Family 148Dynamical systems and ergodic theoryLeanNot yet digested

The entropy-rate dimension formula for self-similar measures

For every self-similar measure on the line generated by finitely many contracting similarities, proves

\dim_{\mathrm H}\mu=\min\{1,h_{\mathrm{RW}}/\chi\}

, where

h_{\mathrm{RW}}

is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.

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

Sep 24, 2026·main.pdfLean

We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.

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