On the Entropy of Flows with Reparametrized Gluing Orbit Property
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON THE ENTROPY OF FLOWS WITH REPARAMETRIZED GLUING ORBIT PROPERTY∗
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Peng SUN (
China Economics and Management Academy, Central University of Finance and Economics, Beijing 100081, China E-mail : [email protected] Abstract We show that a flow or a semiflow with a weak form of reparametrized gluing orbit property has positive topological entropy if it is not minimal. Key words
Flow; gluing orbit property; reparametrization; minimality; topological entropy
2010 MR Subject Classification
1
37B05; 37B40; 37C50
Introduction
The gluing orbit property introduced in [1–3] is a much weaker variation of the well studied specification property. It is satisfied by a larger class of systems while still implying interesting facts, as indicated by a series of recent works. See [4–9]. It is well known that specification property implies positive topological entropy. On the contrary, there are systems that have both gluing orbit property and zero topological entropy (for example, minimal translations on tori [4]). It seems that such systems should be quite simple. In [10], in the setting without reparametrizations, we showed that they must be minimal. In communication with Paulo Varandas, we realize that this result should be more practical for flows if reparametrizations can be allowed. In this article, we introduce an even weaker form of the reparametrized gluing orbit property defined in [4, 5]. We are able to extend our result to this case. Theorem 1.1 Let (X, f t ) be a flow or a semiflow with weak reparametrized gluing orbit property. If (X, f t ) is not minimal, then it has positive topological entropy. The original motivation to study specification and gluing orbit properties was to understand the behavior of hyperbolic systems. In the smooth setting, positive topological entropy implies existence of nonzero Lyapunov exponents for some invariant measures. Theorem 1.1 indicates that in the non-minimal case, weak reparametrized gluing orbit property is still related to some weak form of hyperbolicity. In [11], we obtain a complete description of systems with gluing orbit property and zero entropy: they are minimal and equicontinuous, hence they are conjugate to minimal rotations on compact abelian groups. It is still an open question whether zero-entropy semiflows with (weak) reparametrized gluing orbit property are equicontinuous. Moreover, we have a work in ∗ Received February 11, 2019; revised May 19, 2019. Peng Sun is supported by National Natural Science Foundation of China (11571387) and CUFE Young Elite Teacher Project (QYP1902).
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
progress whose result implies that a minimal semiflow with reparametrized gluing orbit property must have zero topological entropy. We also have a similar result [12] on zero-entropy systems satisfying properties that are even weaker than gluing orbit. Preliminaries and definitions will be provided in Section 2. Then, we prove Theorem
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