Abstract
Abstract
A
$D_{\infty }$
-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group
$D_{\infty }$
. It is defined by two zero-one square matrices A and J satisfying
$AJ=JA^{\textsf {T}}$
and
$J^2=I$
. A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a
$D_{\infty }$
-conjugacy invariant. We introduce natural
$D_{\infty }$
-actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural
$D_{\infty }$
-actions are not
$D_{\infty }$
-conjugate. We also discuss the notion of
$D_{\infty }$
-shift equivalence and the Lind zeta function.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics