Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $\Lambda$ over the circle embedded in $\mathbb{R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing \emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$ strong stable foliation, we prove that Hausdorff dimension ${\rm HD}(\Lambda\cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else ${\rm HD}(\Lambda)=t_0+1$. We prove also that for the packing measure $0<\Pi_{t_0}(\Lambda \cap W^s)<\infty$ but for Hausdorff measure ${\rm HM}_{t_0}(\Lambda\cap W^s)=0$ for all $W^s$. Also $0<\Pi_{1+t_0}(\Lambda)<\infty$ and ${\rm HM}_{1+t_0}(\Lambda)=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure ${\rm HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.

Such a solenoid in the linear case (or at least if η ′ ≡ d) can be called a uniformly thin solenoid 1 . Compare stronger uniform dissipation condition in the Outline subsection. is an invariant hyperbolic set, so called expanding attractor. The assumption f is injective on M can be weakened to the assumption f is injective on Λ, by replacing M by a solid torus being a sufficiently thin neighbourhood of Λ. However, for clarity, we assume the injectivity of f directly on M .
For each p = (x, y, z) ∈ Λ the disc W s x = W s (p) = {(x ′ , y ′ , z ′ ) : x ′ = x} is a (principal) component (in M ) of the stable manifold of p and the interval W ss x,y = W ss (p) = {(x ′ , y ′ , z ′ ) : x ′ = x, y ′ = y} is a (principal) component of strong stable manifold of p. Unstable manifolds W u (p) are more complicated, each is dense in Λ and for each x, x ′ ∈ R 1 the unstable lamination of Λ defines the holonomy map h x,x ′ : W s x/2πZ ∩ Λ → W s x ′ /2πZ ∩ Λ. Sometimes we write W s x in place of W s x/2πZ . Denote by π x the projection (x, y, z) → x. The part of global W u (p), which is the lift of [x, x ′ ] ⊂ R for π x will be denoted W u [x,x ′ ] (p). For [x, x ′ ] equal to [0, 2π] or slightly bigger, clear from the context, we shall write sometimes just W u (p).
Denote by π x,y the projection (x, y, z) → (x, y). We assume in this paper the following transversality assumption : each intersection of two distinct π x,y (W u (p)) and π x,y (W u (q)) is transversal.
Let µ = µ t 0 be the Gibbs measure (equilibrium state) on Λ for the potential t 0 log |λ ′ |, where t = t 0 is zero of the topological pressure t → P (f, t log λ ′ ). The measure µ can be called geometric or SRB in stable direction or just stable SRB-measure. Denote by µ s x its conditional measures on W s x for each x, see explanations following Lemma 3.6.
We prove the following Theorem A. Let Λ be a non-uniformly thin solenoid for f : M → M as in the definition above, which satisfies the transversality assumption. Then, for HD denoting Hausdorff dimension and for every x ∈ S 1 , 1. HD(Λ ∩ W s x ) = t 0 . 2. HD(Λ) = 1 + t 0 .
A theorem used in particular to compare sizes of Λ ∩ W s x for varying x, is Theorem D. Under the assumptions of Theorem A, if the bunching condition is satisfied, then all the holonomies h x,x ′ for x, x ′ ∈ R are uniformly Lipschitz continuous.
If the bunching condition is not assumed, then for each R there exists Lip(R) > 0 such that for each x there is a set L x ⊂ W s x ∩ Λ such that for all x ′ satisfying |x − x ′ | < R the holonomies h x,x ′ are locally bi-Lipschitz continuous with a common constant Lip(R) and µ s x (Λ ∩ W s x \ L x ) = 0. In fact µ s x (N L w ∩ W s x ) = 0 for certain weak non-Lipshitz set N L w invariant under all h x,x ′ for 0 ≤ x, x ′ ≤ 2π, which intersected with W s x is bigger than the complement of Lipschitz L x . Moreover HD(N L w ∩ W s x ) < t 0 . Here "local" means for every p ∈ L x there exists δ such that for all q ∈ W s x ∩ Λ ∩ B(p, δ) and |x ′ − x| < R the Lipschitz condition with the constant Lip(R) holds for h x,x ′ .
Bunching condition above, appeared in a related setting in [13], see also [6,Th. 4.21] with a stronger conclusion that the unstable foliation is C 1 .
Some of the assertions above hold also for the projections to the {(x, y)} plane, in particular Theorem E. Under the assumptions of Theorem A HD(π x,y (Λ ∩ W s x )) = t 0 and HD(π x,y (Λ)) = 1 + t 0 . We do not know if 0 < Π t 0 (π x,y (Λ∩W s x )) < ∞ or 0 < Π 1+t 0 (π x,y (Λ)) < ∞. The assertions of Theorem A almost automatically hold for Hausdorff dimension replaced by upper box dimension BD. Indeed, the estimates from below follow from HD ≤ BD. The estimate BD(Λ ∩ W s x ) ≤ t 0 follows from Lemma 4.3. The estimate BD(Λ) ≤ 1 + t 0 follows from Proof of Theorem 4.2, Step 3.
On the linear case. The mapping f in (1.1) is called lower triangular (because such is the differential Df in the y, z direction) non-linear. Our paper complements the study of the linear diagonal case with 0 < ν < λ < 1/d. It was done by B. Hasselblatt and J. Schmeling in [9], where nevertheless there were hints concerning the non-linear situations, and by M. Rams and K. Simon [18]. Namely, Theorems A,B,D generalize [9] and Theorem C generalizes [18]. By the way Theorem B was proved in [18] only for Lebesgue almost all x. Transversally conformal case. This is the case where f is conformal on every W s , well understood. Theorems A and B hold, though the dimension t 0 can be larger than 1 (in a thick case). Packing and Hausdorff measures on W s in dimension t 0 are equivalent. In fact this is transversally complex 1D situation, whereas our non-conformal case corresponds after π x,y -projection to transversally real 1D situation with overlaps.
