An upper bound of the measure-theoretical entropy

Yuntao Zang

doi:10.3934/dcds.2022052

Article Contents

2022, Volume 42, Issue 9: 4263-4295. Doi: 10.3934/dcds.2022052

This issue Previous Article Achievable connectivities of Fatou components for a family of singular perturbations Next Article Instability results for the logarithmic Sobolev inequality and its application to related inequalities

An upper bound of the measure-theoretical entropy

Yuntao Zang^,

Soochow College, Soochow University, Suzhou, 215006, China

^*Corresponding author: Yuntao Zang
^*Corresponding author: Yuntao Zang

Received: August 31, 2021

Revised: January 31, 2022

Early access: April 2022

Published: August 2022

The author is supported by China Postdoctoral Science Foundation (2020TQ0098, 2021M690057)

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Abstract

Let $ f $ be a $ C^{1+\alpha} $ diffeomorphism on a compact manifold $ M $ and let $ \mu $ be an ergodic measure. We use a special family of fake center-stable manifolds to bound the entropy of $ \mu $ in terms of positive Lyapunov exponents and the so called 'dimensional entropy', a notion related to the topological entropy of submanifolds.

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Mathematics Subject Classification: Primary: 37D25; Secondary: 37A35.

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Figure 1. Standard family

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Figure 2. Maximal separated subset of $ \Lambda $

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Figure 3. Second property in Lemma 2.4

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Figure 4. Expansion on different directions

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Figure 5. Geometric estimate

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