The entropy-rate dimension formula for self-similar measures on the line

1 Introduction

Let \Lambda be a finite nonempty alphabet, and let

The family is indexed: different symbols may specify the same map. Given a probability vector p=(p_i)_{i\in\Lambda} with p_i>0, its self-similar measure is the unique Borel probability measure satisfying

Equivalently, \mu is the law of \lim_{n\to\infty}\varphi_{I_1}\circ\cdots\circ\varphi_{I_n}(0), where the symbols I_j are independent with law p. The limit exists uniformly in the address: if r_{\max}=\max_i|r_i| and T_0=\max_i|t_i|, all coding limits have absolute value at most T_0/(1-r_{\max}).

For a word w=i_1\cdots i_n, write \varphi_w=\varphi_{i_1}\circ\cdots\circ\varphi_{i_n} and p_w=p_{i_1}\cdots p_{i_n}. An exact overlap is an equality \varphi_u=\varphi_v for distinct words of the same positive length. Equality here means equality of the complete affine maps, including both the signed slope and the translation. We allow all such overlaps.

The relevant entropy counts maps rather than addresses. Set

All logarithms below have base two. For a finite-valued random variable Z, its Shannon entropy is H(Z)=-\sum_z\P(Z=z)\log\P(Z=z), with 0\log0=0. Define the random-walk entropy rate and the Lyapunov exponent by

Indeed, a length-(n+m) map is a function of two independent maps with laws G_n,G_m, so H(G_{n+m})\le H(G_n)+H(G_m); subadditivity gives the displayed limit and infimum. Also 0\le h\le H(p):=-\sum_i p_i\log p_i. The ratio h/\chi is independent of the common logarithm base.

We use the lower Hausdorff dimension of a measure:

Feng–Hu’s exact-dimensionality theorem [7, Theorem 2.8] applies to finite self-similar systems without separation. Indeed, the maps preserve a sufficiently large compact interval, extend to contracting smooth diffeomorphisms, and have derivative norm and least singular value both equal to |r_i|. The system is therefore conformal, and its Bernoulli coding law is ergodic. These observations also cover negative ratios and repeated indexed maps. Thus, for the measure \mu above, there is a constant d such that

For an exact-dimensional measure this constant equals both \dim_H\mu and \inf\{\dim_H E:\mu(E)=1\}. To see the equivalence, restrict to countably many sets on which the local bounds s^{d+\varepsilon}\le\mu(B(x,s))\le s^{d-\varepsilon} hold uniformly for all sufficiently small s. The lower mass bound gives a full-measure union of sets of dimension at most d+\varepsilon by a covering argument; the upper mass bound gives dimension at least d-\varepsilon for every positive-mass set. Let \varepsilon\downarrow0. Consequently either Hausdorff measure-dimension convention gives the same statement below.

This proves the entropy-rate dimension conjecture, formulated as Conjecture 3 in Varjú’s survey [22]. When there are no exact overlaps, H(G_n)=nH(p), and the theorem gives the usual entropy-to-Lyapunov formula. Section 6 records this case, the attractor formula, and the homogeneous all-weight consequence, including the three-map family \{\lambda x,\lambda x+1,\lambda x+t\}. The theorem includes critical and supercritical entropy rates. It asserts dimension, without an assertion of absolute continuity.

1.1 Background and proof strategy

The dimension of a self-similar set is classical when its pieces are sufficiently separated. Moran’s construction [13] and Hutchinson’s iterated-function-system framework [11] give the similarity-dimension formula under the open set condition. Hutchinson also established the invariant-measure construction used above [11, Theorem 4.4(1)]. Removing separation leads to the exact-overlap problem: can dimension fall below its natural upper bound when distinct words never define the same map? The set version is associated with Simon [18]; see also the formulation and historical discussion in [22, Section 1]. The entropy-rate formulation addresses the additional loss of information when exact overlaps do occur.

Bernoulli convolutions provide an early and influential model for this question. These are the laws of \sum_{j\ge0}\varepsilon_j\lambda^j, where 0<\lambda<1 and the \varepsilon_j are independent equiprobable signs. Erdős proved singularity for reciprocal Pisot parameters in (1/2,1) [5]. A Pisot number is a real algebraic integer greater than one whose other conjugates have modulus less than one. Solomyak proved absolute continuity with an L^2 density for almost every parameter in (1/2,1) [19]; Peres–Solomyak later gave a simpler proof [14]. Garsia introduced the discrete entropy rate in this setting and related it to singularity [9, Theorem 1.2]. These results illustrate the sensitivity to overlaps and arithmetic; absolute continuity and Hausdorff dimension are distinct questions.

