A FINITE APPROXIMATE LIVŠIC THEOREM FOR ANOSOV DIFFEOMORPHISMS

Rui Zou; Hua Wei

doi:10.11948/20240420

2025 Volume 15 Issue 4

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Rui Zou, Hua Wei. A FINITE APPROXIMATE LIVŠIC THEOREM FOR ANOSOV DIFFEOMORPHISMS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2185-2194. doi: 10.11948/20240420

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Rui Zou, Hua Wei. A FINITE APPROXIMATE LIVŠIC THEOREM FOR ANOSOV DIFFEOMORPHISMS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2185-2194. doi: 10.11948/20240420

A FINITE APPROXIMATE LIVŠIC THEOREM FOR ANOSOV DIFFEOMORPHISMS

Rui Zou^1,,
Hua Wei^1, ,

1.
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Author Bio: Email: zourui@nuist.edu.cn(R. Zou)
Corresponding author: Email: whw5700@163.com(H. Wei)
Fund Project: The authors were supported by National Key R&D Program of China (2022YFA1007800) and National Natural Science Foundation of China (12471185, 12271386)

Abstract

Abstract

In this paper, we prove a finite approximate version of the Livšic theorem for Anosov diffeomorphisms. Let $f$ be a transitive Anosov diffeomorphism and $\varphi\in C^{\alpha}(M)$. We show that there exist $0<\beta\le \alpha, C>0$ and $\tau>0$ such that for any $\varepsilon>0$, if $\left|\sum_{i=0}^{n}\varphi\left(f^i(p)\right)\right|\le \varepsilon$ for each periodic point $ p=f^n(p)$ with $n\le \varepsilon^{-\frac{1}{2}}$, then there exist $u\in C^{\beta}(M)$ and $h\in C^{\beta}(M)$ such that $\varphi=u\circ f-u+h. $ Moreover, $\|u\|_{C^\beta}\le C$ and $\|h\|_{C^\beta}\le C\varepsilon^\tau.$
- Livšic theorem /
- Anosov diffeomorphisms /
- cohomological equations
MSC: 37C25, 37C50, 37D05

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