Motivation. Let us present the geometric picture of the solenoid. It helps to think not of the solenoid itself (which is locally just a Cantor bouquet of almost parallel lines, hard to analyze with an untrained eye) but of its approximations f n (M ). Each of those is a tube, winding around along S 1 , going around d n times. Thus, every section of f n (M ) with a disc W s x = {x} × D is a disjoint union of d n components, each of those being a (slightly deformed) ellipse, with exponentially increasing ratio of the large semiaxis to the small semiaxis, and all the large semiaxes pointing roughly in the same {y} direction. See Figure 1. As already mentioned, those aproximate ellipses are disjoint, but there are plenty of sections in which some of the ellipses are very close to each other, in a distance exponentially smaller than their diameters. One of the main concerns in our study will be the understanding of the way those ellipses 'move' as we move the section plane around S 1 .
This picture resembles very much the picture of an affine iterated function system, or maybe even better: the averaged picture of many different but similar affine iterated function systems (as each ellipse in the section comes with different backward path, and is thus produced by a different collection of nonconformal contracting maps). An important element of the picture is that those contracting maps in the sections satisfy a version of the domination property, that is the strong contracting direction in one iteration stays close to the strong contracting direction in the following iterations. That is, we clearly have two different negative Lyapunov exponents in our system. This picture lets us expect the behaviour similar to the known generic behaviour of affine iterated function systems. We expect the Hausdorff and packing and box dimensions of each section to be given by the Falconer's singular value pressure formula, which in our 'thin' case should depend only on what happens in the expanding and weakly contracting directions (in particular, the dimensions should be preserved by the projection to the {(x, y)} plane). Moreover, like in simpler solenoid cases, we expect that the situations when two ellipses pass nearby (which are unavoidable by the very geometry of the solenoid) should have an effect strong enough to zero the Hausdorff measure, but the packing measure should stay positive and finite. And those are exactly the statements we eventually prove in Theorems A-C and E.
Hasselblatt and Schmeling stated in [9] 2 the following Conjecture. The fractal dimension of a hyperbolic set is the sum of those of its stable and unstable slices, where "fractal" can mean either Hausdorff or upper box dimension.
For solenoids, in [9] and here, an affirmative answer on Hausdorff dimension has been proven. Hausdorff dimension in the stable direction is t 0 and in unstable 1, that is 1 + t 0 together. Notice that this is dimension t 0 of conditional measures of µ geometric (SRB) in stable direction (see above) and of dimension 1 of an SRB measure in the unstable direction. Both SRB measures are usually different (even mutually singular), unless (e.g.) in the diagonal linear case, where both measures coincide with the measure of maximal entropy.
For any invariant hyperbolic measure ν indeed HD(ν) = HD(ν s )+HD(ν u ), see [1], but even supremum of HD(ν) over invariant ν on Λ can be less than HD(Λ). See e.g. [17]. So in a general case one is forced to use the both SRB measures.
Finally note that Hasselblatt and Schmeling relax the assumption of transversality to the assumption the intersections of π x,y projections of W u 's are non-flat. This in particular holds for all real analytic (that is with the functions u, v real analytic) linear solenoids, see [9]. A natural challenge would be to generalize our non-linear theory to a general real-analytic (nontransversal) case.
Outline. In Section 3 we prove a part of Theorem D saying that the points in W s x where a holonomy h x,x ′ is not locally bi-Lipschitz has measure µ equal 0. This follows (and clarifies) [9]. In fact we prove that a bigger set has measure 0, the set of p which are not strong Lipschitz, called above weak non-Lipschitz. For such a p the projection π x,y (W u (p)) intersects some π x,y (W u (q))'s for q / ∈ W u (p) arbitrarily close to W u (p). Equivalently p is strong Lipschitz if W ss loc (p) = {p}, hence counting Hausdorff dimension only E s /E ss counts so HD(Λ ∩ W s ) = h µ (f )/ − χ µ (λ ′ ) = t 0 , where h µ (f ) is the measure (Kolmogorov's) entropy, [11]. This is done in Section 4, and yields Theorem A. Again we roughly follow [9].
Note that Lipschitz property is related with Theorem A on Hausdorff dimension a little bit by chance, saying however that HD(W s x ∩ Λ) does not depend on x. In fact holonomy being Lipschitz is a weak condition, e.g. it holds for all holonomies h x,x ′ provided η ′ > λ ′ /ν ′ as in Theorem D (well known), as twisting, hurting Lipschitz property, cannot develop if f −n squeezes (by (η ′ ) −1 ) too much. Lipschitz property is crucial to conclude HD(W s x ∩ Λ) = t 0 ⇒ HD(Λ) = 1 + t 0 in Theorem A. Compare Conjecture above.
Theorem B is proved in Section 5. The proof has common points with [18]. Analysis is more delicate than in the proof of Theorem A. We prove that for µ-a.e. p for a sequence of n's the π x,y -projection of the tube f n (M ) (truncated to [0, 2π]), called of order n containing p, intersects only a bounded number of other projections of tubes of order n.
Theorem C is proved in Section 6, again using an idea from [18]. It uses the fact of arbitrarily high multiplicity of overlapping of projections of tubes of order n for µ-a.e. p.
The estimate HD(N L w x ) < t 0 in Theorem D is proved in Section 7, together with a more precise estimate, following from large deviations estimate concerning Birkhoff averages.
Theorem E follows from other theorems because the assertions on dimensions are verified on the sets where the projection π x,y is finite-to-one.
Section 7 contains also a remark on general Williams 1-dimensional expanding attractors and a remark on a possibility of integrating general solenoids to triangular as in (1.1) ones.
Acknowledgements. We wish to thank Adam Abrams for making pictures to this paper. We are grateful to Aaron Brown and Jörg Schmeling for useful discussions. All the authors are partially supported by Polish NCN grant 2019/33/B/ST1/00275.

