Fractal dimension from the information that exact overlaps leave behind

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Family 148 · Dynamical systems and ergodic theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

A self-similar fractal is built by repeatedly choosing a shrinking copy of the whole picture. If the copies stay separate, counting choices and measuring their shrinkage predicts the dimension. When copies overlap, different sequences of choices may describe the same small piece, and that count can overstate how much information survives.

Family 148 claims a complete formula for the dimension of finite self-similar probability measures on the real line. The numerator is the rate at which distinct composed maps retain information; the denominator is the average logarithmic contraction. Their ratio is capped at the ambient dimension one.

The claim allows exact coincidences, arbitrarily close distinct copies, unequal contraction ratios, and orientation-reversing maps. It imposes no arithmetic restriction on the coefficients and no quantitative separation between different pieces.

The result concerns dimension. A measure of dimension one need not have a density with respect to length, so the formula is not an absolute-continuity theorem.

2. Background and essential definitions

Randomly applying shrinking affine maps

Take a finite nonempty indexed family

$$\varphi_i(x)=r_i x+t_i, \qquad t_i\in\mathbb R,\quad 0<|r_i|<1,$$

and probabilities $p_i>0$ summing to one. A negative $r_i$ reverses orientation as well as shrinking distances. Different indices are allowed to name the same map.

Choose independent symbols $I_1,I_2,\ldots$ with probabilities $p_i$. The limit of

$$\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0)$$

exists and has the self-similar probability law $\mu$. Equivalently,

$$\mu=\sum_i p_i(\varphi_i)_*\mu,$$

where the pushforward $(\varphi_i)_*\mu$ is the distribution obtained by applying $\varphi_i$ to a $\mu$-distributed point. The compact set of all possible limits is the attractor $K$.

Count complete maps, not symbolic addresses

A word $w=i_1\cdots i_n$ describes the composed map $\varphi_w$. An exact overlap occurs when two distinct words of the same positive length define exactly the same affine map. Equality includes both the signed slope and the translation.

Let

$$G_n=\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}.$$

This random variable takes values in the finite set of composed maps. If several words produce one map, their probabilities are added before its entropy is computed.

Using base-two logarithms, the Shannon entropy of a finite random variable $Z$ is

$$H(Z)=-\sum_z\mathbb P(Z=z)\log_2\mathbb P(Z=z).$$

The random-walk entropy rate and average contraction exponent are

$$h_{\mathrm{RW}}=\lim_{n\to\infty}\frac{H(G_n)}n, \qquad \chi=-\sum_i p_i\log_2|r_i|>0.$$

The entropy limit exists by subadditivity: a longer composition is determined by two independent shorter compositions. The ratio does not depend on the logarithm base, provided both quantities use the same one.

Dimension of a measure

Hausdorff dimension quantifies how the number and sizes of small covering intervals scale. For a probability measure, one can ask for the smallest dimension of a Borel set carrying positive mass, or of one carrying full mass. These conventions agree here because self-similar measures are exact dimensional by the earlier Feng–Hu theorem.

Exact dimensionality means that, for some constant $d$,

$$\lim_{s\downarrow0}\frac{\log\mu(B(x,s))}{\log s}=d \qquad\text{for }\mu\text{-almost every }x.$$

The interval $B(x,s)$ has center $x$ and radius $s$. Informally, typical mass scales approximately like $s^d$. The constant $d$ is the dimension in the claimed formula.

3. Development history and previous partial results

Moran’s work and Hutchinson’s iterated-function-system framework established dimension formulas when the smaller copies satisfy suitable separation conditions. The basic ratio of symbolic entropy to logarithmic contraction is natural in that setting.

Overlaps create a subtler problem. The exact-overlaps conjecture asks whether a dimension drop below the natural upper bound must reflect genuine equality between composed maps. The entropy-rate version incorporates the information already lost through such equalities and asks whether anything further can lower dimension.

