergodic under T 1/α and by Lemma 2.7, α is not an eigenvalue of Z.
2.6. Shannon entropy. For a probability measure µ on R d and any measurable for the conditional entropy of µ with respect to E, given F . Let D n = D n (R d ) denote the partition of R d into dyadic cubes of side-length 2 −n which we call the "level-n" dyadic partition. When D ∈ D n and µ is a measure with µ(D) > 0, let for the element of D n that contains x. For entropy with respect to the dyadic partition we use the short-hand notation H n (µ) = H(µ, D n ) = − D∈Dn µ(D) log µ(D). In the following, we record some elementary properties of entropy.
Lemma 2.12. Let µ be a probability meausure on R d , and let E and F be partitions of R d such that each element of E intersects at most k elements of F and vice versa.
Lemma 2.13 (Chain rule). Let µ be a probability measure on R 2 , and let E, F be partitions of R 2 such that E refines F . Then Let D n (θ) denote the level-n dyadic partition of θ ∈ RP 1 . The following is a simple application of the chain rule; let R θ denote the "shortest" rotation which takes θ ∈ RP 1 onto the y-axis, with R x-axis given by the clockwise rotation.
Lemma 2.14. Let µ be a probability measure on R 2 , and let θ ∈ RP 1 . Then, Lemma 2.15 (Concavity and almost-convexity). For any probability measures µ 1 , . . . , µ k and a probability vector (p 1 , . . . , p k ), Taking a convolution can decrease entropy at most by an additive constant that depends only on the dimension of the ambient space.
Proof. The claim follows by approximating ν in weak- * sense with a convex combination of Dirac measures, and then applying Lemmas 2.15 and 2.12.
Lemma 2.17. Let µ and ν be probability measures on [0, 1], and suppose that µ is non-atomic. Then for every r > 0 and ε > 0, there exists N 0 ∈ N such that for any interval I with µ(I) ≥ r, we have is continuous in (a, b) ∈ K r (in the subspace topology) and the continuity is uniform in N.
Let r, ε > 0 be given, and let δ > 0 be small with respect to ε and r. Fix (a 0 , b 0 ) ∈ K r and let I δ be the largest interval contained in , applying bilinearity of convolution and Lemma 2.15 in the first and second-to-last inequalities, we have by Fatou's lemma. By uniform continuity, this convergence is uniform in K r , which is what we wanted to prove.

2.7.
Magnifying measures. It turns out that the magnifications of a strictly selfaffine measure along a conformal partition have a very special structure. This structure makes it much easier to analyse the convolutions of magnifications instead of the convolution of self-affine measures directly, which in turn is put to use in the proof of Theorem 1.1 through the local entropy averages. In the following, we recap the standard terminology of magnifying measures, and some statistical properties regarding sequences of magnifications. For x ∈ R d and r ≥ 0, we let T x : y → y − x denote the translation taking x to the origin, and S r the exponential "magnification" operation on measures, given for a measure µ whose support contains the origin by for every Borel A ⊆ B(0, 1). The following property of S r -ergodic measures appears also in [20].
There is a natural way in which measures on R d give rise to measures on P(R d ). Namely, consider the sequence (S r T x µ) r≥0 , called the scenery of µ at x. The statistical properties of this sequence are described by the accumulation points of the sequence 1 t t 0 δ SrTxµ dr t≥1 , called the scenery flow of µ at x. The accumulation points of the scenery flow in the weak- * topology are measures on P(R d ), and are called tangent distributions of µ at x. The measure µ is called uniformly scaling if the scenery flow converges almost everywhere to a unique tangent distribution P . It is then said that µ generates P .
A remarkable result of Hochman [15] is that tangent distributions at almost every point are fractal distributions, objects which enjoy strong spatial invariance properties. We will give the definition here for completeness, although we will not use it directly.
Definition 2.19. An S r -invariant measure P on P(R d ) is called a fractal distribution if for any measurable A, P (A) = 1 if and only if for every r > 0, P -almost every ν satisfies S r T x ν ∈ A for ν-almost every x with B(x, e −r ) ⊆ B(0, 1).
Theorem 2.20 (Theorem 1.7, [15]). Let µ be a Radon measure on R d . Then for µ-almost every x, every tangent distribution at x is a fractal distribution.
