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Flip signatures

Part of: Basic linear algebra Topological dynamics

Published online by Cambridge University Press:  26 September 2022

SIEYE RYU Open the ORCID record for SIEYE RYU [Opens in a new window]
SIEYE RYU*
Affiliation:
Independent scholar, South Korea
*
e-mail: sieyeryu@gmail.com
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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.

Keywords

flip signaturesD ∞-topological Markov chainsD ∞-conjugacy invariantseventual kernelsAshley’s eight-by-eight and the full two-shift

MSC classification

Primary: 37B10: Symbolic dynamics 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 10 , October 2023 , pp. 3413 - 3447
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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