On the number of fixed points of a sofic shift-flip system

Abstract

If $X$ is a sofic shift and $\varphi : X\rightarrow X$ is a homeomorphism such that ${\varphi }^{2} = {\text{id} }_{X} $ and $\varphi {\sigma }_{X} = { \sigma }_{X}^{- 1} \varphi $, the number of points in $X$ that are fixed by ${ \sigma }_{X}^{m} $ and ${ \sigma }_{X}^{n} \varphi , m= 1, 2, \ldots , n\in \mathbb{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 $X$.

Counting finite orbits for the flip systems of shifts of finite type

“…For shifts of finite type, they obtained the general form of their flip systems and also the formula for their zeta function. Kim & Ryu [10] then extended the results for the flip systems of sofic shifts. Despite such extensive results, the asymptotic behaviour of the prime orbit counting function for these flip systems was left undetermined.…”

The Lind Zeta functions of reversal systems of finite order

“…Similarly, when (X, σ X , ϕ) is either a shift-flip system of finite type or a sofic shift-flip system, the number of fixed points p σ X ,ϕ (m, n) can be expressed in terms of matrices [14,16]. To present it, we need notation.…”

“…As we will see in Section 3, Theorem A implies that it suffices to consider the numbers of fixed points of actions G X either by automorphisms or by reversals of order 2. The case when actions are by reversals of order 2 is investigated in [14,16]. (See also Section 2.)…”

“…(See [23,3,8] for the definition of an N-rational function.) In [16], it is proved that h 4k−2 (t 2k−1 ) is N-rational. However, neither g 2k (t 2k ) 2k…”

Sofic-Dyck shifts

“…An N-rational expression of the zeta function of a sofic shift has been obtained by Reutenauer in [39] (see also [12]). Formulas for zeta functions of flip systems of finite type are given in [24], and for sofic flip systems in [25]. The zeta functions of the Dyck shifts were determined by Keller in [23].…”