arXiv subjects

Sieye Ryu

Publications and source records attributed to Sieye Ryu.

5 recordsLinked to original sources

arXiv2021

Flip Signatures

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 $J^2=I$. 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 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.

math.DS↗

arXiv2019

Predictability, topological entropy and invariant random orders

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.

math.DS↗

arXiv2017

On Conjugacy Invariants of $D_{\infty}$-Topological Markov Chains

A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are equivalence relations between flip pairs. We investigate the relationships between the existence of a $D_{\infty}$-conjugacy, the existence of a $D_{\infty}$-strong shift equivalence, the existence of a $D_{\infty}$-shift equivalence and the coincidence of the Lind zeta functions.

math.DS↗

arXiv2017

The Lind Zeta functions of reversal systems of finite order

A decomposition theorem for the Lind zeta function of a reversal system $(X, T, R)$ of finite order is established. A reversal system can be regarded as an action of a certain group $G$ on $X$. To establish an explicit formula for the Lind zeta function of $(X, T, R)$, we need to consider finite index subgroups $H$ of $G$ with induced actions given by automorphisms or by flips. When the underlying dynamical system $(X, T)$ is either a shift of finite type or a sofic shift, we express the Lind zeta function of $(X, T, R)$ in terms of matrices.

math.DS↗

arXiv2011

On the number of fixed points of sofic flip systems

In the case when $X$ is a sofic shift and $ϕ: X \to X$ is a homeomorphism such that $ϕ^2 = \text{id}_X$ and $ϕσ_X = σ_X^{-1} ϕ$, the number of points in $X$ that are fixed by $σ_X^m$ and $σ_X^n ϕ$, $m=1,2,...$, $n\in\Bbb Z$, is expressed in terms of a finite number of square matrices: The matrices are obtained from Krieger's joint state chain of a sofic shift which is conjugate to $X$.

math.DS↗