Holonomy along unstable lamination
Definition 2.1. We will now introduce a symbolic description on the attractor, defining an 'almost bijection' ρ : is the usual two-sided full shift space on d symbols. Why it is only an 'almost bijection' will be explained soon. Moreover, it will be done in such a way that ρ semi-conjugates the left shift ς acting on Σ d to f | Λ , namely Let us start closer explanations from the x coordinate. Looking at the formula (1.1) we see that if (x ′ , y ′ , z ′ ) = f (x, y, z) then x ′ does depend only on x, not on y nor z. The restriction of f to the first coordinate is the d-to-1 expanding map η. Denote by a 0 , ..., a d−1 , a d = a 0 the points of η −1 (0) ∈ R/2πZ = S 1 numbered in the increasing order. We can assume that a 0 = 0 = η(0). For i = 0, 1, . . . , d − 1 we denote V |i := [a i , a i+1 ] × D; those sets will be called vertical cylinders of level 1. We can then define for every n = 1, 2, . . . the vertical cylinders of level n, by For every sequence (i 1 , i 2 , . . .) ∈ {0, . . . , d − 1} N there exists exactly one x ∈ S 1 such that and vice versa: for all except countably many points x ∈ S 1 there exists exactly one sequence (i 1 , i 2 , . . .) ∈ {0, . . . , d − 1} N such that (2.1) holds. The exceptions are the points x such that η k (x) = 0 for some k ∈ N. For each of those points one can find exactly two sequences satisfying (2.1). We note that f k−1 (p) ∈ V |i k is equivalent to η k−1 (x) ∈ [a i k , a i k +1 ], so what we described up to this point is the usual construction of symbolic description for an expanding map of the circle.
Let us now define the horizontal cylinders of level n = 0, 1, 2, . . . by the formula: ..,i 0 ), and then define The fact that ρ semi-conjugates ς to f | Λ is clear from the definitions.
We will denote by V (n) and H(n) the sets of all vertical (resp. horizontal) cylinders of level n. For a given p ∈ Λ we will denote by V n (p) and H n (p) the vertical and horizontal cylinder of n, containing p. Sometimes we will just write H n and V n if we do not specify p.
We note that ρ(i) = (x, y, z), with x depending on i 1 , i 2 , . . . and (y, z) depends on x and on i 0 , i −1 , . . .. It makes thus sense to write denoting the one-sided shift spaces on d symbols, the former one given by nonpositive entries and the latter one by the positive entries. We then denote Clearly, ρ(i) = W u (i) ∩ W s (i). We will also use the notation W u (p), W s (p) for p ∈ Λ, as the shortcut for W u (ρ −1 (p)), W s (ρ −1 (p)). We note that as η is expanding and λ, ν are contracting, W u (p) and W s (p) are pieces of the unstable and stable manifolds at p ∈ Λ -which explains the notation used. This notation has been already introduced in Introduction, together with the notation W s x for x ∈ S 1 .
This definition makes sense, since f preserves the foliation {W ss x } (vertical intervals here). To simplify notation denote sometimes objects being the projection of objects in M by π x,y by adding a hat over them, e.g. Λ := π x,y (Λ) or p = π x,y (p).
Remind that the projection (x, y, z) → x is denoted by π x , see Introduction. For any p ∈ M the point π x (p) will be sometimes denoted by x(p).
, where the 'margins' Lη −1 n will be defined in Notation 2.4 and Definition 2.5.
In words, Γ is a Cantor set, consisting of the intersections of the π x,yprojections of W u 's belonging to different (slightly extended) horizontal cylinders of level 0. It discounts intersections of the projections of W u 's being in the same cylinder of level 0. See Figure 3.
(Unstable) transversality assumption. All intersections of these lines (i.e. projections by π x,y of unstable manifolds) with different i 0 's are in this paper assumed to be mutually transversal.
Remark 2.3. Notice that by the compactness argument and the continuity of the sub-bundle E u Λ , by the transversality assumption all the intersection angles are bounded away from 0, say by α 0 .
Also by compactness and continuity of E u on Λ there exists r 0 > 0 such that if for p, p ′ ∈ Λ ∩ W s their mutual Euclidean distance is r < r 0 and their i 0 's are different then the distance of their π x,y projections from Γ, more precisely from the intersection W u (p) ∩ W u (p ′ ) which is in particular nonempty, is bounded by 2r/ tan α 0 .
Remember that we consider f in the form (1.1) and write η ′ := ∂η ∂x , λ ′ := ∂λ ∂y and ν ′ := ∂ν ∂z . Then ). Definition 2.5. A point p ∈ Λ is said to be strong locally Lipschitz if there is L > 0 such that for all n big enough, denoting (η − n (p) by η n , n ] (f −n (p)) ≥ Lη −1 n , with the distance in W u measured between the projections by π x in R.
Equivalently we could replace here V n (f −n (p)) by f −n (p). It would influence the constant L only.
By the unstable transversality and transversality of intersection of stable and unstable foliations, this is equivalent to the distance in the {(x, y)}-plane satisfying We call all points p which are strong locally Lipschitz with the constant L such that (2.2) holds for all q ∈ W u (p) in place of p strong locally bi-Lipschitz.
Notice that this definition allows to say that the whole W u (p) is strong locally bi-Lipschitz and write 2) satisfied at p with L > 2 Const strong locally Lipschitz condition holds for all q ∈ W u (p), with L = L/2. So p is strong locally bi-Lipschitz.
Here at p means that for every Proof. We repeat (adjust) the calculations in [9]. Consider q ∈ W u (p) and Lipschitz continuity of the holonomy h x(p),x(q) at p would follow from the existence of a uniform upper bound of for p ′ close enough to p, i.e. n defined above, large. It is comfortable to consider the distance d = d 1 + d 2 , the distances in the y and z coordinates.