Bernoulli convolutions became a central testing ground. They are distributions of random sums $\sum_{j\geq0}\varepsilon_j\lambda^j$, with independent signs and $0<\lambda<1$. Erdős proved singularity for certain arithmetic parameters, while Solomyak established absolute continuity for almost every parameter in $(1/2,1)$. Garsia introduced the discrete entropy rate in this context. These developments exposed the interaction between overlaps, arithmetic, dimension, and densities.

Hochman showed that a dimension drop forces superexponential concentration of cylinder maps, using inverse theorems for entropy growth. This yields powerful conclusions under appropriate separation or arithmetic assumptions. Breuillard and Varjú developed approximation methods for Bernoulli convolutions, and Varjú obtained full dimension for every transcendental parameter in $(1/2,1)$.

Rapaport established the no-exact-overlap formula for algebraic contraction ratios and arbitrary real translations, including signed unequal ratios. Rapaport and Varjú treated important homogeneous three-map cases and an entropy-rate formula with rational translations. Feng and Feng handled algebraic translations with a common signed contraction. Bárány and Verma supplied a version with exact collisions under weak exponential separation.

A key obstacle remained: absence of exact overlaps does not guarantee any useful minimum distance between distinct maps. Baker and, independently, Bárány and Käenmäki constructed systems without exact overlaps whose distinct cylinders become superexponentially close at prescribed rates. The new manuscript claims to control such closeness without adding a separation assumption.

4. What the new manuscripts claim

The main theorem states that every finite indexed system described above satisfies

$$\boxed{\displaystyle \dim_H\mu=\min\left\{1,\frac{h_{\mathrm{RW}}}{\chi}\right\}.}$$

Exact overlaps, repeated generators, negative contraction ratios, and unequal contraction ratios are all permitted. The probabilities are strictly positive. The theorem includes the threshold $h_{\mathrm{RW}}=\chi$ and the regime where the numerator is larger.

The no-overlap consequence

If there are no exact overlaps, distinct words of a given length remain distinct complete maps. Consequently,

$$H(G_n)=nH(p),\qquad H(p)=-\sum_i p_i\log_2p_i,$$

and the formula becomes $\dim_H\mu=\min\{1,H(p)/\chi\}$.

The manuscript also explains why forbidding exact equalities between words of equal length is equivalent, for these nonzero contractions, to forbidding them between arbitrary distinct nonempty words. Thus its convention does not hide a separate class of different-length coincidences.

For the attractor, under the no-exact-overlap hypothesis, it claims

$$\dim_H K=\min\{1,s_*\}, \qquad \sum_i|r_i|^{s_*}=1.$$

The number $s_*$ is the similarity dimension. This simple set formula is stated with the no-overlap condition; the unrestricted main theorem is formulated for measures using their map entropy rate.

For a positive common contraction $\lambda$, the denominator simplifies to $\log_2(1/\lambda)$. The paper records the all-weight consequence for homogeneous systems, including the three maps $\lambda x,\lambda x+1,\lambda x+t$, whenever they have no exact overlaps.

A simple example of why addresses overcount

Use the map $x/3$ twice, each indexed copy with probability $1/4$, and the map $x/3+2/3$ with probability $1/2$. After merging the repeated map, these are the ordinary equal-weight Cantor choices.

The original three-symbol entropy is $3/2$ bits per step, but the surviving map entropy is one bit per step. The formula gives $\log 2/\log 3$, the dimension of the usual Cantor probability measure. The extra symbolic label carries no geometric information. The main theorem claims that the same accounting principle remains exact for much more complicated collisions between long words.

5. Why the advance matters

Provisional importance: 5/5. The claim supplies a universal dimension formula in a central class of overlapping fractal measures. It would remove both arithmetic and quantitative separation assumptions while retaining exact coincidences in the correct information quantity.

The formula gives a precise interpretation of dimension loss: after accounting for the ceiling imposed by the real line, the relevant loss is the entropy lost when different addresses become the same map. Distinct maps may be astonishingly close, but the theorem claims that this closeness causes no additional asymptotic dimension deficit.