Hochman showed that \dim_H\mu<\min\{1,H(p)/\chi\} forces superexponential concentration of cylinder maps [10, Theorem 1.1], using inverse theorems for entropy growth under convolution. For unequal ratios his entropy formulation already keeps track of the full affine map, including its slope [10, Theorem 1.4]. The entropy-rate formula under weak exponential separation follows from his work; the version allowing exact collisions is proved in Bárány–Verma [3, Theorem 3.5]. For Bernoulli convolutions, Breuillard–Varjú developed approximation by algebraic parameters with controlled entropy [4]. Building on these and earlier entropy methods, Varjú proved full dimension for every transcendental parameter in (1/2,1) [21, Theorem 3]. Rapaport proved the no-exact-overlap formula for algebraic contraction ratios and arbitrary real translations, allowing signed unequal ratios [15, Theorem 2].

Rapaport–Varjú extended entropy and approximation methods to homogeneous three-map systems [16]. For equal weights, their results include an exceptional parameter set of Hausdorff dimension zero and the no-overlap formula when \lambda>2^{-2/3} [16, Corollary 1.4 and Theorem 1.8]. Rapaport–Varjú proved the entropy-rate formula, including exact overlaps, for rational translations and a positive common ratio [16, Theorem A.1]. Feng–Feng proved the no-exact-overlap formula for algebraic translations and a signed common ratio [6, Theorem 1.2]. The distinction in Theorem 1.1 is that the entropy rate of the full affine-map walk is sharp even when exact collisions occur, with no arithmetic or separation assumption.

Absence of exact overlaps does not itself supply a quantitative separation estimate. Baker [1, Theorem 1.3] and, independently, Bárány–Käenmäki [2, Theorem 2.1] constructed systems without exact overlaps whose distinct cylinders approach one another at arbitrarily prescribed superexponential rates. Thus the entropy carried by extremely close maps must be handled without bounding the smallest positive separation.

Our proof combines the entropy retained by this map walk with a finite-law estimate that detects information across arbitrarily fine scales. It uses translation averaging, developed by Wang [23, Section 4.1], and the entropy calculus and fair-pair arguments of Varjú [20, Section 2.2, Propositions 20–21, and Lemma 22]. Uniform nonsaturation has an antecedent for Bernoulli convolutions in Breuillard–Varjú [4, Lemma 13]. We prove the required entropy estimates here; the only external dimension theorem used in the proof is exact dimensionality.

The argument has three steps. First, for a finite real law \nu, compare its exact entropy with the entropy of its grid cell at a fixed scale. The difference is information hidden inside those cells. A bound for the sum of several independent copies forces this information to be witnessed by fair (equal-weight) two-point laws at smaller scales. The estimate is independent of the smallest distance between support points.

Second, group the coding symbols into blocks and condition on their symbol counts. This is the block-type disintegration of Galicer–Saglietti–Shmerkin–Yavicoli [8, Section 6.4 and Lemma 6.6], also developed by Saglietti–Shmerkin–Solomyak [17, Lemma 6.2]; Käenmäki–Orponen [12, Section 2.2] explicitly allow repeated indexed maps in this construction. Here the counts fix each block’s signed contraction. For a sequence of such blocks, its translation then identifies its complete composed map. Conditioning costs only the entropy of the counts, so the map-entropy rate gives a lower bound for the remaining translation entropy even when addresses collide. The same fixed contractions give a small sumset bound. A hypothetical dimension deficit therefore yields a linear amount of fair-pair mass across a finite band of very fine scales.

Third, each pair produces a positive entropy gain against the unobserved tail. The bands may end at arbitrarily large depths. We choose successive block lengths only after the preceding endpoints are known. At each observation scale their retained blocks are disjoint, so their gains telescope to a bounded total. Averaging over observation scales gives a fixed positive contribution from every band, a contradiction as the number of bands grows.

Section 2 proves the entropy calculus, nonsaturation, and two-point gain. Section 3 proves the finite-law estimate. Section 4 applies it to conditional map laws, and Section 5 arranges the disjoint gains and completes the proof. Section 6 derives the no-overlap measure and attractor formulas, followed by the homogeneous and three-map specializations. No quantitative bound on the fine-scale endpoints is needed.

