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Math-2691 · mathematics · Dynamical systems

Entropy-rate dimension formula for all real self-similar measures

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a finite nonempty indexed family $\phi _i(x)=r_i$ $x+t_i$ with real $t_i$, $0<|r_i|<1$ and strictly positive weights $p_i$ summing to one, let $\mu =sum_i$ $p_i(\phi _i)_*\mu $. Let $G_n$ be the random complete affine map $\phi _I1$ composed with ... composed with $\phi _In$ for independent letters with law p, $h_RW=lim_n$ $H(G_n)/n$ and $chi=-sum_i$ $p_i$ $\log |r_i|$, using the same logarithmic base. Is the lower Hausdorff dimension of mu always $min(1,h_RW/chi)$, allowing signed unequal ratios and arbitrary exact overlaps?

Short English statement

For a finite nonempty indexed family $\phi _i(x)=r_i$ $x+t_i$ with real $t_i$, $0<|r_i|<1$ and strictly positive weights $p_i$ summing to one, let $\mu =sum_i$ $p_i(\phi _i)_*\mu $. Let $G_n$ be the random complete affine map $\phi _I1$ composed with ... composed with $\phi _In$ for independent letters with law p, $h_RW=lim_n$ $H(G_n)/n$ and $chi=-sum_i$ $p_i$ $\log |r_i|$, using the same logarithmic base. Is the lower Hausdorff dimension of mu always $min(1,h_RW/chi)$, allowing signed unequal ratios and arbitrary exact overlaps?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2691-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2691

Manuscript 1 (family 148, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Entropy-rate dimension formula for all real self-similar measures

OpenAI's claim

For any finite real self-similar probability measure, its dimension is claimed to equal the smaller of one and map-walk entropy rate divided by Lyapunov exponent, even with exact overlaps and signed unequal contractions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Consider a nonempty finite indexed family $\Phi=(\varphi_i)$ consisting of similarities $\varphi_i(x)=r_ix+t_i$ of the real line, with arbitrary translations and $0<|r_i|<1$, and positive probability vector $p$. Let $\mu$ be its self-similar measure, $\chi=\sum_i p_i\log(1/|r_i|)$, and $h_{\mathrm{RW}}=\lim_{n\to\infty}H(G_n)/n$, where $G_n$ is the random complete affine composition of $n$ independent $p$-distributed generators. OpenAI claims \[ \dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}. \] No separation is imposed; repeated generators and exact overlaps are allowed. The attractor corollary assumes absence of exact overlaps (distinct finite words give distinct complete maps) and claims $\dim_HK_\Phi=\min\{1,s_*\}$, with $\sum_i|r_i|^{s_*}=1$. The measure assertion is stronger than that no-overlap set corollary.

Outline of the argument

Manuscript 1 (family 148, 2026-09-24): Finite-law comparisons expose entropy hidden within grid cells as fair two-point information at finer scales. Blocking and conditioning on symbol counts fixes the composed contractions while preserving map entropy; a hypothetical dimension loss would create too much fair-pair mass in a fine-scale band, contradicting the entropy estimates.

Formal verification

For family 148, the scope page documents the following selections. The selected dimension identity permits signed unequal contractions and exact coincidences. Further declarations treat homogeneous measures and overlap-free attractors; density and absolute continuity are not selected. The documented statement covers the main claim of manuscript 1. Comparator files: SelfSimilar.lean, SelfSimilarCorollaries.lean. Declarations: OAI.EntropyRateDimension.entropy_rate_dimension, OAI.EntropyRateDimension.Extensions.homogeneous_dimension_direct, OAI.CorSetReference.cor_set.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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