Lemma 2.21. Let P be a fractal distribution. Then for P -a.e. ν, any line L with ν(L) > 0 must contain the origin.
Proof. We first show that P -a.e. atomic measure is the point mass at the origin. This is almost immediate from the results of [15].
For a contradiction, let A = {ν : ν({x}) > 0, x = 0} and suppose that P (A) > 0. Let P ′ be an ergodic component of P with P ′ (A) > 0, and let η ∈ A be a uniformly scaling measure generating P ′ . Indeed, it is not difficult to verify from the definition of a fractal distribution that P ′ -almost every measure is uniformly scaling. Let x be such that η({x}) > 0. Since η| {x} ≪ η, also η| {x} generates P ′ by an application of the Lebesgue-Besicovitch differentiation theorem, whence P ′ is supported on the point mass at the origin. In particular, P ′ (A) = 0, a contradiction. Now, to prove the statement of the lemma, suppose that there exists a set B with P (B) > 0 such that for every ν ∈ B, there exists a line L ν with ν(L ν ) > 0 and 0 ∈ L ν . Let P ′ be an ergodic component of P with P ′ (B) > 0, and let η ∈ B be a uniformly scaling measure generating P ′ . Now, since η(L η ) > 0, also the measure η| Lη generates P ′ . Let L denote the line L η translated so that it contains the origin. Clearly, all tangent measures of η| Lη are supported on L, whence P ′ is supported on measures which are supported on L. Since any line not containing the origin intersects L in at most one point, such a line has P ′ -almost surely zero measure by the above. Thus P ′ (B) = 0 which is a contradiction.
We can also choose to magnify µ along a discrete sequence of scales. For N ∈ N, we call the sequence 1 nN n k=1 δ S kN Txµ n∈N the N-scenery process of µ at x. Similarly, weak- * accumulation points of this sequence we call N-tangent distributions. While these are a-priori different objects from the accumulation points of the continuoustime scenery flow, most properties that are typical for tangent distributions are also typical for N-tangent distributions.
Lemma 2.22. Let µ be a Radon measure on R d . For µ-almost every x, any N ∈ R and any N-tangent distribution P of µ at x, the following holds: For P -almost every ν, any line L with ν(L) > 0 that intersects the interior of B(0, 1) must contain the origin.
Proof. It follows from [15,Proposition 5.5] that for µ-almost every x, if P is an N-tangent distribution at x, then the measure is a fractal distribution. Therefore by Lemma 2.21, for P -a.e. ν and L-almost every r ∈ [0, N], any line L with S r ν(L) > 0 must contain the origin. Taking r → 0 along a countable sequence, we see that for P -almost every ν, any line L with ν(L) > 0 that intersects the interior of B(0, 1) must contain the origin.

Conditional measures on lines.
For a measure µ on R 2 and θ ∈ RP 1 , let µ = µ θ x dπ θ µ(x) denote the disintegration of µ with respect to π θ . It is well-known that for almost every x ∈ θ, the measure µ θ x is supported on the line x + θ ⊥ , and that µ θ x is the limit of normalized restrictions of µ on thinner and thinner tubes centered at x + θ ⊥ . It will be useful for us to know that these tubes can replaced by preimages of sets of relatively large measure.
Lemma 2.23. Let µ be a measure on R 2 , let δ > 0 and θ ∈ RP 1 . For every r > 0, let Then for π θ µ-almost every x, we have Proof. Let ε > 0, and let E n ⊆ θ be the compact set given by Lusin's theorem with π θ µ(E n ) ≥ 1 − 1/n, on which the function y → µ θ y is continuous. Since π θ µ( n∈N E n ) = 1, it suffices to prove the statement for almost every x ∈ E n , for every n ∈ N. Now, for almost every x ∈ E n , if B := B(x, r) and I ∈ I(x, r, δ), by an application of the Lebesgue-Besicovitch differentiation theorem. Therefore, for any Borel set A ⊆ R 2 , I ∈ I(x, r, δ) and ε > 0, if r is small enough, by continuity of y → µ θ y . Similarly,

On local entropy averages
In this section, we recall the local entropy averages of [19] and introduce the different notions of magnifactions of measures that we use. Let µ and ν be selfaffine measures as in the statement of Theorem 1.1, and denote byμ,ν the associated Bernoulli measures.
