OpenAI research collection · 1 manuscript · Source checked 2026-10-19
For every self-similar measure on the line generated by finitely many contracting similarities, proves \dim_{\mathrm H}\mu=\min\{1,h_{RW}/\chi\}, where h_{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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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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