scholarly journals The dynamical zeta function for commuting automorphisms of zero-dimensional groups

2016 ◽  
Vol 38 (4) ◽  
pp. 1564-1587
Author(s):  
RICHARD MILES ◽  
THOMAS WARD
Keyword(s):  
Zeta Function ◽  
Open Problem ◽  
Counting Function ◽  
Positive Entropy ◽  
Point Counting ◽  
Natural Boundary ◽  
Compact Metric Space ◽  
Dimensional Group ◽  
Dynamical Zeta Function ◽  
Zero Entropy

For a $\mathbb{Z}^{d}$-action $\unicode[STIX]{x1D6FC}$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when $\unicode[STIX]{x1D6FC}$ is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point counting function is also examined and, in the zero entropy case, we show how this may be severely restricted for subgroups of prime index in $\mathbb{Z}^{d}$. We also consider a related open problem concerning the appearance of a natural boundary for the dynamical zeta function of a single automorphism, giving further weight to the Pólya–Carlson dichotomy proposed by Bell and the authors.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

scholarly journals Dynamical Zeta Function for Several Strictly Convex Obstacles

2008 ◽  
Vol 51 (1) ◽  
pp. 100-113 ◽  
Author(s):  
Vesselin Petkov
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Absolute Convergence ◽  
Strictly Convex ◽  
Dynamical Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Analytic Singularities

AbstractThe behavior of the dynamical zeta function ZD(s) related to several strictly convex disjoint obstacles is similar to that of the inverse Q(s) = of the Riemann zeta function ζ(s). Let Π(s) be the series obtained from ZD(s) summing only over primitive periodic rays. In this paper we examine the analytic singularities of ZD(s) and Π(s) close to the line , where s2 is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ZD(s), Π(s) has a singularity at s = s2.

scholarly journals On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Magdalena Foryś-Krawiec ◽  
Jana Hantáková ◽  
Piotr Oprocha
Keyword(s):  
Complete Picture ◽  
Partial Characterization ◽  
Limit Set ◽  
Positive Entropy ◽  
Minimal Set ◽  
Backward Trajectory ◽  
Limit Sets ◽  
Backward Trajectories ◽  
Zero Entropy

<p style='text-indent:20px;'>In the paper we study what sets can be obtained as <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets are <inline-formula><tex-math id="M4">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets and for all but finitely many points <inline-formula><tex-math id="M5">\begin{document}$ x $\end{document}</tex-math></inline-formula>, we can obtain every <inline-formula><tex-math id="M6">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limits set as the <inline-formula><tex-math id="M7">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory starting in <inline-formula><tex-math id="M8">\begin{document}$ x $\end{document}</tex-math></inline-formula>. For zero entropy maps, every <inline-formula><tex-math id="M9">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.</p>

EIGENVALUES FOR HIGH ORDER ELLIPTIC OPERATORS IN A FRACTAL STRING

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 267-275 ◽  
Author(s):  
HONG DENG ◽  
GONGWEN PENG
Keyword(s):  
Zeta Function ◽  
Elliptic Operator ◽  
Riemann Zeta Function ◽  
Counting Function ◽  
High Order ◽  
Minkowski Dimension ◽  
Fractal Boundary ◽  
Open Set ◽  
Riemann Zeta ◽  
Fractal String

In this paper, we study the spectrum of order 2m (m≥1) elliptic operator A in a bounded open set Ω∈R1, with fractal boundary Γ=∂Ω and Minkowski dimension D∈(0, 1), thus proving the corresponding modified Weyl-Berry conjecture to be true, namely [Formula: see text] where N(λ, A, Ω) is the counting function, [Formula: see text], C1, D =2-(1-D) π-D(1-D)(-ζ(D)), ζ(D) is the classical Riemann–zeta function, and ℳ(D, Γ) is the Minkowski measure of Γ.

scholarly journals Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

2018 ◽  
Vol 07 (03) ◽  
pp. 1850007
Author(s):  
O. Khorunzhiy
Keyword(s):  
Graph Theory ◽  
Zeta Function ◽  
Long Range ◽  
Riemann Hypothesis ◽  
Counting Function ◽  
Symmetric Matrices ◽  
Ihara Zeta Function ◽  
Unique Measure ◽  
Edge Probability ◽  
Average Vertex

We consider the ensemble of [Formula: see text] real random symmetric matrices [Formula: see text] obtained from the determinant form of the Ihara zeta function associated to random graphs [Formula: see text] of the long-range percolation radius model with the edge probability determined by a function [Formula: see text]. We show that the normalized eigenvalue counting function of [Formula: see text] weakly converges in average as [Formula: see text], [Formula: see text] to a unique measure that depends on the limiting average vertex degree of [Formula: see text] given by [Formula: see text]. This measure converges in the limit of infinite [Formula: see text] to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

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