For real numbers a and b, write a ≈ b to indicate that they are multiplicatively comparable. By Lemmas 2.1, 2.3 and basic geometry, B j|n ≈ |π θ(j) B j|n B(0, 1)| ≈ |π θ * (j) B j|n B(0, 1)| for all j ∈ Λ N and n ∈ N, where the constant of comparability depends only on the IFS Ψ. In order to simplify notation, suppose that 1/2 ≤ B j|n −1 |π θ * (j) B j|n B(0, 1)| ≤ 2 for every j and n.
For every i ∈ Γ N and j ∈ Λ N , we define the "stopping times" Although the stopping times on Γ N and Λ N are denoted by the same letter i k , the choice of domain will always be clear from the context: for example, i| i k := i| i k (i) and j| i k := j| i k (j) . Then for every k, We now define the different notions of scale-k magnifications of µ and ν.
Notation 3.1. For each i ∈ Γ N , j ∈ Λ N and k ∈ N, let Note that µ i| i k and ν j| i k are measures supported on a thin ellipses whose major axes have length comparable to 1 and are oriented in the directions θ(A i| i k ) and θ(B j| i k ), respectively. On the other hand, µ i,k is a measure supported on [−1, 1] 2 , the "dyadic magnification" of µ.
We require the following form of the local entropy averages of [19]: The proof is essentially in [19], but we provide a sketch for the proof of the first half of the statement for the convenience of the reader. The second half goes through similarly. We begin by recalling some terminology of [19]. In the following, let X and Y be finite sets, let 0 < ρ < 1, and let d be a metric on X N such that d(x, x ′ ) ≈ ρ |x∧x ′ | . Following [19], we say that i) a map g : X N → Y N is a tree morphism if, for every n ∈ N and length-n cylinder [x 1 . . . Proof of Theorem 3.2. Let N ∈ N. As in [19], the idea is to lift µ × ν to a measure η on the tree Σ := {1, . . . , 2 N } N × Λ N , and then find a tree morphism g N : Σ → Y N N and a faithful map h N : On Σ, let distance between pairs (k, j) and (k ′ , j ′ ) be defined as the number Then the topology on Σ is generated by sets of the form [k| n ]×[j| in ], k ∈ {1, . . . , 2 N } N , j ∈ Λ N . Write D 2 N (R 2 ) = {D 1 , . . . , D 2 N } and let F k denote the map which sends [−1, 1] onto the closure of D k . Let Π 1 denote the map {1, . . . , 2 N } → [−1, 1] 2 , k → lim n→∞ F k|n (0), and Π 2 the map Λ N → R 2 , j → lim k→∞ ψ j| k (0). Using the maps Π 1 and Π 2 , the measure µ × ν can be lifted to a measure η on Σ (by possibly translating the dyadic partition so that the boundaries of dyadic squares have µ-measure 0). Let Π 0 denote the map Σ → R 2 × R 2 , (k, j) → (Π 1 (k), Π 2 (j)) and * the map We will now construct a tree Y N N , a tree morphism g N : Σ → Y N N and a faithful map h N : Y N N → R 2 such that the following diagram commutes: Supposing that the sequence a 1 . . . a n has been determined, we choose a n+1 so that Π 1 [k| n+1 ] + Π 2 [j| i n+1 ] ⊆ Q a 1 ...ana n+1 . Since diam(Q a 1 ...an ) → 0, we may define g N (k, j) = a 1 a 2 . . .. It is evident from the construction that g N ([k| n ] × [j| in ]) ⊆ [a 1 a 2 . . . a n ], that is, sets of the form [k| n ] × [j| in ] are mapped into cylinders of Y N N by g N . While this property of g N is strictly speaking weaker than that of the tree morphism defined earlier, it is enough for our purposes. By construction, we have h N • g N = * • Π.
Since h N is faithful, [19,Proposition 5.3] asserts that it preserves entropy in the sense that for every k ∈ N. Since rescaling a measure does not change its entropy much if we change the scale of the entropy by the same amount, by Lemma 2.12, we have where i ∈ Γ N is such that Π(i) = Π 1 (k). Therefore, by the assumption and the local entropy averages for measures on trees [19,Theorem 4.4], we have dim g N η ≥ c. We remark here that in the hypothesis of [19,Theorem 4.4] it is assumed that g N is a tree morphism, but it is not difficult to verify that just having the property for sets of the form [k| n ] × [j| in ] is enough to apply the theorem in our setting. Finally, taking g N η through the faithful map h N yields µ * ν and distorts the dimension by at most O(1/N), by [19,Proposition 5.2], which completes the proof.

