Rudolph-Johnson without dynamics: endomorphic rigidity on the circle
Abstract
We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms x ↦ mx. Here, we refer to an infinite set of multipliers m as sublacunary if its consecutive ratios tend to $1$. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers ⌊ nlog n ⌋ and the semigroup generated by $2$ and $3$ which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure μ has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of μ. These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of $3$. The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.
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BibTeX
@misc{burton2026rudolphjohnson,
title = {{Rudolph-Johnson without dynamics: endomorphic rigidity on the circle}},
author = {Peter Burton and Kate Juschenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07317},
eprint = {2610.07317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07317},
}
RIS
TY - UNPB
TI - Rudolph-Johnson without dynamics: endomorphic rigidity on the circle
AU - Peter Burton
AU - Kate Juschenko
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07317
UR - https://arxiv.org/abs/2610.07317
AB - We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms x ↦ mx. Here, we refer to an infinite set of multipliers m as sublacunary if its consecutive ratios tend to $1$. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers ⌊ n^log n ⌋ and the semigroup generated by $2$ and $3$ which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure μ has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of μ. These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of $3$. The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.
ER -