Similarity Dimension And The Open Set Condition

From Contraction Ratios to a Dimension Candidate

A finite family of similarities assigns a contraction ratio to each map. These ratios suggest a dimension, but they do not by themselves determine the Hausdorff dimension of the resulting set: the maps might overlap so extensively that several pieces contribute less geometric complexity than the ratios predict. The open set condition is a precise separation hypothesis under which the candidate is valid.

Work in Euclidean space \(\mathbb{R}^d\), with its usual distance and Lebesgue measure \(m\). Let \(S_1,\ldots,S_N:\mathbb{R}^d\to\mathbb{R}^d\) be similarities with ratios \(0<r_i<1\), meaning that \(|S_i(x)-S_i(y)|=r_i|x-y|\) for all \(x,y\). A word \(w=(i_1,\ldots,i_k)\) specifies \(S_w=S_{i_1}\circ\cdots\circ S_{i_k}\) and \(r_w=r_{i_1}\cdots r_{i_k}\). The compact invariant set \(K\), also called the attractor, consists of the limits \(S_{i_1}\circ\cdots\circ S_{i_k}(x_0)\) along infinite sequences of indices. The limit exists and is independent of \(x_0\): the distance between two such iterates is at most \(r_{\max}^k|x_0-y_0|\), where \(r_{\max}=\max_i r_i<1\). In particular \(K=\bigcup_{i=1}^N S_i(K)\).

The OSC separates the first-generation open pieces. Their further images are separated as well whenever their words are incomparable (neither word is an initial segment of the other). The attractor itself need not consist of disjoint pieces: boundaries may meet. This distinction is why the condition is stated for open sets.

Why the Open Set Condition Gives the Dimension

We use the Hausdorff dimension convention from the earlier material on Hausdorff measure: \(\dim_H K\) is the critical exponent at which the \(t\)-dimensional Hausdorff measure changes from infinite to zero. For the upper bound, cylinders give covers whose diameters shrink by their word ratios. For the lower bound, the OSC lets us distribute mass among cylinders in proportion to \(r_w^s\), and prevents too many cylinders of a given scale from clustering in one small ball.

Proof. First, \(K\) is bounded and compact. Indeed, its points are limits of images of a fixed point under words, and the contraction ratios are bounded above by \(r_{\max}<1\); the coding map from the compact sequence space \(\{1,\ldots,N\}^{\mathbb{N}}\) to \(\mathbb{R}^d\) is continuous. Write \(D=\operatorname{diam}K\). At level \(k\), the sets \(S_w(K)\), for words \(w\) of length \(k\), cover \(K\), and \(\operatorname{diam}S_w(K)=r_wD\). For any \(t>s\), \(\sum_i r_i^t<1\), so the sum of the \(t\)-powers of the diameters in this cover is $$ D^t\sum_{|w|=k}r_w^t =D^t\left(\sum_{i=1}^N r_i^t\right)^k\longrightarrow0. $$ The largest diameter is at most \(r_{\max}^kD\), which also tends to zero. Thus the \(t\)-dimensional Hausdorff measure of \(K\) is zero, and \(\dim_H K\le s\).

For the lower bound, construct a probability measure \(\mu\) on \(K\) that assigns mass \(r_w^s\) to each coded cylinder of word \(w\). On the sequence space, a cylinder \([w]\) consists of sequences beginning with \(w\). Set \(P([w])=r_w^s\). This assignment is consistent because \(\sum_i r_{wi}^s=r_w^s\sum_i r_i^s=r_w^s\). It defines a premeasure on finite unions of cylinders: refine any such union to a common word length, where consistency gives the same sum. Countable additivity whenever a union is still in this cylinder algebra follows from compactness: cylinders are both open and closed, and a countable cylinder cover of a compact cylinder has a finite subcover. The Caratheodory extension theorem therefore gives a probability measure on the sequence space. Push it forward by the coding map to obtain \(\mu\) on \(K\); then \(\mu(S_w(K))=r_w^s\) when counted by codes. If different codes represent the same point, this equality is understood as the mass of the coded cylinder before pushforward; the ball estimate below uses the sum of those cylinder masses and does not require distinct points to have unique codes.