Proof of Theorem 1.1
In this section, we state our key technical results, and apply them to prove Theorem 1.1. We begin with some notation.
Recall that we defined RP 1 as the collection of one-dimensional subspaces of R 2 . Through the identification RP 1 ∼ = [0, π) we use θ ∈ [0, π) to denote both angles and lines (making that angle with the positive x-axis). Recall that R θ denotes the "shortest" rotation which takes θ onto 0 ⊥ (the y-axis), and for θ = 0, we choose the clockwise rotation. For almost every i ∈ Γ N and θ ∈ RP 1 , we let µ i,θ denote the probability measure supported on the y-axis, obtained by taking the conditional measure µ θ Π(i) from the disintegration µ = µ θ x dπ θ µ = µ θ Π(i) dμ supported on the line θ ⊥ + Π(i), translating it by T Π(i) and finally rotating it by R θ . That is, The measures µ i,θ are occasionally called slices of µ.
Fix a large integer N, and for i ∈ Γ N and k ∈ N, let ℓ k = ℓ k (i) be any increasing sequence such that lim k→∞ ℓ k = ∞ and for every k. Such a sequence exists for every i by the strong separation condition. Our main geometric observation is that for any self-affine measure µ with dim µ > 1, the measures µ i,k have a fiber structure in the sense that π θ(i) ⊥ µ i,k is very close to a slice of the original measure µ, in a direction typical to the Furstenberg measure µ F . This is true also when dim µ ≤ 1, but in this case the proof is slightly different. Throughout the paper, we adopt the convention that −µ denotes the push-forward of µ through the map x → −x. Recall that π 2 denotes the orthogonal projection onto the y-axis. Proposition 4.1 (Fiber structure). Let µ be as in Theorem 1.1 with dim µ > 1, and suppose that the domination condition is satisfied. For any ε > 0, the following holds after an affine change of coordinates: Forμ-almost every i ∈ Γ N and all θ in a set of positive µ F -measure, there exists m ∈ N, a sequence of intervals (I k ) k of length 2 −m and a set N ε ⊆ N with lim inf n→∞ The domination assumption can be removed from the proposition at the cost of having the change of coordinates depend on the word i ∈ Γ N ; Furthermore, we leave it for the interested reader to verify that if π 2 is replaced by π θ(i) ⊥ and one chooses to magnify along balls centered at Π(i) instead of dyadic squares, then no domination condition, change of coordinates nor restriction on the intervals (I k ) k is required. Such a modification is not required in the proof of Theorem 1.1, however. The proof of the proposition is given in Section 5.
Let us briefly compare Proposition 4.1 with the existing results on the scenery of self-affine measures. First, the result of Ferguson et. al. [13] deals with measures on Bedford-McMullen carpets. In this case, the Furstenberg measure µ F equals a point mass, and the defining matrices involve no reflections, whence the cocycle ρ above may be replaced by the identity and the map M is just the left shift on the first argument. Using the carpet structure, the authors also describe the fibers of µ i,k with respect to π 2 , which seems to be difficult in our setting.
The result of Kempton [22], on the other hand, deals with self-affine measures under the irreducibility and domination conditions. In addition, the methods of [22] assume that the orthogonal projection of µ in µ F -almost every direction is absolutely continuous, and that the defining affine contractions preserve orientation. The projection condition allows Kempton to deduce that each fiber of µ i,k with respect to π 2 is just the Lebesgue measure, while assuming the defining contractions to be orientation-preserving allows the cocycle ρ to be replaced with the identity. Finally, Proposition 4.1 considers magnifications of µ along the dyadic squares instead of balls centered at Π(i), which brings in some additional technical considerations regarding the distribution of µ near the boundaries of dyadic squares.

Estimating the local entropy averages. Recall from Section 3 that we have to find a lower bound for either
, for most k. Bounding the second quantity is easier, but the bound we obtain is only efficient when dim µ ≥ dim ν ≥ 1 or 1 ≥ dim µ ≥ dim ν: The proof is given in Section 6. Applying Theorem 3.2, we get that dim(µ * ν) ≥ min{1, dim µ} + min{1, dim ν}.