1.2 The entropy-rate upper bound

We prove the upper bound before turning to the contradiction argument. Fix a positive integer n. Use the finite set of distinct maps in the support of G_n as a new alphabet, giving each map g its exact probability q_g=\P(G_n=g)>0. Grouping independent coding symbols into n-blocks shows that this new system has the same measure \mu. Its one-symbol entropy is H(G_n) and its mean contraction depth is n\chi, since coincident maps have the same absolute slope.

For 0<\varepsilon<n\chi, the strong law shows that almost every address in this block system eventually has, at every length j, a prefix of probability at least 2^{-j(H(G_n)+\varepsilon)} and absolute contraction at most 2^{-j(n\chi-\varepsilon)}. There are at most 2^{j(H(G_n)+\varepsilon)} such prefixes. For each fixed eventual starting length J, let E_J be the closed set of addresses satisfying both conditions for every j\ge J. Its compact coding image is covered by intervals of diameter at most C2^{-j(n\chi-\varepsilon)}, where C bounds the attractor diameter. The total ath power of these diameters tends to zero when a>(H(G_n)+\varepsilon)/(n\chi-\varepsilon). The countable union over eventual starting lengths is a full-measure Borel set with this dimension bound. The case of zero attractor diameter is immediate. Letting \varepsilon\downarrow0, then taking the infimum over n, gives

In particular, the conclusion of Theorem 1.1 already holds when h=0, including a single-map system. To prove equality in all other cases, it suffices to rule out

These inequalities are the contradiction hypothesis in Sections 4 and 5.

2 Entropy between two scales

We first establish the entropy estimates that turn a separated pair of translations into a definite gain. The decisive property of the self-similar measure is uniform nonsaturation: if its dimension is less than one, every one-bit scale interval leaves a positive entropy deficit. The intermediate entropy identities apply to arbitrary compactly supported probability measures on the line.

For a finite-valued variable Z, write H(Z)=-\sum_z\P(Z=z)\log\P(Z=z), with 0\log0=0. Conditional entropy is the average entropy of the conditional law, and I(Z;Y)=H(Z)-H(Z\mid Y); conditional mutual information is defined similarly. We use the entropy chain rule and nonnegativity of conditional mutual information. For a bounded real random variable X and s>0, define

We use the same notation for the law of X. For s,t>0, put

As throughout the paper, all logarithms and entropies are in base 2. Write h(f)=-\int f\log f for differential entropy and V_s for an independent uniform random variable on [0,s].

The smoothing and nested-grid identities are the averaged-entropy calculus of [20, Section 2.2, Lemmas 5, 6, and 10]; we include their proofs.

The integer-ratio hypothesis is essential to our use of concavity. In the argument below it is applied only with ratios 2, a chosen integer K, or the square of a chosen integer M.

For Bernoulli convolutions, the following nonsaturation property appears in Breuillard–Varjú [4, Lemma 13]. The stopping-time argument below extends it to unequal contractions of either sign.

The next estimate explains how the deficit produces gain. Unlike the concavity statements, it does not require an integer ratio of scales. The fair-pair comparison follows the method of [20, Proposition 20 and Lemma 22].

3 Pair mass below a fixed scale

We next consider a finite probability law without any separation assumption. Its exact entropy may greatly exceed its entropy at a prescribed scale. The following lemma detects that excess through the mass that can be placed in fair pairs at smaller scales. An upper bound for a sumset will make the lemma useful for the block laws in the next section. Varjú’s scale-local decomposition into fair two-point submeasures [20, Proposition 21] is a predecessor of this approach. Here the sumset term controls the total capacity over all finer bands; we prove the needed estimate by randomized nested partitions.

Thus c_\ell(\nu) is the largest total weight of fair two-point laws in the indicated distance band whose weighted sum is a submeasure of \nu. Summing the constraints gives \sum_e w_e\le1; the feasible set is compact, so the maximum exists and 0\le c_\ell(\nu)\le1. For example, if \nu=p\delta_x+(1-p)\delta_y with x\ne y, then the capacity in the band containing |x-y| is 2\min\{p,1-p\}, and all its other capacities vanish.

For a fixed finite law only finitely many pair capacities are nonzero. Later we will apply Lemma 3.2 to a finite family of finite laws for each fixed block count. Their nonzero capacities therefore lie in a common finite range, although the upper endpoint need not have any effective bound.

