Jeroen S.W. Lamb:

k-Symmetry and return maps of space-time symmetric flows

abstract:

A diffeomorphism $U:\Omega\mapsto\Omega$ is called a (reversing)
$k$-symmetry of a dynamical system in $\Omega$ 
represented by the diffeomorphism $f:\Omega\mapsto\Omega$ 
if $k$ is the smallest
positive integer for which $U$ is a (reversing) symmetry of $f^k$ (the
$k$-times iterate of $f$), i.e.\ $U\circ f^k=f^{\pm k}\circ U$. 

We show how $k$-symmetry naturally arises in the context of return maps of
flows with mixed space-time symmetries. 
In this context we explain the occurrence of dual (representations of)
reversing $k$-symmetry groups in $k$-symmetric maps in relation 
to different choices for the position of the surface of section. 
We furthermore discuss the connection between periodic orbits of 
$k$-symmetric maps and symmetric periodic orbits of the flows they 
represent. 

Finally, we discuss the application of our results in local bifurcation 
theory and provide a geometric interpretation of formal (Birkhoff) normal form
symmetry as a time-shift symmetry of a locally constructed space-time
symmetric flow.