From this it readily follows that if dim(µ * ν) < min{2, dim µ + dim ν}, then dim µ > 1 > dim ν. This was one of the statements of Theorem 1.1. From now on, we suppose that dim µ > 1 > dim ν, and that Φ and Ψ satisfy the domination condition.
In this case, bounding the quantity 1 N H N (µ i,k * ν j| i k ) is more efficient. Applying Lemmas 2.13 and 2.16 we see that Here, the average of the terms 1 N H N (µ i,k ) over k is close to dim µ. Thus, Theorem 1.1 follows from the following inequalities together with the local entropy averages: These claims are also proved in Section 6. We show how to conclude the proof of Theorem 1.1 from this.
Proof of Theorem 1.1. Let µ and ν be as in the hypothesis, with dim µ > 1 > dim ν. For a contradiction, suppose that is not an arithmetic set. It is not difficult to see that there must then exist a pair forμ-almost every i ∈ Γ N andν-almost every j ∈ Λ N . It now follows from Theorem 3.2 that dim(µ * ν) ≥ min{2, dim µ + dim ν} − 2ε, which is a contradiction if ε is small enough.

The scenery of the self-affine measure
In this section, our aim is to prove Proposition 4.1, whence we assume throughout the section that µ is a self-affine measure as in Theorem 1.1 with dim µ > 1. The arguments of this section were inspired by the work of Kempton [22] on a similar result for a more special class of self-affine measures, and some of our lemmas are analogous to those in [22].

5.1.
Restrictions of µ on thin rectangles. The strong separation condition asserts the existence of a bounded open set V = ∅ whose closure is mapped into disjoint subsets of V by the maps ϕ i . For the sake of notational simplicity, we suppose that V = B(0, 1); since all the statements in the following are local in nature, everything works also for a general V by restricting onto small balls centered in the point of interest.
Let i ∈ Γ N , let B be a small ball centered at Π(i) and let n be the largest integer so that ϕ i|n (B(0, 1)) ⊇ B. By the strong separation condition, we have µ B = ϕ i|n ϕ −1 i|n µ B = ϕ i|n µ ϕ −1 i|n B . Therefore, in order for us to understand the magnifications µ B , we have to understand the measures µ ϕ −1 i|n B , the restrictions of µ on the thinner and thinner ellipses ϕ −1 i|n B whose major axes have length comparable to 1. It is intuitive that these measures should approximate the slice measures of µ in different directions, as n → ∞. But since slices can be defined as limits of the measures supported on thinner and thinner tubes, it is better to work with rectangles instead of ellipses for a moment.
For i ∈ Γ N , θ ∈ RP 1 , r 2 ≥ r 1 ≥ 0, write Y i,θ,r 1 ,r 2 for the rectangle centered at Π(i) with sidelengths 2 −r 1 ≥ 2 −r 2 and the longer side oriented in the direction θ. For a rectangle Y with center at x and major side oriented in the direction θ, write H Y for the map which translates x to the origin and stretches T x Y onto R −1 θ [−1, 1] 2 . It follows from Lemma 2.23 and standard geometric measure theory that the measures H Y i,θ,r 1 ,r 2 µ Y i,θ,r 1 ,r 2 have a fiber structure in the following sense.
Lemma 5.1. Forμ × µ F -a.e. (i, θ) ∈ Γ N × RP 1 and any δ, ε, c, r 1 > 0, there exists t > 0 such that the following holds: If (Q r 2 ) r 2 ≥0 is a sequence of squares that contain the origin with one side parallel to θ, |Q r 2 | ≥ c > 0 and For the proof, we record the following elementary observation.
We will next relate the measures µ i,k to the measures H Y i,θ,r 1 ,r 2 µ Y i,θ,r 1 ,r 2 . To make statements more economic, we introduce some additional notation.

Magnifications of µ.
For a ∈ Γ * , let A a = U a D a V −1 a denote the singular value decomposition, where U a , V a are orthogonal and D a = diag(α 1 (A a ), α 2 (A a )). Fix a large integer N, and for i ∈ Γ N and k ∈ N, let ℓ k = ℓ k (i) be any increasing sequence such that lim k→∞ ℓ k = ∞ and for every k. Such a sequence exists for every i by the strong separation condition. Throughout the following, we will use the short-hand notation Q i,θ,k := Y σ ℓ k i,θ,kN +log α 1 (i| ℓ k ),kN +log α 2 (i| ℓ k ) .