That is a structural conclusion, not necessarily a computational shortcut. Determining the entropy rate of a complicated system may still be difficult. The importance lies in identifying the exact quantity that governs dimension, including in systems where a simple separation argument cannot work.

6. The main idea

Look for information hidden below the current resolution

Observe a finite distribution only through a grid of small intervals. Several distinct values may fall in one cell and become indistinguishable. The difference between exact entropy and grid-cell entropy measures the information hidden below that resolution.

The manuscript develops an estimate that detects this hidden information through equal-weight two-point distributions at finer scales. Such a pair is the simplest unit of uncertainty: a fair choice between two locations. Crucially, the estimate does not depend on the smallest positive distance between support points.

Averaging over translations of the grid avoids making the argument depend on a fortunate placement of cell boundaries.

Freeze contraction by conditioning on symbol counts

Unequal contractions complicate the relation between symbolic length and physical size. The proof groups symbols into blocks and conditions on how often each symbol occurs in a block. These counts fix the product of the signed contraction ratios, even though the order of the symbols can vary.

With the slope fixed, the translation identifies the complete affine map. This is why the theorem tracks full maps rather than just the endpoint $G_n(0)$: before conditioning, two maps could share that endpoint and still have different slopes.

Recording symbol counts has a controlled entropy cost. The remaining translation entropy can therefore be bounded using the map entropy rate, including when different words coincide exactly. A hypothetical dimension smaller than both one and $h_{\mathrm{RW}}/\chi$ would leave a linear amount of hidden two-point uncertainty across a finite band of finer scales.

Make those hidden choices produce incompatible gains

If the measure’s dimension is below one, its entropy is not saturated at the maximum allowed by the line. The manuscript shows that each suitable two-point choice then produces a positive entropy gain against the still-unobserved tail of the random composition.

The fine-scale bands may extend to arbitrarily deep resolutions. Instead of asserting a bound on their endpoints, the proof chooses later block lengths after earlier endpoints are known. It arranges disjoint windows so the gains can be added without counting the same information twice.

The total entropy increase over the relevant scale intervals is bounded. Yet the proposed deficit supplies a positive contribution from each of arbitrarily many disjoint bands. The resulting contradiction is the mechanism claimed to prove the lower bound. The upper bound comes from grouping equal composed maps and applying ordinary covering estimates.

7. Scope, limitations, and open questions

The theorem is for finite systems of affine similarities on the real line. It does not automatically extend to higher-dimensional self-similar systems, general self-affine maps, or nonlinear contractions.

Dimension one does not imply absolute continuity. A measure can have full Hausdorff dimension while being supported, in the measure-theoretic sense, on a set of zero length. The formula does not assert an integrable density, a smooth density, or a quantitative rate of convergence of scale entropies.

The entropy rate itself is a limit over exact composed-map laws. The paper does not give a universal effective procedure or error bound for computing it. Nor does its no-overlap corollary provide a positive separation estimate between distinct maps.

The assumptions $0<|r_i|<1$ and finiteness of the family remain part of the statement. Repeated maps are explicitly allowed and their probabilities are combined at the map level; they must not be treated as independent geometric branches when applying the formula.

8. Sources and further reading

Hutchinson’s Fractals and self-similarity gives the foundational construction. Feng and Hu’s exact-dimensionality theorem is an explicit input to the new manuscript. Hochman’s entropy and overlap theorem and Rapaport’s algebraic-contraction result describe major preceding advances.

For the obstruction to inferring separation from the absence of exact overlaps, see Baker’s superexponentially close cylinder construction. Varjú’s survey of entropy rates states the conjectural framework. The distinction between dimension and singularity also appears in family 153, devoted to Bernoulli convolutions.

Source manuscripts

• M1. OpenAI, The entropy-rate dimension formula for self-similar measures on the line (September 24, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.