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April  1998, 4(2): 301-320. doi: 10.3934/dcds.1998.4.301

Minimal sets of periods for torus maps

Boju Jiang 1, and  Jaume Llibre 2,

1. 

Department of Mathematics, Peking University, Beijing 100871, China
2. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Spain

Received  April 1997 Revised  February 1998 Published  February 1998

Let $T^r$ be the $r$-dimensional torus, and let $f:T^r\to T^r$ be a map. If $\Per(f)$ denotes the set of periods of $f$, the minimal set of periods of $f$, denoted by $\MPer(f)$, is defined as $\bigcap_{g\cong f}\Per(g)$ where $g:T^r\to T^r$ is homotopic to $f$. First, we characterize the set $\MPer(f)$ in terms of the Nielsen numbers of the iterates of $f$. Second, we distinguish three types of the set $\MPer(f)$ and show that for each type and any given dimension $r$, the variation of $\MPer(f)$ is uniformly bounded in a suitable sense. Finally, we classify all the sets $\MPer(f)$ for self-maps of the $3$-dimensional torus.
Keywords: torus maps, Nielsen fixed point theory., torus homeomorphisms, periodic points, three dimensional torus, Minimum set of periods.
Mathematics Subject Classification: Primary: 55M20; Secondary: 57N05, 57N1.
Citation: Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301
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