Proposition 5.3. For every (i, θ) ∈ Γ N × RP 1 , there exists an integer m ≥ 0 and a sequence of non-singular linear maps (L i,θ,k ) k∈N such that for all large enough k.
An important point of the proposition is that the direction θ can be chosen arbitrarily. In proving this we require the following geometric lemma, analogous to [22,Lemma 8.2]. Write E i,θ,k ⊆ Q i,θ,k for the largest ellipse contained in Q i,θ,k . Lemma 5.4 (Figure 2). For µ × µ F -almost every (i, θ), there exists 0 < C < 1 such that for all large enough k. Proof. For any θ, i ∈ Γ N and n ∈ N, we have See Figure 3.
i|n . If the side of length 1 is parallel to θ(A i|n ) ⊥ (which is the major expanding direction of A −1 i|n ), then its image through A −1 i|n is of length α 1 (i| n ) −1 .
Proof of Proposition 5.3. Let (i, θ) ∈ Γ N × RP 1 , let k be large, and let ℓ k be defined as in (5.1). Note that with the above notation, By the strong separation condition, with this notation we may write by switching the order of scaling and rotation in the last equality.
Let 0 < C < 1 be the constant given by Lemma 5.4 so that Figure 4.
we have obtained the representation This proves the statement with m := ⌈− log C⌉.
tion, then all that would be left to obtain the fiber structure of µ i,k+m were to apply Lemma 5.1. However, it turns out that this is not necessarily the case, and U i| ℓ k V −1 i| ℓ k L i,θ,k instead takes squares onto parallelograms in a way which depends on i and θ.
To overcome the difficulties brought by this additional distortion, we begin by breaking U i|n V −1 i|n into three components: The reflection, and two rotations which do not reflect.
For a linear map A : R 2 → R 2 and θ ∈ RP 1 , let A| θ : θ → R 2 denote the map obtained by restricting A onto θ. Let O ⊂ Γ N × RP 1 be the open set of those (i, θ) for which π 2 (x) | π 2 • A −1 i 0 (x) < 0 for some (equivalently every) x ∈ θ. Define the function ρ : N × Γ × RP 1 → {−1, 1}, This map captures the reflections done by A −1 i|n on the line θ, or by A i|n on the line A −1 i|n θ. Indeed, for any x ∈ θ, we may decompose A −1 i|n as (5.4) In other words, first rotate x to the y-axis, apply the possible reflections, rotate the y-axis onto the line A −1 i|n θ, and finally scale. The map ρ is easily seen to satisfy the cocycle equation ρ(n + k, (i, θ)) = ρ(n, M k (i, θ))ρ(k, (i, θ)).
Lemma 5.5. Let i ∈ Γ N and let U i|n D i|n V −1 i|n be the singular value decomposition of A i|n . For any θ ∈ RP 1 \ {θ(i)}, we have Proof. Let B i|n := U i|n D −1 i|n U −1 i|n and note that In particular, U i|n V −1 i|n = B i|n A i|n . Now, by (5.4), On the other hand, as n → ∞, for some sequence (k n ) n ∈ N, by Lemma 2.4. Since B i|n is positive definite and θ = θ(i), the sequence (−1) kn has to be eventually constant. By absorbing the eventual value of this sequence to the definition of ρ(n, (i, θ)), we may without loss of generality suppose that It remains to study the behaviour of linear map L i,θ,k as k → ∞, in the statement of Proposition 5.3. The content of the following lemma is that 2 onto a parallelogram of bounded eccentricity and one side in direction θ(i).
Lemma 5.6. For every i ∈ Γ N and θ ∈ RP 1 \ {θ(i)}, . As the map L i,θ,k acts on the set Figure 5), and then stretched onto a parallelogram by H Q

5.4.
Fiber structure for µ i,k . We will now explain how to combine Proposition 5.3 and Lemmas 5.1 and 5.6 to prove Proposition 4.1 and to obtain the fiber structure of µ i,k . Recall that the problem is the additional distortion brought by U i| ℓ k V −1 i| ℓ k L i,θ,k ; see Figure 6. However, because of our freedom in choosing the direction θ, this issue can be resolved in the following way. Essentially, we perform a change of coordinates so that µ F gives positive mass to a small neighbourhood of the y-axis, and all of the directions θ(i) are pulled very close to the x-axis. As a result of this change, the parallelogram in Figure 6 is taken within Hausdorff distance ε of the unit square.
