Box dimension of stable sub-slices of fractal graphs over Anosov diffeomorphisms

The following proposition exhibits an easy way to calculate the box dimension of a set and will be used a few times in the arguments.

In the special case that $X$ is the graph of a continuous function $f\colon I=[c,d]\to\mathbb{R}$, Proposition 2.2 can be applied to calculate the box dimension considering similar inequalities for the variation of the function $f$ defined by

The essence of the proof above is that the number of intervals of size $\delta$ needed to cover the domain are proportional to $\delta^{-1}$ and on each of these intervals the number of intervals of size $\delta$ needed to cover the image is, by hypothesis, proportional to $\delta^{a-1}$, so the number of squares of size $\delta$ needed to cover the graph is proportional to $\delta^{a-2}$. This can also be done for a $n$-variable function with the only difference being that the number of $n$-cubes needed to cover the domain is proportional to $\delta^{-n}$. Then we can state the following result.

A formal proof is omitted since it is similar with the proof of Proposition 2.3 with the only adaptation explained above.

The Stable Manifold Theorem ensures that if $\epsilon$ is small enough, then the sets

are $C^{1}$ embedded disks tangent to $E^{s}$ and $E^{u}$, respectively. In particular, there is a local product structure close to any $x\in\Lambda$: there exists $\delta>0$ and a map

where $W^{s}_{\epsilon,\delta}(x)=W^{s}_{\epsilon}(x)\cap B(x,\delta)$ and $W^{u}_{\epsilon,\delta}(x)=W^{u}_{\epsilon}(x)\cap B(x,\delta)$, defined by

Thus, for each $x\in\Lambda$, the map $[,]$ defines a homeomorphism between the product

and $\Lambda\cap B(x,\delta^{\prime})$ (for some $\delta^{\prime}\in(0,\delta]$). The sets in this product are called the local stable and local unstable slices of $\Lambda$ at $x\in\Lambda$. If the map $[,]$ is bi-Lipschitz, then the dimension of $\Lambda$ equals the sum of the dimensions of the stable and the unstable slices at any point, since Hausdorff and box dimensions are preserved by bi-Lipschitz functions. This fact is used by McCluskey and Manning [8] to calculate the Hausdorff dimension of two dimensional horseshoes.

For hyperbolic sets on manifolds of higher dimensions, the map $[,]$ is always Hölder continuous but not necessarily Lipschitz (see Shub [9]). Thus, it is really difficult to compute the dimension in this case. Hasselblatt and Schmeling [5] stated the following conjecture:

They prove this conjecture for the Smale Solenoid, that is a hyperbolic set on a manifold of dimension three (see [5] for more details). We hope we have motivated the importance of the stable/unstable slices and the calculus of their dimensions, and finish this sub-section.

Following Kaplan et al. [6] we consider the skew product $F\colon\mathbb{T}^{k}\times\mathbb{R}\rightarrow\mathbb{T}^{k}\times\mathbb{R}$, where $\mathbb{T}^{k}$ is the $k$-torus, defined by

where $A$ is a linear Anosov diffeomorphism, $0<\lambda<1$, and $p\colon\mathbb{T}^{k}\to\mathbb{R}$ is a smooth function. This skew-product has an attractor that can be seen as a graph over $\mathbb{T}^{k}$ (see [6]) such that any point $(x,y)$ in the attractor satisfies

For simplicity, consider the function $\phi\colon\mathbb{T}^{k}\to\mathbb{R}$ defined by

Thus, $y(x)=\phi(A^{-1}x)$, and since $A$ is a linear Anosov diffeomorphism, the box dimensions of the graphs of $y$ and $\phi$ are the same. Assume that $A$ has $k$ distinct eigenvalues $B_{1},\dots,B_{k}$ and that the respective eigenvectors $v_{1},\dots,v_{k}$ form a basis of $\mathbb{R}^{k}$. Write points of $\mathbb{T}^{k}$ on this basis as $\underline{t}=(t_{1},\ldots,t_{k})$ and write $\phi$ on this basis as

At any point of the graph $(\underline{t},\phi(\underline{t}))$, each direction $v_{i}$ gives a slice of the graph given by the graph of the function $\phi_{i}$ given by

(with $t$ sufficiently small). These slices are called the sub-slices of the graph $\phi$ at the point $(\underline{t},\phi(\underline{t}))$. If $E^{s}$ denotes the space spanned by all the eigenvectors whose respective eigenvalues have modulus smaller than 1, and $E^{u}$ denotes the space spanned by all the eigenvectors whose respective eigenvalues have modulus bigger than 1, then the slices of the graph given by $E^{s}$ and $E^{u}$ are exactly the stable and the unstable slices defined in the previous sub-section. In this paper we discuss how to calculate the box dimension of each sub-slice of the graph. The case we deal in this subsection is when the absolute value of the eigenvalue is bigger than the contraction rate $\lambda$ of the skew product. The following proposition ensures that the box dimensions of these sub-slices equal one.

