Ergodic measures of intermediate entropy for affine transformations of nilmanifolds

Wen Huang; Leiye Xu; Shengnan Xu; Wen Huang; Leiye Xu; Shengnan Xu

doi:10.3934/era.2021015

Electronic Research Archive

2021, Volume 29, Issue 4: 2819-2827. doi: 10.3934/era.2021015

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Ergodic measures of intermediate entropy for affine transformations of nilmanifolds

CAS Wu Wen-Tsun Key Laboratory of Mathematics, and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received: 01 August 2020 Revised: 01 January 2021 Published: 22 February 2021
37A35, 37B40

In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation $ \tau $ of nilmanifold has a periodic point, then for every $ a\in[0, h_{top}(\tau)] $ there exists an ergodic measure $ \mu_a $ of $ \tau $ such that $ h_{\mu_a}(\tau) = a $.
- Intermediate entropy,
- affine transformation,
- nilmanifold
Citation: Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds[J]. Electronic Research Archive, 2021, 29(4): 2819-2827. doi: 10.3934/era.2021015

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Abstract

In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation $ \tau $ of nilmanifold has a periodic point, then for every $ a\in[0, h_{top}(\tau)] $ there exists an ergodic measure $ \mu_a $ of $ \tau $ such that $ h_{\mu_a}(\tau) = a $.

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