Visible part of dominated self-affine sets in the plane

For $e\in S^{1}$, let $\ell(e)$ denote the half line starting from origin and propagating to direction $e$. That is $\ell(e)=\{te:t\geq 0\}$. For a compact set $E\subset\mathbb{R}^{2}$ the visible part of $E$ in direction $e\in S^{1}$ is the set of points in $x\in\mathbb{R}^{2}$ that satisfy

This set is denoted by $\operatorname{Vis}^{e}E$. Let $\operatorname{proj}^{e}$ denote the orthogonal projection along the direction $e$. (Note that $\operatorname{Vis}^{e}E$ may be different from $\operatorname{Vis}^{-e}E$, but $\operatorname{proj}^{e}E=\operatorname{proj}^{-e}E$ always.) Consider the Hausdorff dimension of the visible part of a compact set $E$. If $\operatorname{dim_{H}}E<1$ then $\operatorname{dim_{H}}\operatorname{proj}^{e}E=\operatorname{dim_{H}}E$ for almost all $e\in S^{1}$ by Marstrand’s projection theorem [16]. Since $\operatorname{Vis}^{e}E\subset E$ and $\operatorname{proj}^{e}\operatorname{Vis}^{e}E=\operatorname{proj}^{e}E$, it follows that $\operatorname{dim_{H}}\operatorname{Vis}^{e}E=\operatorname{dim_{H}}E$ for almost all $e\in S^{1}$. If $\operatorname{dim_{H}}E\geq 1$, then still we have that $1\leq\operatorname{dim_{H}}\operatorname{Vis}^{e}E$ for almost all $e\in S^{1}$, but the upper bound $\operatorname{dim_{H}}\operatorname{Vis}^{e}E\leq\operatorname{dim_{H}}E$ should no longer be optimal for most $e\in S^{1}$. The visibility conjecture states that $\operatorname{dim_{H}}\operatorname{Vis}^{e}E=1$ for almost all $e\in S^{1}$. Obviously one can not hope this to hold for all directions, since a graph of a function can have dimension greater than $1$ for example. Further, an example of Davies and Fast [6] shows that $\operatorname{dim_{H}}\operatorname{Vis}^{e}(K)=2$ is possible for a dense $G_{\delta}$ set of directions. This is the furthest one can go, since recently Orponen [20] showed that it is impossible to have $\operatorname{dim_{H}}\operatorname{Vis}^{e}(K)=2$ for set of directions of positive measure. It is rather easy to see that the visibility conjecture is false for the box counting dimension and thus for the Assouad dimension as well. This follows, since a countable set equals to its visible part for almost all directions and there exist compact countable sets with full box dimension. For example, one can simply consider $K=A\times A$, where $A=\{0\}\cup\{(S_{n})^{-1}\}_{n=1}^{\infty}$ and $S_{n}=\sum_{k=1}^{n}1/k$. For details, see Example 5.2.

The visibility conjecture has been confirmed in a few special cases: Järvenpää et.al. [8] proved the conjecture for quasi-circles, Arhosalo et al. [1] confirmed that for fractal percolation the conjecture holds almost surely, and Falconer and Fraser [7] showed that the conjecture holds for self-similar sets satisfying a projection condition and the open set condition so that the open set can be chosen to be convex. In all these cases, the authors actually verified the conjecture for the box dimension and for all directions $e\in S^{1}$. See also the recent work of Järvenpää et.al. [9].

One obvious variant of the problem is to consider the visible set from a given point instead of a direction. O’Neil [18] showed that for compact connected subsets of $\mathbb{R}^{2}$, the Hausdorff dimension of the visible part from a point $x\in\mathbb{R}^{2}$ is strictly less than the Hausdorff dimension of the original set, and it is uniformly bounded away from $2$, for almost all viewpoints $x$. An other related problem is to determine when $\operatorname{Vis}^{e}E=E$. Orponen [19] showed that if $\operatorname{dim_{H}}E>1$, then the set of directions for which $\operatorname{Vis}^{e}E=E$ has Hausdorff dimension at most $2-\operatorname{dim_{H}}E$. On the other hand, it follows from the main result of [22], that if $\operatorname{dim_{B}}E<1/2$, then $\operatorname{Vis}^{e}E=E$ holds outside a set of directions of box dimension $2\operatorname{dim_{B}}E$. For other related results, see for example [5, 4, 23].

