Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates

Qingsong Gu; Jiaxin Hu; Sze-Man Ngai

doi:10.3934/cpaa.2020030

Article Contents

2020, Volume 19, Issue 2: 641-676. Doi: 10.3934/cpaa.2020030

This issue Previous Article Next Article Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates

1.
Department of Mathematics, The Chinese University of Hong Kong, China
2.
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
3.
Key Laboratory of HPCSIP, Ministry of Education of China, College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China
4.
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA

^*Corresponding author
^*Corresponding author

Received: February 28, 2017

Revised: April 30, 2019

Published: October 2019

This research was supported in part by the National Natural Science Foundation of China, grants 11871296, 11771136 and 11271122, by Tsinghua University Initiative Scientific Research Program, and by a Hong Kong RGC grant. SN is also supported in part by Construct Program of the Key Discipline in Hunan Province, the Hunan Province Hundred Talents Program, and the Center of Mathematical Sciences and Applications of Harvard University

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Abstract

We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than $ 2 $ and are closely related to the spectral dimension of fractal Laplacians.

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Mathematics Subject Classification: Primary: 28A80, 35K08; Secondary: 35J05.

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Figure 1. (a) The IFS {S₀, S₁} has overlaps. (b) The auxiliary IFS {T₀, T₁, T₂} does not have overlaps

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Figure 2. Points $ x, y $ and cells $ K_{\omega ^{\prime }}, K_{\tau ^{\prime }} $, where $ \omega = \omega ^{\prime }0 $

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Figure 3. Positions of three points $ x, y, z $ when $ \omega ^{\prime} = \omega $

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Figure 4. A chain of $ k+1 $ cells with length $ | \omega | $ contained in $ [x, z] $

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Figure 5. (a) The IFS $\{S_i\}_{i = 0}^3$ has overlaps. (b) The auxiliary IFS $\{T_i\}_{i = 0}^2$ does not have overlaps

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