We prove a uniform ball bound. Choose \(a>0\) and \(u\in U\) with \(B(u,a)\subset U\), and let \(D_U=\operatorname{diam}U<\infty\). Fix \(0<\rho<1\). Stop each infinite word at its first prefix \(w\) for which \(r_w\le\rho\). These stopping words are incomparable and satisfy \(r_{\min}\rho<r_w\le\rho\), where \(r_{\min}=\min_i r_i>0\): the parent has ratio greater than \(\rho\), and the last factor is at least \(r_{\min}\).

The open sets \(S_w(U)\) for stopping words are pairwise disjoint. To check this, take two distinct stopping words and remove their common initial segment; at the first differing index their images lie in disjoint first-generation sets, and applying the common preceding similarity preserves disjointness. Each \(S_w(U)\) contains a ball of radius \(a r_w>a r_{\min}\rho\). Also \(K\subset\overline U\): for any code and any \(u\in U\), its finite images of \(u\) lie in \(U\) and converge to the coded point. Consequently, if \(S_w(K)\) meets \(B(x,\rho)\), then \(S_w(U)\) lies inside \(B(x,(1+D_U)\rho)\). This follows by taking a point of intersection, approximating it by points of \(S_w(U)\), and using \(\operatorname{diam}S_w(U)\le D_U r_w\le D_U\rho\).

Lebesgue measure now bounds the number of stopping sets that can meet \(B(x,\rho)\). The balls of radius \(a r_{\min}\rho\) inside them are disjoint and lie in \(B(x,(1+D_U)\rho)\). Since a Euclidean ball of radius \(q\) has measure \(m(B(0,q))=m(B(0,1))q^d\), their number is at most $$ C_0=\left(\frac{1+D_U}{a r_{\min}}\right)^d. $$ The same bound holds if the radius-\(a r_{\min}\rho\) balls are open, since their boundaries have Lebesgue measure zero. Every coded point in \(B(x,\rho)\) belongs to a stopping cylinder whose attractor piece meets that ball. Each such cylinder has mass \(r_w^s\le\rho^s\). Summing over at most \(C_0\) cylinders gives \(\mu(B(x,\rho))\le C_0\rho^s\).

To apply the mass distribution principle, let \((E_j)\) be any countable cover of \(K\) by sets of diameter at most \(\delta<1/2\). For each \(E_j\) meeting \(K\), choose \(x_j\in E_j\cap K\). If \(\operatorname{diam}E_j>0\), then \(E_j\cap K\subset B(x_j,2\operatorname{diam}E_j)\), so its outer \(\mu\)-measure is at most \(C_0 2^s(\operatorname{diam}E_j)^s\). For a set of diameter zero, the same bound follows by enclosing it in balls of radii tending to zero; the ball estimate implies that its \(\mu\)-measure is zero. Countable subadditivity and \(\mu(K)=1\) yield $$ 1\le C_0 2^s\sum_j(\operatorname{diam}E_j)^s. $$ Thus the \(s\)-dimensional Hausdorff measure of \(K\) is positive, and \(\dim_H K\ge s\). Combining the two bounds proves the theorem. \(\square\)

What the Condition Does—and Does Not—Say

A common mistake is to treat the sum of the ratios as the dimension. For the middle-third construction, that calculation would give \(1/3+1/3=2/3\), and incorrectly report dimension \(2/3\). The ratios must instead be raised to the unknown exponent: \(2(1/3)^s=1\), which gives \(s=\ln2/\ln3\). A second mistake is to infer the dimension from this equation without checking overlaps. The theorem supplies a sufficient condition—the OSC—not a claim that every system lacking OSC has a dimension different from \(s\). Some overlapping systems can still have the candidate dimension, but it is not guaranteed by the ratio equation.

Thus, for a finite system with at least two maps, first solve \(\sum_i r_i^s=1\), then verify that there is a nonempty bounded open set whose images are contained in it and pairwise disjoint. Under those hypotheses the candidate equals \(\dim_H K\). Without such separation, the candidate remains a useful prediction, but further analysis of the overlaps is needed.