4 Conditioning on block types

Let \Phi=(\varphi_i)_{i\in\Lambda} be a finite affine iterated function system on \R, where \varphi_i(x)=r_ix+t_i and 0<|r_i|<1. Let p=(p_i)_{i\in\Lambda} be a strictly positive probability vector, and let \mu be its self-similar measure. Write G_n for the random complete affine map obtained from n independent symbols with law p, and let h=\lim_{n\to\infty}H(G_n)/n be its entropy rate. Throughout this section assume

All logarithms have base 2. We will find finite conditional coding laws whose exact entropy exceeds their entropy at a prescribed scale by a positive multiple of the number of blocks. Lemma 3.2 will then turn this deficit into pair mass across finite bands of arbitrarily fine scales.

We condition on the number of occurrences of each symbol in a short block. These counts fix its signed contraction. Once all block contractions are fixed, the translation determines the complete composed map. The entropy cost of recording the counts is small, so the conditional translations retain almost all the map entropy. This argument allows distinct words to define the same map. The block-type disintegration is due to Galicer–Saglietti–Shmerkin–Yavicoli [8, Section 6.4, Lemma 6.6] and is developed further by Saglietti–Shmerkin–Solomyak [17, Lemma 6.2]; Käenmäki–Orponen explicitly allow repeated indexed maps [12, Section 2.2, Proposition 2.8]. We retain the signed contraction and estimate the entropy of the complete map after collisions have been combined.

4.1 Parameters and conditional coding laws

Write m=|\Lambda|. The second inequality in (4.1) implies h>0, hence m\ge2. Choose \alpha>\chi such that d\alpha<h. For d>0 we may take \chi<\alpha<h/d; for d=0 we may take \alpha=\chi+1. Choose an integer b\ge1 sufficiently large that

and set

Then

These parameters remain fixed throughout the rest of the proof.

Group the iid symbols with law p into words W_0,W_1,\ldots\in\Lambda^b. For w\in\Lambda^b let \operatorname{type}(w)=(\kappa_i)_{i\in\Lambda}, where \kappa_i counts the occurrences of i in w. The set \mathcal T_b of types consists of the nonnegative integer vectors with \sum_i\kappa_i=b, and has cardinality at most (b+1)^m. For \kappa\in\mathcal T_b, define

Every word of type \kappa has probability \prod_i p_i^{\kappa_i}, so Q_\kappa is uniform on its N_\kappa words. Equivalently, we can generate the original coding by first choosing independent types \omega_0,\omega_1,\ldots with distribution q, then choosing the words independently with respective conditional laws Q_{\omega_0},Q_{\omega_1},\ldots. This follows by multiplying the probabilities of any finite sequence of blocks. In particular it preserves the original, possibly unequal, symbol weights p_i.

Write \omega=(\omega_h)_{h\ge0} for the type sequence and \theta\omega=(\omega_{h+1})_{h\ge0} for its shift. For j\ge0, let \mathcal F_j be the sigma-algebra generated by \omega_0,\ldots,\omega_{j-1}; \mathcal F_0 is trivial. For a type \kappa and an environment \omega, define

with R_0=1 and \sigma_0=0. Thus R_j is \mathcal F_j-measurable, including its sign. Let

The increments of \sigma_j are iid bounded positive random variables of mean b\chi, and 0\le\sigma_j\le Dj.

Given \omega, use the independent conditional block draws above to define

Here \Law_\omega denotes the law under those conditional word draws. The infinite sum converges uniformly, since the block translations form a finite set and \max_i|r_i|<1. The law \nu_{\omega,n} depends only on \mathcal F_n, whereas \mu_\omega can depend on the entire environment. Averaging over the types gives \mu=\E\mu_\omega. For any real u\ne0 and any law \lambda, write \mathsf D_u\lambda=\Law(uX) when X has law \lambda; this notation retains the sign of u.

For n\ge1, let \Omega_n=(\omega_0,\ldots,\omega_{n-1}) be the tuple of the first n types, and put

4.2 From the deficit to bands of pair mass

The preceding lemma produces entropy that is invisible at scale \rho_n. To apply the finite-law estimate, we must also control the number of possible sums of several independent copies of a conditional prefix. Fixing the types makes that count uniform in the environment.

The bands in Proposition 4.2 can extend far beyond the scale 2^{-An}. Their finiteness is enough: the next step will select successive block lengths after the earlier upper endpoints B_n are known. No quantitative separation of distinct cylinder translations has been used.

5 Disjoint windows of entropy gain

Proposition 4.2 supplies a linear amount of fair-pair weight in a finite interval of logarithmic depths for every sufficiently long block segment. We now turn this weight into entropy gain. The difficulty is that the depth interval may extend arbitrarily far. We choose several segments whose depth intervals are separated, and show that their gains occupy disjoint windows of block indices. At each physical scale the total gain is bounded, whereas every chosen interval contributes the same positive amount after averaging over scales.