Let ε 1 > 0 be a small number. We will later see how small it has to be. Let A be a linear map such that 0 ⊥ ∈ A(spt µ F ) and d(Aθ(i), 0) < ε 1 for all i ∈ Γ N . Choosing such an A is possible since as was seen in the proof of Lemma 2.3, the domination condition ensures that spt µ F = θ − (Γ N ) = θ(Γ N ). This map will be the change of coordinates in the statement of Proposition 4.1.
Upon changing coordinates by A we replace the IFS Φ by the conjugate IFS {Aϕ i A −1 } i∈Γ , µ by the self-affine measure Aµ, and the Furstenberg measure µ F by the measure Aµ F (which is the Furstenberg measure induced by Aµ). Afterwards, omitting the map A from the notation, we have Let F k θ,i be as in Lemma 5.6. It follows from (5.5) and (5.6) that for every i ∈ Γ N and for θ in a set of positive µ F -measure, the map F k θ,i is within distance ε 1 of a k 0 0 1 , and thus the map i| ℓ k θ for large k, by Lemma 5.6; see Figure 7. Figure 7. After our change of coordinates, the parallelogram θ,k is close to the unit square.
We are now ready to prove Proposition 4.1.
Proof of Proposition 4.1. Let ε > 0, let ε 1 > 0 be small with respect to ε and and apply the change of coordinates A that was described above. Let m be the constant given by Proposition 5.3. Our first goal is to ensure that the origin is often close to the center of the square T 2 kN Π(i) mod 1 D m (2 kN Π(i) mod 1), so that the square has relatively large mass which is required to apply Lemma 5.1. In this proof, write i k := 2 k Π(i) mod 1 and note that For any δ > 0 We begin by showing that this inclusion holds for most k, for small enough δ. By Fubini's theorem, by applying a random translation to µ (which does not affect the dimension of µ * ν), we may suppose that forμ-almost every i, the sequence (i kN +m ) k∈N equidistributes for the Lebesgue measure on [−1, 1] 2 . In particular, for a small enough δ 0 > 0 there exists N 1 ε ⊆ N with lim inf n→∞ On the other hand, by [15,Proposition 1.19], forμ-a.e. i, so there exists a δ > 0 and a set N 2 ε ⊆ N such that lim inf n→∞ The remaining task is now to ensure that the measure U i| ℓ k V −1 does not give too large mass around the boundaries of squares, which was another requirement of Lemma 5.1. For s, t > 0 define the set A s,t = {ν ∈ P(R 2 ) : there exists a line ℓ with d(ℓ, 0) > δ 0 and ν(ℓ s ) ≥ t}.
By Lemma 2.22, forμ-almost every i, there exists ε ′ > 0 such that Indeed, if this was not the case, we could find a sequence of N-tangent distributions (P n ) n∈N such that P n (A 1/n,ε 1 ) ≥ ε/4 by the fact that A 1/n,ε 1 is closed. In particular, any accumulation point P of this sequence (which is again an N-tangent distribution) would have . Let X be the set given by Lemma 5.1 (applied with c = δ 0 /2), such thatμ × µ F (X) > 1 − ε 1 and for all large enough r 2 and every (i, θ) ∈ X, we have Finally, for every (i, θ) we may let N 4 ε ⊆ N be the set of those k for which M ℓ k (i, θ) ∈ X. Forμ×µ F -almost every (i, θ), we have lim inf n→∞ #{0≤k≤n: M k (i,θ)∈X} n ≥ 1−ε 1 by Birkhoff's ergodic theorem, and since it is not difficult to verify from the definition of ℓ k (in (5.1)) that ℓ k ≤ O N (k), we have lim inf n→∞ Let now i ∈ Γ N be chosen from the set of fullμ-measure and θ ∈ RP 1 from a set of positive µ F -measure such that for (i, θ), all of the sets N j ε as above exist, and let By Proposition 5.3 and Lemma 5.6, for every large enough k. Also, for every k ∈ N ε we have S kN T Π(i) µ(T i kN D m (i kN )) ≥ δ by (5.7) and (5.8), and by the properties (5.5) and (5.6) of our change of coordinates, we have by Lemma 5.2, by the definition of N 4 ε and (5.9) that for every k ∈ N ε , when I k := π 2 T i kN D m (i kN ).