We shall use the triangular form of the differential Df | {y,z} = λ ′ 0 a ν ′ .
We estimate for a constant A depending on the angle between W u and W s . Here λ n , a n , ν n are averages of derivatives λ n , a n , ν n respectively, on appropriate intervals, namely integrals divided by the lengths of the intervals, horizontal along y for two first integrals and vertical along z for the last one.
On the other hand To obtain an upper bound of (2.5) it is sufficient to assume the existence of an upper bound of the ratio of the above quantities, namely (omitting (f −n (p)) to simplify notation) 1 + Aλ n /η n λ n ∆ 1 p + |a n ∆ 1 p + ν n ∆ 2 p| .
We needed bars over λ, ν, a to reduce above a fraction to the summand 1. From now on these bars (integrals) are not needed.
We conclude calculations with sufficiency to assume the existence of an upper bound of (2.6) 1 or to assume that the inverse Thus, Lipschitz property follows from either of The condition (2.8), in the diagonal case a n = 0, means that the contraction in the space of stable leaves W s by f −n , along the coordinate x, due to small η −1 n is strong enough to bound the twisting effect caused by log ν n / log λ n , hence implying the Lipschitz continuity of all the holonomies at p along unstable foliation of a bounded length leaves (e.g. by 2π). This is for ∆ 1 (p) ≈ 0 (hence ∆ 2 (p) large). Otherwise Lipschitz condition holds automatically.
The condition (2.7) is equivalent to strong locally Lipschitz (2.2) in the Definition 2.5 by transversality condition, see Remark 2.3 and (2.2) and (2.3). This implies that the distance between W s (f −n (p)) and By Remark 2.6 for Const above large enough we obtain strong bi-Lipschitz property.
We denote the set of all strong locally bi-Lipschitz points in Λ by L s and L s ∩ W s (p) with x(p) = x by L s x . Sometimes we write Λ s (τ, L, n) for specified n, see (2.2).
As we already mentioned in Theorem 2.8 The holonomy is locally Lipschitz on L s . Notation 2.10. We call the set complementary to L s in Λ: weak non-Lipschitz, and denote it by N L w . By Lemma 2.8 this condition is weaker than non-Lipschitz. It includes some Lipschitz (e.g. if bunching condition holds, see Theorem D).
Later on we shall need the following fact easily following from the definitions Lemma 2.11. For every p ∈ L s there exists n such that If it happens for n arbitrarily large, it contradicts p ∈ L s .
A geometric meaning of this, is that for an arbitrary W s , replacing λ ′ by a function having logarithm cohomologous to log λ ′ (denote it also by λ ′ ), not depending on future (|i 1 , ...), the quantities log λ n (f −n (p)) for p = ρ(...i 0 |i 1 , ...) are approximately diameters of f n (W s (f −n (p))) provided ν ′ < λ ′ . The quantity t 0 which would be Hausdorff and box dimensions in the conformal case, here in the non-conformal case is only the upper bound of the dimensions of W s ∩Λ, so-called "affinity dimension", [8]. The aim of this and the next sections is to prove that t 0 is in fact the Hausdorff dimension of all W s ∩ Λ.
We start now with where L 1 is the constant from Theorem 2.8. Define also (3.1) h n := sup h n (i) and h := lim sup h n .
The following follows easily from the definitions and the transversality assumption The opposite inequality also holds if sup λ ′ < 1/ sup η ′ and ν ′ (p) < λ ′ (p) for every p ∈ Λ, the property we name: uniform dissipation.
Lemma 3.5. Assume transversality and χ µt 0 (λ ′ )) < − log sup η ′ (call it half-uniform dissipation). Then Proof. For an arbitrary ε > 0 and n large enough, denoting by BD the upper box dimension, we easily get To prove this we cover W u by pairwise disjoint (except their end points) arcs of the same length equal to 1/(sup η ′ ) n (up to a constant) and use the definition of box dimension.
A difficulty we shall deal with below, is however to pass in (3.4) to a uniform over i version, that is with sup i h ∞ n (i) in (3.2) Notice that The first inequality uses the "Lipschitz holonomy" along W u 3 (in fact only local) between an arbitrary W s and W u = W u (p). We shall prove it more precisely below: Denote by A(f ) supremum over all p, p ′ as above of the number of the intersection points of W u (p) and W u (p ′ ). It is finite by the transversality assumption, see e.g. [15,Prop. 4.6].
For every n ≥ 0 we have, due to ν < λ, for n large enough to kill a twisting effect which may be caused by partialν ∂y . Hence due to the transversality assumption, for every component Comp of the intersection.
By the definition of t 0 we have, summing over all (i ′ −n , ..., for t > t 0 and constant C(t) or t < t 0 respectively. For each r > 0 and q ∈ W u (p)) ∩ Γ, where q ∈ ρ(..., i ′ 0 |) find n = n(q) the least integer such that the length satisfies l( W u ∩ H i ′ −n ,...,i ′ 0 ) ≤ r; by the length (denoted above by diam) we mean here the length of the projection by π x to R (of course we can alternatively consider the lengths in W u (p) or W u (p)). 3 Formally this is not even a holonomy, because of intersections of the leaves. However it is Lipschitz in the sense of varying all the lengths of the uniformly transversal sections of each strip Hn for all n, by at most a common factor. Denote W u ∩ H i ′ −n ,...,i ′ 0 by I(q, r). Consider in W u the ball (arc) J(q, r) = B( q, r). Choose a family J(q k , r) of the arcs of the form J(q, r) covering W u ∩Γ, having multiplicity at most 2, namely that each point in W u belongs to at most 2 arcs. Then I(q k , r) ⊂ J(q k , r) for all k. On the other hand by the definition of n(q) there is a constant K such that Kl(I(q k , r)) ≥ l (J(q k , r)).
