Jeroen S.W. Lamb & G. Reinout W. Quispel:

Cyclic reversing k-symmetry groups

abstract:

We consider discrete invertible dynamical systems $L$ with the
property that the $k$th iterate $L^k$ possesses (reversing)
symmetries that are not possessed by $L$. 
A map $U$ is called a (reversing) $k$-symmetry of $L$ if $k$ is the
smallest positive integer for which $U$ is a (reversing) symmetry of $L^k$.

In this paper we discuss the particular case that $L$ possesses a 
cyclic reversing $k$-symmetry group. We derive a decomposition property of
maps that possess a cyclic reversing $k$-symmetry group and
we classify the occurrence of such groups in invertible dynamical systems.

We discuss the occurrence of nonsimultaneously linearizable nonisomorphic 
reversing
$k$-symmetry groups in maps possessing cyclic reversing $k$-symmetry 
groups,
illustrated by an example of a diffeomorphism on the plane $\Real^2$. We also 
construct examples of diffeomorphisms with cyclic reversing $k$-symmetry
groups on the circle $S^1$, on the two-torus $T^2$, and on the cylinder
$S^1\times\Real$.