The function $\phi$ being $C^{1}$ in the direction $v_{i}$ implies that the slice given by the graph of $\phi_{i}$ is a $C^{1}$ curve and, hence, has box dimension one. We end this subsection noting that this applies to all unstable sub-slices since, in this case, the eigenvalues have modulus bigger than 1, while $\lambda<1$.

In this section, we consider the case $k=2$, that is, the skew product is of the form

In this case, at any point of the graph there is only one stable slice and we discuss how to calculate its box dimension following Kaplan et al. [6]. This will be used in Section 3 to calculate the box dimension of all stable sub-slices in the higher-dimensional case associated to directions $v_{i}$ with $|B_{i}|<\lambda$. Using the notation of the previous subsection, when $k=2$ the hyperbolic matrix $A$ has two eigenvalues $B_{1}<1$ and $B_{2}>1$. If $v_{1}$ is the eigenvector associated to $B_{1}$, then the stable slice at any point $\underline{t}\in\mathbb{T}^{k}$ is given by the graph of the function $\phi_{1}$ defined by

Letting $q(t)=p(tv_{1}+\underline{t})$, the stable slice is given by the graph of the function

The following theorem calculates the box dimension of this stable slice:

Actually, the conclusion holds for every function $q$ that is quasi-periodic and satisfies hypothesis (H1) (see Definitions 2.10 and 2.11). In this subsection, we explain the proof of this statement. First, we recall the necessary definitions and explain their consequences.

Even though $q$ is not periodic it is possible to define a formal Fourier series (see Corduneanu [2]) such that

where

There is only a countable set of $a\in\mathbb{R}$ such that $q_{a}\neq 0$.¹¹1Notice that if $q$ was periodic this definition would coincide with the standard definition of Fourier Series. The main idea to prove Theorem 2.9 is to formally consider

that contains the function $f$ but also the negative part of the series. Ignoring the fact that the function $g$ may not be well defined, if we try to calculate its Fourier series coefficients we get:

Regardless of the lack of rigor in the calculation above, all that is truly necessary is that

converges. The Hypothesis (H1) of Kaplan et al. [6] gives a sufficient condition to ensure this convergence.

Although this condition may appear to be too technical, Kaplan et al. [6] proved that if $p$ is a $C^{3}$ function then $q$ will satisfy (H1) (see Proposition 3.3 for our version in higher dimensions). So now that $g_{\sigma}$ is well defined, there are two possible cases: either 1) $g_{\sigma}=0$ for every $\sigma$, and in this case hypothesis (H1) ensures that $g(t)\equiv 0$ as shown in Proposition 2.2 from Kaplan et al. [6], and, hence, that

This ensures that $f$ is $C^{1}$ by a similar argument as in Proposition 2.8 (see Lemma 2.1 in [6]); or 2) there is a Fourier coefficient $g_{\sigma}\neq 0$ and it can be used to ensure the existence of $C_{1}>0$ such that

for every interval $J$ with $L=\text{length}(J)$ small enough. Indeed, the following proposition is the main step in proving this.

Kaplan et al. [6] use this proposition for any translation of the form $\gamma(t)=f(t+\theta)$, since the variation of $f(t+\theta)$ on $[0,2\pi n/\rho]$ equals the variation of $f(t)$ on $[\theta,2\pi n/\rho+\theta]$, so they can cover any interval of the domain. Using Lemma 2.6 and Lemma 2.7 in [6] they shown that for large enough $k,n\in\mathbb{N}$ there is $0<C_{0}<|g_{\sigma_{0}}|$ such that

Setting $C_{1}$ as $|g_{\sigma_{0}}|-C_{0}$ and using Proposition 2.13 we obtain

whenever $\text{length}(J)\leq(2\pi n/\sigma_{0})B^{-k}$. For the upper bound the fact that $q$ and $q^{\prime}$ are bounded let us obtain $C_{2}>0$ such that

if $L=\text{length}(J)$ is small enough (see Proposition 2.9 in [6]). Then Proposition 2.3 ensures that

After calculating the box dimensions of the stable and unstable slices, it is possible to calculate the box dimension of the graph of $\phi$. We expand some details in the proof given in [6].

In Proposition 2.8 we proved that if $\lambda<|B_{i}|$ then the box dimension of the graph of $\phi_{i}$ is 1. When $|B_{i}|<\lambda$, the function $\phi_{i}$ can be written in the form of the function $f$ in Theorem 2.9. Indeed, letting

it follows that

Thus, the argument of subsection 2.4 can also be applied for $\phi_{i}$, using the function

associated to $q_{i}$ as in (3), and we obtain the following theorem.

The hypothesis on $q_{i}$ are satisfied with enough regularity on $p$. Indeed, $q$ will be almost periodic as long as $p$ is continuous and the Hypothesis (H1) is satisfied when $p$ is $C^{k+1}$. This is proved in the next results but analogous results in the case $k=2$ are announced in [6].