Finally notice that for two different q k and q k ′ it may happen that n = n(k) = n(k ′ ) and the n-th codings i −n , ..., i 0 are the same; in other words the n-th horizontal cylinders coincide. Then however J(q k ) and J(q k ′ ) intersect so the coincidence of these codings may happen only for at most two different k and k ′ .
Thus, for all t > t 0 , Hence, as our estimates hold for every r > 0, we obtain the first inequality in (3.4) BD This has been Moran covering type argument.
Another variant of this proof would be to consider for each n the partition of W u into d n := 2π/r n arcs of length r = r n = 1/([sup η ′ n ]+1) and consider the family of those arcs which intersect Γ. Denote them by J k . For each k choose an arbitrary q k ∈ Λ such that q k ∈ J k ∩Γ and q k belongs to some ρ(i ′ ) with i 0 = i ′ 0 . Then consider I k as above. Finally notice that each interval I k as shorter than r n for n large enough, can appear at most twice.
Finally, by (3.6), the estimate (3.4) is uniform, that is n for which it holds is independent of i. ndeed, for each i and n we obtain for r = 1/(sup η ′ ) n , denoting ε = 2(t − t 0 ), The simplest uniform dissipation, namely if η −1 n ≡ d −n , provides the partitions of S 1 into arcs of equal lengths to be used in estimating BD. In more general cases the partitions of S 1 into arcs between consecutive η n preimages of a fixed point cause difficulties and a necessity to assume the half-uniform dissipation assumption using sup η ′ rather than η ′ . They will be overcome in Section 4 in Proof of Theorem 4.1 by replacing h by certain h reg by restricting in the definition to µ t 0 -Birkhoff regular points with respect to log η ′ , restricting Γ. Now, assuming the uniform dissipation, using h < h * following from Lemma 3.5 and Proposition 3.4, we can the following Lemma 3.6. Assume that the transversality and the uniform dissipation condition (introduced in Proposition 3.4) hold. Then µ s x ′ has been already defined as a conditional measure.
Remark that for µ-almost all p, p ′ the holonomy h x, Note that these measures coincide also with the factor measure µ − := Φ * (µ t 0 ) where Φ maps the two sided to the one-sided shift to the right, on the space of sequences (..., i −n , ..., i 0 ) (projected to f −1 by ρ). In fact we can write µ in place of µ − considering its restriction to the σ-algebra generated by horizontal cylinders.
Given arbitrary ε > 0 and n, denote the set of p's ( a union of H n (p)'s) where (3.7) does not hold, by Y ε,n . Thus, the irregular set has measure µ equal to 0. Its ε-regular complement lim inf n→∞ X ε,n = n k≥n X ε,k for X ε.k = Λ \ Y ε,k has full measure µ for each ε.
Remark that for our Gibbs measure we can use Birkhoff's Ergodic Theorem for f −1 and log λ ′ in place of Shannon-McMillan-Breiman: Denote by Z ε,n the family of all horizontal cylinders H 2n , whose π x,yprojections H 2n have "horizontal extensions" to (−L2π, (L + 1)2π) intersecting f n (Γ) and else which are in X ε,2n . The constant L is the one that appeared in (2.2).
Claim. The upper bound of #Z ε,n is roughly exp(nh+(n+1)h * ) ('roughly' means: up to a factor of order at most exp nε).
Indeed, the number exp nh comes from f n (H) for each H ∈ H(n), more precisely from f n -images of the rectangles H ∩ V |i 1 ,...,in counted in Definition 3.2. The number exp(n+1)h * comes from the number of 'regular' H ∈ H(n) roughly as in (3.7), that is being f n+1 -images of 'regular' V n . More precisely of those H's for which some H 2n+1 ⊂ f n (H) are in X ε,2n+1 ∩ X ε,n 4 . The measure µ of each such H is lower bounded for p ∈ H by (4.4), and Gibbs property of µ (used already above to reformulate (3.7) to the language of λ ′ ).
One can even replace liminf by limsup.
Proof. Lemma 4.3 follows from (4.1) µ s πx(p) (B s (p, λ n (f −n (p)))) ≥ Const(λ n (f −n (p))) t 0 . One uses the definition of µ s (Gibbs property) and the fact that the diameter of each B s (p, r) is comparable to λ n (f −n (p)), by bounded distortion. The conditional measures µ s were discussed after the statement of Lemma 3.6 More sofisticated is the opposite inequality: Proof. One uses Ledrappier-Young formula [11] h µ (f ) = δ ss (−χ µ (ν ′ )) + (δ s − δ ss )(−χ µ (λ ′ )) and the fact that δ ss = 0 since for p ∈ L s the local manifold W ss consists only of the point p, see Lemma 2.11. Remember also, Lemma 3.6, that Proof of Theorem A, uniformly dissipative setting. Step 2. Since by Lemma 3.5 and Proposition 3.4 h < h * := h µt 0 (f ), we know by Lemma 3.6 that there exists x (in fact for all x) µ s x (L s ) = 1. By Lemma 2.9 all the holonomies h x,x ′ for 0 ≤ x ′ ≤ 2π are locally bi-Lipschitz on L s (x). Change the coordinates on Λ by F (x ′ , y, z) := (x ′ , h −1 x,x ′ (y, z)), mapping Λ to the cartesian product [0, 2π) × Λ ∩ W s x . Then this change is locally Lipschitz on 0≤x ′ ≤2π h x,x ′ (L s x ) = L s . Hence HD(Λ) ≥ HD(L s ) = 1 + HD(L s x ) = 1 + t 0 . More precisely F is locally Lipschitz, in the sense that there exists L > 0 such that for every p ∈ L s there exists measurable r(p) > 0 such that for every r ≤ r(p) and q ∈ B(p, r), dist(F (p), F (q)) ≤ Ldist(p, q). This is sufficient to non increase dimension by splitting the space into a countable number of pieces.
