Phase transition for piecewise linear fibonacci bimodal map

Haoyang Ji; Wenxiu Ma; Haoyang Ji; Wenxiu Ma

doi:10.3934/math.2023424

AIMS Mathematics

2023, Volume 8, Issue 4: 8403-8430. doi: 10.3934/math.2023424

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Research article

Phase transition for piecewise linear fibonacci bimodal map

Haoyang Ji ^{1
,
,},
Wenxiu Ma ²

1.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
2.
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Received: 22 September 2022 Revised: 17 January 2023 Accepted: 19 January 2023 Published: 03 February 2023
MSC : 37E05, 37A10

In this paper we concern with the Fibonacci bimodal maps. We first study the topological properties of the Fibonacci bimodal maps in the context of kneading map and give an equivalent description of Fibonacci combinatorics. Then we construct a one-parameter family $ f_{\lambda} $ of countably piecewise linear Fibonacci bimodal maps depending on the parameter $ \lambda $ which are all odd functions. By a random walk argument on its induced Markov map, we will show that a phase transition occurs from Lebesgue conservative to Lebesgue dissipative behaviors.
- Fibonacci combinatorics,
- piecewise linear modal,
- invariant measure,
- Cantor attractor,
- Markov process
Citation: Haoyang Ji, Wenxiu Ma. Phase transition for piecewise linear fibonacci bimodal map[J]. AIMS Mathematics, 2023, 8(4): 8403-8430. doi: 10.3934/math.2023424

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Abstract

In this paper we concern with the Fibonacci bimodal maps. We first study the topological properties of the Fibonacci bimodal maps in the context of kneading map and give an equivalent description of Fibonacci combinatorics. Then we construct a one-parameter family $ f_{\lambda} $ of countably piecewise linear Fibonacci bimodal maps depending on the parameter $ \lambda $ which are all odd functions. By a random walk argument on its induced Markov map, we will show that a phase transition occurs from Lebesgue conservative to Lebesgue dissipative behaviors.

References

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