The entropy-rate dimension formula for self-similar measures on the line
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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— Constantin Kogler (@ConstantinKogl) Oct 7, 2026
I was stunned by the claimed solution to determining the dimension of self-similar measures on the line (Item 148 in the Review) and related problems. I am heartened that the proof appears to stand on the shoulders of earlier mathematicians and live within the framework they developed. I will…
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The Exact Overlaps Conjecture for Self-Similar Measures on the Real LinearXiv.orgSamuel Kittle, Constantin KoglerOct 7, 2026in the index
A generalization of the exact overlaps conjecture predicts a formula for the dimension of self-similar measures in terms of the random walk entropy and the Lyapunov exponent. We prove the exact overlaps conjecture and its generalized version in dimension one. The proof relies on a new entropy…