Step 3. The opposite inequality is implied by Lemma 4.3. Indeed, notice that for every r = λ n (p) and x ′ ∈ B(x(p), r) we have and in conclusion µ(B(p, Cr)) ≥ Const r 1+t 0 yielding the required upper estimate HD(Λ) ≤ 1 + t 0 . Analogously p ∈ Λ is said to be Birkhoff (ξ, ε, N )-forward regular if the above estimates hold for ξ + n (p)) in place of ξ − n (p)). When we mean just (4.3) we say (ξ, ε, n)-forward (backward) regular, omitting "Birkhoff". Compare Shannon-McMillan-Breiman property in Proof of Lemma 3.6 By bounded distortion the property (4.3) for p = ρ(...i −n , ..., i 0 , ..., i n , ...) depends only on (i −n , ..., i 0 ), provided we insert constant factors before exp, so it can be considered as a property of a horizontal cylinder H n . Analogously for the forward regularity this is a property of vertical cylinders V n . We call these cylinders (ξ, ε, n)-forward or backward regular and all other points or level n cylinders irregular.
Proof of Theorem A. We shall modify (generalize) the definition of irregular sets Y ε,n in Lemma 3.6 and follow the strategy of the proof of that lemma.
Write also Y ε,n,m := Λ \ X ε,n,m for irregular sets. Now, as in Section 3, Proof of Lemma 3.6, the idea is to remove 5 for each n the irregular set Y ε,n,m for m to be defined later on, and estimate the number of remaining cylinders H n+m which are regular contaminated by other regular cylinders in the sense below (4.7).
A point (and cylinder) p ∈ H n+m regular as above is said to be (Γ reg n,m )contaminated if forp := f −n (p) (4.7) π x,y (p) ∈ B u (Γ reg , L 1 η −1 n (p)), compare Definition 2.5. B u denotes a ball in W u (p). The set Γ reg is defined as Γ in Definition 2.2, but restricted to p being π x,y image of q = ρ(..., i 0 |) and q ′ = ρ(..., i ′ 0 |) such that f n (q) and f n (q ′ ) are in X ε,n,m . As in Definition 2.5 we can say equivalently that V n (p) is Γ reg n,m -contaminated if it does not satisfy (2.2), with Γ replaced by Γ reg n,m . We can say also that the rectangle H m ∩ V n is contaminated, as in Definition 3.2. See also Subsection 7.4.
Summarizing: for given H m (p) with p = f n (p) ∈ X ε,n,m we define h reg n := 1 n+1 log Z n where Z n is the number of Γ reg n,m -contaminated V n in H m (p) (by H m (q) with the i 0 symbols different from the one for H m (p), with q = f n (q) ∈ X ε,n,m ). The number Z n is bounded by a constant times the number of H m above, taking in account L in (2.2) and the observation that regular H m , as "thinner" than V m can intersect at most two (neighbour) V m 's. So exp nh reg n ≤ Const exp mh * , hence using (4.9), The rest of the proof repeats Proof of Theorem 4.2 In particular by Birkhoff ergodic theorem for an arbitrary ε > 0 µ(lim sup n→∞ Y ε,n ) = 0 and the complementary set in N L w , for ε > 0 small enough, where h reg < h * , has measure µ also equal to 0.
The above proof finishes also the proof of Theorem D in the general setting, saying that µ s x (N L w ) = 0, compare Lemma 3.6 in the uniform dissipation case. Compare also (3.8).

Packing measure
For the definition of packing measure we refer the reader to [14,Section 8.3]. Denote packing measure in dimension t by Π t .
We shall prove the following , for the Gibbs measure µ = µ t 0 on Λ, then for every p ∈ Λ it holds Moreover the density dΠ t 0 /dµ s t 0 is positive µ s πx(p) -a.e. (recall that µ s πx(p) is the conditional measure on W s (p)). Also and moreover dΠ t 0 +1 /dµ t 0 is positive µ t 0 -a.e. on Λ.
This generalizes the analogous theorem proved for linear solenoids in [18]. Proof.
Given an arbitrary ε > 0 denote by H(ε, t) the union of all H t containing points in Λ satisfying the backward regularity condition (4.3) for t (denoted there n) and ξ = λ, ν.
Analogously denote by V(ε, t) the union of all V t containing points in Λ satisfying the forward regularity condition analogous to (4.3) for t (denoted there by n) and ξ = η and by V(ε, t).
We sometimes call H t and V t as above, just regular.
Consider an arbitrary m ∈ N and given ε > 0 define n = n(ε, m) as the biggest integer n such that, compare (4.9) and (4.8), We consider ε small enough that the latter fraction is bigger than 1. Later on we shall consider an arbitrary α : 0 < α ≤ 1 and the integer [αn] in place of n (the square bracket means the integer part), sometimes writing just αn. Finally we shall specify α. Of course (5.3) is satisfied for [αn] in place of n.
Notice that for an arbitrary (λ, ε, m)-backward regular p ∈ H m ⊂ H(ε, m), therefore for regular H m (p); the diameter of its intersection with any W s is at most exp(m(1 − ε)χ µ (λ ′ )) (up to a constant related to distortion).
For all i = (..., i 0 |), writing ρ(i) = W u (i) = W u we obtain the uniform (over i) estimate (3.4) on h ∞ n (i) as in Lemma 3.5 for W u restricted to the intersection with V(ε, n). We write h ∞,reg n (i). Indeed, we can use then for every forward regular V n the property diam(W u ∩ V n ) ≥ exp −n(χ µ (η ′ ) + ε). (We accept that one ε can differ from another if it does not lead to a confusion.) In (3.5) we use then χ µ (λ ′ ) < −(χ µ (η ′ ) + ε).
Defining h ∞,reg := lim n→∞ lim sup i h ∞,reg n analogously to Definition 3.3, we get for ε small enough h ∞,reg < h * .
