# American Institute of Mathematical Sciences

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

## Minimal sets of periods for torus maps

 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.
Citation: Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301
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