Minimal sets of periods for torus maps

Boju Jiang; Jaume Llibre

doi:10.3934/dcds.1998.4.301

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1998, Volume 4, Issue 2: 301-320. Doi: 10.3934/dcds.1998.4.301

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Minimal sets of periods for torus maps

Boju Jiang^1, and
Jaume Llibre^2,

1.
Department of Mathematics, Peking University, Beijing 100871
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona

Received: April 1997

Revised: February 1998

Published: April 1998

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Abstract

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.

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Mathematics Subject Classification: Primary: 55M20; Secondary: 57N05, 57N10.

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