We obtain the same estimates, in particular h reg < h * , if in place of W u , thicken it to H m restricting ourselves to backward regular p ∈ H m ⊂ H(ε, m), because then, if p ∈ V [αn] backward regular, for n = n(ε, m) see Definition 3.2 and the transversality.
In words, the number of forward regular vertical cylinders V [αn] whose π x,y projections V [αn] intersect the "rhombs" Step 2. Close cylinders.
We keep n, m and arbitrary α ≤ 1 as above and consider an arbitrary integer 0 < k ≤ m. We take care of intersections of H m with H ′ m 's with Step 3. Remote cylinders. (p ′ ) = ∅ for arbitrary constant C > 0, m, n large enough, ε small, provided that is we assume Step 4. Irregular sets.
Denote H(m, reg) : To conclude the proof of our theorem use now [18,Lemma 3] for the conditional measure µ s x on W s x . It yields in our case that due to µ(H(m, reg)) ≥ 1 2 −3· 1 8 = 1 8 for m large enough, hence µ s x (H(m, reg) ∩ W s x ) ≥ Const · 1 8 , for a positive measure µ s x subset W of W s x , for every q ∈ W there is a sequence m j such that H m j +[αn j ] (q) ∈ H(m j , reg). In particular there is a sequence of 'regular' horizontal cylinders containing q of level tending to ∞, whose π x,y -projections are each at most boundedly intersecting the family of projections of other horizontal cylinders of the same level, provided they are both in H [αn] (q).
, see e.g. [14,Theorem 8.6.2]. The density dΠ t 0 /dµ s x is positive µ-a.e. since the set W can be found of measure µ arbitrarily close to 1. This can be achieved by replacing the constants 1 2 and 1 8 by arbitrarily small positive constants, by increasing m 0 adequately. This increases the allowed bound of the multiplicity of intersections of H m+[αn] 's, thus increasing C.
Another variant of this part of the proof is to use ergodicity of f .
Finally the existence of an upper bound of dΠ t 0 /dµ s x , in particular finiteness of Π t 0 (W s ∩ Λ) follows from the uniform boundedness from below of µ s x B(q,r) r t 0 for r small enough, see Lemma 4.3. We again refer to [14,Theorem 8.6.2].
To prove 0 < Π 1+t 0 (Λ) in (5.2) notice that for an integer n 0 and every q ∈ W, every k j := m j + [αn j ] as at the beginning of Step 4 and every p 1 , p 2 ∈ W u (q) we have the following inclusion of intervals In words: each square of sides of order r ′ j , namely is a subset of a piece of H k j (q) of length r ′ j (along x-axis), with vertical (along y) sections of length of order r j , where r j := λ − k j (q). Hence, for "skew product" µ as in (4.2) We used here the fact that for Const > 0 small enough . Applying Frostman lemma finishes the proof of left-hand side inequality of (5.2). The right-hand side inequality follows from Lemma 4.3.

Remark 5.2.
When we take f k or f n+k image, the conditional measures stay the same by the f -invariance of µ.
The phenomenon which manifests and helps is the affinity of the mapping when we measure distances with respect to invariant measures after passing to conditional measures on unstable foliation. Remark 5.3. Notice that in estimating from below the local dimension δ s of Λ ∩ W s (p) for a.e. p we referred to Ledrappier-Young formula, using W ss loc (p) ∩ Λ = {p}, Lemma 2.11. In fact we knew there only that H 2n (p) did not intersect H 2n (p ′ ) such that H 2n (p ′ ) ⊂ H n−1 (p) \ H n (p), but we did not exclude the intersecting for H 2n (p ′ ) ⊂ H n (p). To avoid intersections we split H n (p) into H n+1 (p) and the complement, splitting both into H 2(n+1) and getting disjointness for H 2(n+1) (p ′ ) ⊂ H n (p) \ H n+1 (p). Etc., splitting H n+1 (p), H n+2 (p) ... . This allowed the local disjointness of W u 's as in the preceding paragraph.
We coped with the disjointness of entire H n 's in Section 5 on packing measure, but for each W u the disjointness of the consecutive cylinders containing it has been proved only for a sequence of n's.
The set A(d) of these p has positive measure µ and is invariant under holonomies h x,x ′ . Therefore, invoking also ergodicity of µ, (6.3) holds in every W s x for µ s x -a.e. p ∈ W s x and r = r(p, d). If we consider A = d∈N n∈N f n (A(d)) then using Frostman lemma we prove (6.1) for Λ replaced by A. Then, for any Hölder continuous potential Φ : X → R let µ Φ denote the unique Gibbs invariant measure for Φ, see [4]. Consider arbitrary Hölder functions φ, ψ : X → R. Then, for every t ∈ R, lim n→∞ 1 n log µ ϕ x ∈ X : sgn(t)S n ψ(x) ≥ sgn(t)n X ψ dµ ϕ+tψ where by P top we denote topological pressure, see e.g. [14].
A basic example of such F is ς : Σ + d → Σ + d being the left shift map on the one-sided shift space with the standard metric dist(i, i ′ ) = n∈N |i n −i ′ n |d −n . Symmetrically one considers the right shift map ς −1 : Σ − d → Σ − d on the space of sequences (..., i n , ..., 0|). We can consider two-sided sequences or e.g. our solenoid Λ identifying sequences with the same future, or past as for our W s 's and f −1 . Compare Definition 2.1 In particular the following holds Lemma 7.2. For every Hölder φ and ψ, for every ε > 0 there exist C > 0 and τ > 0 such that for every n ∈ N In Sections 3 and 4, proving e.g. that µ s x (N L w ∩ W s x ) = 0 in Lemma 3.6 we did not use large deviations. In Section 5 we already did (the qualitative version of Lemma 7.2. Now we shall show how the usage of large deviations, Lemma 7.1, allows to estimate from above Hausdorff dimension of the set in each W s x where the holonomy is not locally Lipschitz, thus strengthening Lemma 3.6. See also notation in and after Lemma 2.9 and the same with log λ ′ replaced by − log η ′ . Also with the latter fraction above replaced by its inverse.
