Dimension of ergodic measures projected onto self-similar sets with overlap

Thomas Jordan; Ariel Rapaport

doi:10.1112/plms.12337

Dimension of ergodic measures projected onto self-similar sets with overlap

Thomas Jordan, Ariel Rapaport

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Abstract

For self-similar sets on R satisfying the exponential separation condition we show that the dimension of natural projections of shift invariant ergodic measures is equal to min{1, h−χ}, where h and χ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin’s recent result on the Lq dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

Original language	English
Pages (from-to)	1-16
Number of pages	16
Journal	Proceedings of the London Mathematical Society
Volume	0
Early online date	29 Apr 2020
DOIs	https://doi.org/10.1112/plms.12337
Publication status	E-pub ahead of print - 29 Apr 2020

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Dr Thomas M Jordan
- Thomas.Jordanbristol.acuk
- Probability, Analysis and Dynamics
- School of Mathematics - Senior Lecturer in Pure Mathematics
- Pure Mathematics
- Ergodic theory and dynamical systems
Person: Academic , Member

Cite this

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title = "Dimension of ergodic measures projected onto self-similar sets with overlap",

abstract = "For self-similar sets on R satisfying the exponential separation condition we show that the dimension of natural projections of shift invariant ergodic measures is equal to min{1, h−χ}, where h and χ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin{\textquoteright}s recent result on the Lq dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.",

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AU - Jordan, Thomas

AU - Rapaport, Ariel

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N2 - For self-similar sets on R satisfying the exponential separation condition we show that the dimension of natural projections of shift invariant ergodic measures is equal to min{1, h−χ}, where h and χ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin’s recent result on the Lq dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

AB - For self-similar sets on R satisfying the exponential separation condition we show that the dimension of natural projections of shift invariant ergodic measures is equal to min{1, h−χ}, where h and χ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin’s recent result on the Lq dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

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DO - 10.1112/plms.12337

M3 - Article (Academic Journal)

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JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

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Dimension of ergodic measures projected onto self-similar sets with overlap

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