In this paper I study the visible parts of self-affine sets. Domination and projection condition are standing assumptions throughout the paper. Theorem 2.4 is the main result and it says that the Assouad dimension of the visible part equals to $1$ for all directions outside the set of the limit directions given by the affine dynamics. This theorem then has several corollaries. Corollary 4.3 says that for dominated self-affine carpets the Assouad dimension of the visible part equals to $1$ for all but two exceptional directions (that span the same line). Corollary 4.4 says that if the self-affine system satisfies the strong cone separation, then the Assouad dimension of the visible part equals to $1$ for almost all directions. These results can be seen as rather strong, considering how easily the Assouad dimension jumps up in different situations. For example, it is well known that the fractal percolation has equal Hausdorff and box dimension $<2$ but full Assouad dimension, and Assouad dimension also tends to be maximal in projections in a way that is impossible for Hausdorff or box dimension [21]. Corollary 4.5 studies the case where the limit directions of the cylinders do not overlap too much, and states that the Hausdorff dimension of the visible part equals to $1$ for all directions in this case.

The purpose of this section is only to fix the setting of the paper and state the main result. In the next sections, along the course of the proof, I give more insight by explaining the geometry behind the assumptions and the result.

Throughout the paper, a direction means a unit vector $e\in S^{1}$ and orientation is an element of the projective space $\mathbb{P}^{1}$, that is, the metric space of lines in $\mathbb{R}^{2}$ that go through origin, and where the distance is measured by the angle between the lines. For a vector $e\in\mathbb{R}^{2}$, let $\left<e\right>=\{te:t\in\mathbb{R}\}$ denote the corresponding element of the projective space. It is sometimes more intuitive to think $S^{1}$ as a set of angles instead of unit vectors. This justifies the use of notations $\langle\theta\rangle:=\{t(\cos\theta,\sin\theta):t\in\mathbb{R}\}$, $\operatorname{proj}^{\theta}:=\operatorname{proj}^{(\cos\theta,\sin\theta)}$, $\operatorname{Vis}^{\theta}:=\operatorname{Vis}^{(\cos\theta,\sin\theta)}$, and $\ell(\theta):=\ell((\cos\theta,\sin\theta))$ for $\theta\in\mathbb{R}$.

Let $A\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ be an invertible linear map, so that $A(B(0,1))$ is not a ball. Then $A(B(0,1))$ is an ellipse whose semiaxes have different lengths. Let $\alpha_{1}(A)$ be the length of the longer one of the semiaxes and let $\alpha_{2}(A)$ be the length of the shorter one. Equivalently $\alpha_{k}(A),k=1,2$ are the square roots of the eigenvalues of $A^{T}A$ (ordered so that the larger is $\alpha_{1}$). Also, set $\vartheta_{1}(A)\in\mathbb{P}^{1}$ to be the orientation of the longer semiaxes of $A(B(0,1))$. That is, $\vartheta_{1}(A)=\left<A\eta_{1}(A)\right>$, where $\eta_{1}(A)$ is the normalized eigenvector of $A^{T}A$ associated to the eigenvalue $\alpha_{1}(A)^{2}$. Likewise, set $\vartheta_{2}(A)=\left<A\eta_{2}(A)\right>$, where $\eta_{2}(A)$ is the normalized eigenvector corresponding to $\alpha_{2}(A)^{2}$. It is a basic fact that $\vartheta_{1}(A)\perp\vartheta_{2}(A)$ and $\eta_{1}(A)\perp\eta_{2}(A)$.

Let $\{A_{i}\}_{i=1}^{\kappa}$ be a collection of contractive invertible linear maps, let $\{c_{i}\}_{i=1}^{\kappa}$ be a collection of vectors in $\mathbb{R}^{2}$, and let $\varphi_{i}(x)=A_{i}x+c_{i}$, for all $i\in\{1,\ldots,\kappa\}$. It standard that there exists a unique compact set $E$ satisfying

The set $E$ is called self-affine.