Here A ε bounds Hausdorff dimension of the irregular part and B ε bounds Hausdorff dimension of the regular non-Lipschitz part.
Proof. First we prove the estimate (7.2) for B ε . We rely on Section 5: Proof of Theorem 5.1, Step 1. The coefficient α is not needed, since N L w is a local property and overlappings of remote cylinders do not count (see Proof of Theorem 5.1, Step 3).
We obtain the uniform estimate for every (λ, ε, m)-backward regular H m , with n satisfying (5.3) Its "contaminated part" can be estimated as follows, see (5.4) and (4.10), where C(n) grows sub-exponentially. We used here, as already e.g. in (5.4), the fact that Const −1 µ(H m ∩ V n )/µ(H m )µ(V n ) < Const following from Gibbs property of µ. Now by summing over regular H m with weights µ(H m ) we get the same estimate for µ Hm (V n (H m ) ∩ H m ) and by the f -invariance of µ the same estimate for H ′ (n) := f n Hm (V n (H m ) ∩ H m ) for n = n(ε, m), built of cylinders H m+n . Now we shall translate the measure estimate above for all m, n(ε, m), to an estimate of Hausdorff dimension.
Denote µ(H ′ (n)) by µ n . By Gibbs property of µ = µ t 0 and using normalized restrictions µ n := µ| H ′ (n) /µ n , considering conditional measures on W s (not changing notation), we get for each H n+m ⊂ H ′ (n) and p in it . with m expressed by n maximal possible to satisfy (5.3). So, for any x ∈ S 1 , By an arbitrarily small change of κ we can assure the bound by 1 replaced by numbers tending exponentially to 0 as m → ∞, allowing summing over m.
All the theorems in this paper hold also for hyperbolic expanding attractors in dimension 3 with 1-dimensional unstable manifolds, non-uniformly thin (see definition in Section 1) and satisfying the transversality assumption, our solenoids are example of. The only exception is thr theorem on singularity of Hausdorff measures Theorem 6.1, where the assumption that for some p, q ∈ Λ there is a non-empty intersection of projections W u (p) and W u (q) is needed. For our solenoids it holds automatically but for extensions to R 3 of say Plykin or DA attractor it is not so. See [19].
Proofs are the same since these attractors are extensions of expanding maps on branched 1-manifolds and Markov coding can be used.
of class C 1+ε , injective, such that f (cl M ) ⊂ M , satisfying λ(x, 0, 0) = ν(x, 0, 0) = 0, with hyperbolic attractor Λ, and satisfying transversality, the non-conformal form more general than f in the triangular in (1.1). Indeed, we are interested in non-conformal solenoid, so we assume that the tangent bundle on M , or at least on Λ, splits into T Λ M = E u ⊕ E s , Df invariant, where E s , the stable one, splits further into weak stable and strong stable T Λ M = E u ⊕ E ws ⊕ E ss , or at least E s contains a strong stable E ss . Note that E s is dynamically defined on the whole M , not only on Λ, by E s (p) := lim Df −n (C s (f n (p))) where C s denote a stable cone taken equal to a cone at a point in Λ near f n (p). Similarly one proves that the bundle E s on M is integrable to a stable foliation W s of M . As having codimension 1 it is C 1+ε , see [12]. Therefore under an appropriate C 1+ε change of coordinates it becomes the foliation of M by vertical discs W s x = {x} × D. Also strong stable foliation W ss (of the whole M as obtained as a limit from the future) can be made consisting of vertical intervals, that is with x, y constant. This foliation is known to be C 1+ε in W s , see [5], hence after a change of coordinates so that it becomes vertical, our diffeomorphism is C 1+ε in each W s . We do not know however what is the smoothness of f in the new coordinates in the whole M .
Therefore to deduce this general case from our triangular case by change of coordinates we just assume W ss is C 1+ε in M . A question stays open whether this assumption is needed, i.e. whether we really use f being C 1+ε in the triangular coordinates.
The following completes the topological picture. Suppose f is already in the triangular form.
Lemma 7.4. There exists on M a change of coordinates Ψ(x, y, z) = (x, y, ψ(y, z), bi-Lipschitz continuous, such that the foliation into the sets x, z constant is invariant and its Ψ −1 -image is a central stable foliation W sc with leaves C 1 smooth.
Proof. Extend f tof : S 1 × R 2 → S 1 × R 2 so that λ and ν are linear with respect to y and z respectively, far from M .
Next find W sc as a limit off n (W y ), where W y is the foliation of M into the intervals x, z constant. By bounded distortion one gets Lipschitz property of the limit and in particular a true foliation (leaves do no glue partially to each other in the limit).
Our Df in these coordinates would be diagonal which would ease estimates. Unfortunately this central stable foliation and therefore f in the new coordinates seems usually not C 1+ε . 7.4. Summary of our strategies. The key objects in the paper are "rectangles" being intersection of horizontal and vertical strips H m and V n , "cylinders" of level m and n respectively, projections to the plane (x, y) of tubes and thickened discs. Such Markov rectangles are basic objects in hyperbolic dynamics.
Horizontal strips can intersect transversally other horizontal strips. An issue is to estimate how large part of any horizontal strip is intersected, "contaminated", by other horizontal strips, measured in a number of contaminated (with margins) rectangles. The tool is going backward by f −m or forward by f n to large scale, so that the rectangles become full (that is over [0, 2π]) horizontal strips and results do not depend on sections by stable discs W s . We distinguish Birkhoff irregular sets among full unstable manifolds (over [0, 2π]) and prove they have stable SRB-measure 0 and even Hausdorff dimension in each W s less than the dimension of Λ ∩ W s . We estimate also the size of the contaminated set of Birkhoff regular unstable manifolds. In each section the choise of m to n (or vice versa) and auxiliary k is different, depending on our needs.