We will first show that the semiflow Z ′′ Φ × Z ′′ Ψ is ergodic, and then use this to prove the ergodicity of Z ′ Φ × Z ′ Ψ . Borrowing an idea of Bowen [7], let r : Γ Z → Γ Z denote the function which replaces all the negative coordinates of . . . i −1 ; i 0 i 1 . . . by i 0 . Then f ′ is cohomologous to h := f ′ • r, that is, where u is the continuous function defined by ). The sum converges since f ′ is Hölder. In particular, the flow Z ′′ Φ is conjugate to the suspension of Γ Z over h, denoted by , and the advantage of this is that the function h depends only on the positive coordinates of Γ Z . Proof. Since Z ′′ Φ is conjugate to Z h Φ and conjugate flows have the same eigenvalues, it suffices to prove the statement for Z h Φ . Moreover, by Lemma 2.11, we may replace Z h Φ by (Z h Φ ) + which we will in this proof continue to denote by Z h Φ , for simplicity of notation.
Let i ∈ Γ, and let i 0 = . . . iii . . . ∈ Γ Z . Now, In particular, Let β = 0 be an eigenvalue of Z h Φ . By Proposition 2.10, there exists a continuous eigenfunction φ for β on Z h Φ , so we have (6.3) φ(T s (i, t)) = e(βs)φ(i, t) for every (i, t) ∈ Z h Φ and s ≥ 0. Since φ = 0, we may let ψ : Z h Φ → R be the real-valued function defined by φ(i, t) = e(ψ(i, t)) for every (i, t) ∈ Z h Φ . We obtain from (6.3) that ψ(T s (i, t)) = βs + ψ(i, t) + n(i, t) for some integer-valued function n : Z h Φ → Z. Inserting the value s = h(i), we obtain ψ(σi, t) = βh(i) + ψ(i, t) + n(i, t) which is equivalent to In particular, h(i 0 ) = β −1 n(i 0 , t) ∈ β −1 Z since i 0 is a fixed point for σ. However, we saw above that h(i 0 ) = − log |λ 1 (A i )|, whence it follows that β ∈ (log |λ 1 (A i )|) −1 Q. Proof of Proposition 6.1. By Lemmas 6.2 and 6.3 and the assumption of Proposition 6.1, the flows Z ′′ Φ and Z ′′ Ψ have no common eigenvalues. Thus by Proposition 2.9, the product Z ′′ Φ × Z ′′ Ψ is ergodic. Since it has at most countably many eigenvalues, there exists a real number c > 0 such that it is also ergodic under the discrete-time map T c . By a change of coordinates, we may suppose further that c = N is an integer. Now, since we have the domination assumption in place, the system Z ′ Φ × Z ′ Ψ is a factor of the system Z ′′ Φ × Z ′′ Ψ through the map (i, t, j, s) → (i + , θ − (i − ), t, j + , θ * (j − ), s).
Since factor maps preserve ergodicity, this proves the statement. 6.2. Dynamics of the magnifications of µ. We claim that the flows Z Φ and Z Ψ capture the dynamics of the sequences (π 2 µ i,k ) k∈N and (π 2 ν j| i k ) k∈N .
In particular, S t • F = F • T t .
Therefore, for almost every (i, θ) we have 1 n n−1 k=0 1 N H N (F (i, θ, 1, kN + log C(i, θ)) I k ) = 1 n n−1 k=0 1 N H N ((F ′ (i, θ, kN + log C(i, θ))) I k ) ≤ dim µ − 1 + ε for large enough n, where C(i, θ) is as in Lemma 6.4. The above also holds after the change of coordinates of Proposition 4.1 for the same value of N, since as the measure F (i, θ, 1, kN + C(i, θ)) is supported on a line, an affine change of coordinates only results in adding something to the fourth argument of F which we were free to choose to begin with. Finally, using (6.6), we obtain 1 n n−1 k=0 1 N H N (π 2 µ i,k+m ) ≤ dim µ − 1 + 2ε forμ-almost every i ∈ Γ N and all large enough n.