Throughout this section, retain the environment and constants from Section 4. Thus \sigma_j=-\log|R_j| is the contraction depth after j blocks, D bounds each block’s depth, and \mathcal F_j=\sigma(\omega_0,\ldots,\omega_{j-1}) records the first j types. In particular, \sigma_j\le Dj. The logarithmic-depth bands are [a_n,B_n], where a_n=\lceil An\rceil-1, and their expected total weight on E_n=\{\sigma_n\le A'n\} is at least cn.

Let \theta denote the left shift of the type sequence. Define the suffix laws, including their contraction from the initial scale, by

The independent conditional block draws give the pathwise identity

All these laws have compact support. The signs of R_j are retained in the laws; only scale changes use |R_j|.

By Lemma 2.3, choose \delta>0 such that

Fix an integer M>4 with 5/M<\delta/2. For an integer target depth L\ge1, write r=2^{-L} and define

Here and below g_j depends on L and \omega. The ratio of the two window scales is the integer M^2. Lemma 2.1 and (5.1) therefore imply, for every environment,

5.1 A gain after averaging the unobserved future

An individual conditional tail need not satisfy the nonsaturation bound for \mu. To use that bound, we condition only on the types through the end of the finite block segment producing a fair pair. The unobserved future types then average its tail law back to a scaled copy of \mu.

5.2 Choosing disjoint block windows

Fix any positive integer q. Choose sufficiently large integers n_1<\cdots<n_q successively so that Proposition 4.2 applies, a_{n_i}\ge1, and

This is possible because the previously chosen B_{n_h} are finite and

In particular, the choice uses no upper bound on how fast B_n grows. Call [a_{n_i},B_{n_i}] band i. Larger i means finer physical scales, since (5.5) gives a_{n_i}>B_{n_h} for h<i.

Choose an even integer T with

For band i, use the candidate starts

For each target L\in\{1,\ldots,T\}, retain the window [j,j+n_i), j\in J_i, exactly on the event

This event belongs to \mathcal F_{j+n_i}, the same sigma-algebra as the conditioning in Lemma 5.1.

The reason for the separation condition is simple. As a block start j moves forward, the adjusted target depth L-\lceil\sigma_j\rceil is nonincreasing. A retained window in a finer band cannot decrease this depth far enough to reach any coarser band. The next lemma makes this order precise for every environment.

Figure 1 shows the order of retained windows from two different bands in this proof.

5.3 Summing over target depths

The disjointness lemma bounds the total gain at one physical scale. Summing over targets recovers every pair scale in each band. Indeed, for a candidate j\in J_i,

Because T is even, this also gives 0\le\lceil\sigma_j\rceil\le T/2. Each integer \ell\in[a_{n_i},B_{n_i}] therefore occurs exactly once as L-\lceil\sigma_j\rceil for 1\le L\le T: take L=\ell+\lceil\sigma_j\rceil. Its lower bound is 1, and its upper bound is at most B_{n_i}+T/2<T.

Each band has at least T/(2Dn_i) candidate starts, and each contributes expected pair mass at least cn_i after summing the targets. The block length cancels, so every band supplies a fixed multiple of T. Take expectations in (5.9). Since the retention indicator is \mathcal F_{j+n_i}-measurable, the tower property and Lemma 5.1 apply to each summand. Sum the result over L=1,\ldots,T and use the preceding exact correspondence to obtain

The second inequality uses Proposition 4.2 and the identical distribution of the type environment under \theta^j; independence between different candidate windows is not needed. The last inequality follows from the exact count

After canceling T, (5.11) bounds the arbitrary integer q by 8D\log M/(\delta c). All the constants in this bound were fixed before q was chosen, so this is impossible.

6 Overlap conventions and dimension of the attractor

We first reconcile the two usual word-length conventions for an exact overlap. The following observation also appears in [10, footnote 3]. The nonzero contraction hypothesis is useful here as well as in the proof of the measure theorem.

Let K_\Phi be the compact attractor, equivalently the set of all coding limits. Its similarity dimension s_* is the unique number s_*\ge0 satisfying

For at least two symbols, existence and uniqueness follow because this sum is continuous and strictly decreasing from |\Lambda|>1 to zero. For one symbol its unique solution is zero.

No claim of absolute continuity, effective separation, or a quantitative rate of convergence follows from these dimension statements. In particular, the finite upper endpoints of the scale ranges used in the proof have no asserted effective bound.