Set $\Sigma^{*}=\bigcup_{k\in\mathbb{N}}\{1,\ldots,\kappa\}^{k}$ and $\Sigma=\{1,\ldots,\kappa\}^{\mathbb{N}}$. Write $\Sigma^{n}$ for $\{1,\ldots,\kappa\}^{n}$ even though this is abusing the notation. Let $|\mathtt{i}|$ denote the length of the word $\mathtt{i}$. That is, $|\mathtt{i}|=n$ whenever $\mathtt{i}\in\Sigma^{n}$ and $|\mathtt{i}|=\infty$, when $\mathtt{i}\in\Sigma$. For $\mathtt{i},\mathtt{j}\in\Sigma$, let $\mathtt{i}\wedge\mathtt{j}$ be the longest common beginning of $\mathtt{i}$ and $\mathtt{j}$ and define a distance function $\varrho$ in $\Sigma$ by setting $\varrho(\mathtt{i},\mathtt{j})=2^{-|\mathtt{i}\wedge\mathtt{j}|}$, with the interpretation $2^{-\infty}=0$. This makes $(\Sigma,\varrho)$ a compact metric space.

For quantities $x$ and $y$, usually depending on $\mathtt{i}$, the notation $x\lesssim y$ means that there is a constant $C>1$, that may depend on the self-affine set $E$, so that $x\leq Cy$. Further, $x\approx y$ means that $x\lesssim y$ and $y\lesssim x$.

For $\mathtt{i}=(i_{1},i_{2},\ldots,i_{n})\in\Sigma^{*}$, let $A_{\mathtt{i}}=A_{i_{1}}A_{i_{2}}\dots A_{i_{n}}$ and for the sake of brevity, write $\alpha_{k}(\mathtt{i})=\alpha_{k}(A_{\mathtt{i}})$ and $\vartheta_{k}(\mathtt{i})=\vartheta_{k}(A_{\mathtt{i}})$ for $k=1,2$. Similarly, also write $\varphi_{\mathtt{i}}=\varphi_{i_{1}}\circ\varphi_{i_{2}}\circ\dots\circ\varphi_{i_{n}}$. The line $\vartheta_{1}(\mathtt{i})$ is called the orientation of the cylinder $\varphi_{\mathtt{i}}(E)$, because the cylinder $\varphi_{\mathtt{i}}(E)$ is “close” to being a line segment that is parallel to the line $\vartheta_{1}(\mathtt{i})$, at least when $|\mathtt{i}|$ is large. This phenomena is examined in more detail in Proposition 3.4. As usual, let $\pi\colon\Sigma\to E$ be the canonical projection defined by

From time to time, the set $\varphi_{\mathtt{i}|_{n}}(E)$ is also denoted by $E_{\mathtt{i}|_{n}}$. The system $\{A_{i}\}_{i=1}^{\kappa}$ is called dominated, or said to satisfy dominated splitting, if there are constants $\tau>1$ and $n_{0}\in\mathbb{N}$, so that $\alpha_{1}(\mathtt{i})>\tau^{|\mathtt{i}|}\alpha_{2}(\mathtt{i})$ for all $\mathtt{i}\in\Sigma^{*}$, with $|\mathtt{i}|\geq n_{0}$. Domination ensures the existence of the limit orientation for all symbols $\mathtt{i}\in\Sigma$. The next lemma records this fact along with other useful properties of the limit orientations.

In addition to domination, a crucial assumption in this paper is the following projection condition.

As said, the purpose is to study the dimension of the visible part. I assume that the reader is familiar with basic notions of dimension. The Hausdorff dimension is denoted by $\operatorname{dim_{H}}$, the box dimension by $\operatorname{dim_{B}}$, and the Assouad dimension by $\operatorname{dim_{A}}$. The definitions of Hausdorff and box dimension one can find from almost any text book of fractal geometry or geometric measure theory (see for example [17]), and for Assouad dimension one can check for example [14]. If the reader is not interested in the Assouad dimension, then I just want to remark that the results are new also for the Hausdorff dimension and that the versions with Assouad dimension are just stronger since $\operatorname{dim_{H}}K\leq\operatorname{dim_{A}}K$ for all sets $K$.

The main result is the following theorem.

The proof goes via weak tangents, defined as follows. Let $F_{n}$ be a sequence of compact sets in $\mathbb{R}^{2}$. Say that $F_{n}$ converges to a compact set $F\subset\mathbb{R}^{2}$ in $B(0,R)$, if

and

For $x\in\mathbb{R}^{2}$ and $r>0$ we write $M_{x,r}$ for the magnification function that shifts $x$ to origin and scales with factor $r^{-1}$. That is

Let $X\subset\mathbb{R}^{2}$ be compact. Then $W\subset B(0,1)$ is said to be a weak tangent of $X$, and written $W\in\operatorname{Tan}(X)$, if $M_{x_{n},r_{n}}(X)$ converges to $W$ in $B(0,1)$ for some sequences $(x_{n})\subset X$ and $r_{n}\searrow 0$. It is typical to consider the weak tangents as subsets of the unit ball (or the unit square), but this is just a convenient choice. One could as well consider the convergence in $B(0,R)$ for any fixed $R>0$ or for all $R>0$ to allow the weak tangents to be unbounded as well.

