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Flip signatures
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 10 / October 2023
- Published online by Cambridge University Press:
- 26 September 2022, pp. 3413-3447
- Print publication:
- October 2023
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- Article
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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.