Assuming Proposition 4.1, which deals with the dimension of the weak tangents of the visible part, its proof of Theorem 2.4 is rather simple.

Considering the proof of Proposition 4.1, it is obvious that if $W$ is a weak tangent of $\overline{\operatorname{Vis}^{e}E}$ with $M_{x_{n},r_{n}}\overline{\operatorname{Vis}^{e}E}\to W$ in $B(0,1)$, then (by passing to a subsequence if necessary) it also holds that $(M_{x_{n},r_{n}}E)\cap B(0,1)$ converges to a weak tangent, say $T$, of $E$. Of course $W\subset T$, but unfortunately, it is not generally true that $W\subset\overline{\operatorname{Vis}^{e}T}$ or $W\supset\overline{\operatorname{Vis}^{e}T}$. In particular, one can not just take the weak tangent $T$ of $E$ of maximal dimension and expect it to have anything to do with the weak tangent of $\overline{\operatorname{Vis}^{e}E}$ of maximal dimension. See example 5.1. Instead, the strategy is to use the structure of the self-affine set and the weak tangents obtained in Section 3 to show that $W\cap\overline{\operatorname{Vis}^{e}T}$ can be covered by graphs of few well behaving functions and that $W\setminus\overline{\operatorname{Vis}^{e}T}$ can be covered by a countable collection of lines. The arguments about visibility rely heavily on the projection condition.

The visibility conjecture asks if $\operatorname{dim_{H}}\operatorname{Vis}^{\theta}(E)=1$ for almost all $e\in S^{1}$. So, the remaining step to confirm the conjecture in some special case, is to show that $\vartheta_{1}(\Sigma)$ is of measure zero (or technically, that the set $e\in\{\langle e\rangle\in\vartheta_{1}(\Sigma)\}$ is of measure zero). Theorem 4.4 deals with this in the case where the self-affine system also satisfies the strong cone separation. In section 5, I give an example where the visible part has large dimension in directions $\vartheta_{1}(\Sigma)$, showing that Theorem 2.4 is sharp.

This section deals with the structure of the weak tangent sets of self-affine sets satisfying the projection condition and dominated splitting. Recall that no separation conditions are required. The structure of tangents of self-affine sets under separation conditions has been studied in [2, 10, 15, 11] for example.

To study the local structure of self-affine sets it is convenient to approximate the cylinders $\varphi_{\mathtt{i}}(E)$ by rectangles. The domination ensures that $\alpha_{2}(\mathtt{i})/\alpha_{1}(\mathtt{i})\to 0$ uniformly as $|\mathtt{i}|\to\infty$. Therefore the approximation of $\varphi_{\mathtt{i}}(E)$ can be done with a “very narrow” rectangle if $|\mathtt{i}|$ is large. This motivates the following definition.

After the previous lemma, it is intuitive that the weak tangent contains lines and half lines pointing in different directions. Due to the obvious connection, it is natural to call such sets Kakeya type sets.

In this section I finish the proof of Theorem 2.4. As discussed earlier, all that is left to do is to prove Proposition 4.1. After this it is time to focus on the Corollaries 4.3, 4.4, and 4.5, that deal with the size of the exceptional set of directions, verifying the visibility conjecture in different special cases.

As mentioned earlier the strategy is to cover $W$ with graphs of nice functions and a collection of vertical lines. With this in mind, recall some basic facts. For $f\colon\mathbb{R}\to\mathbb{R}$, let $\mathcal{G}(f)$ denote the graph of $f$. That is, $\mathcal{G}(f)=\{(x,y)\in\mathbb{R}^{2}:f(x)=y\}$. A function $f\colon\mathbb{R}\to\mathbb{R}$ satisfying $f(t)-f(s)\leq L(t-s)$ for some $L>0$ and for all $t\geq s$ is called semi-decreasing. Also, $f$ is said to be semi-increasing if $-f$ is semi-decreasing and $f$ is called semi-monotone if it is semi-decreasing or semi-increasing. The aim is to use graphs of semi-monotone functions for the coverings, so the first thing to do is to check that their graphs are nice enough.