scholarly journals Subshifts on Infinite Alphabets and Their Entropy

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1293
Author(s):  
Sharwin Rezagholi
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Symbolic Dynamics ◽  
Infinite Graphs ◽  
Complexity Function ◽  
Countably Infinite

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

TOPOLOGICAL ENTROPY VERSUS GEODESIC ENTROPY

1994 ◽  
Vol 05 (02) ◽  
pp. 213-218 ◽  
Author(s):  
GABRIEL P. PATERNAIN ◽  
MIGUEL PATERNAIN
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Exponential Growth ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Betti Numbers ◽  
Exponential Growth Rate ◽  
Sphere Bundle ◽  
Space Of Paths ◽  
Simply Connected

Using Yomdin's Theorem [8], we show that for a compact Riemannian manifold M, the geodesic entropy — defined as the exponential growth rate of the average number of geodesic segments between two points — is ≤ the topological entropy of the geodesic flow of M. We also show that if M is simply connected and N ⊂ M is a compact simply connected submanifold, then the exponential growth rate of the sequence given by the Betti numbers of the space of paths starting in N and ending in a fixed point of M, is bounded above by the topological entropy of the geodesic flow on the normal sphere bundle of N.

scholarly journals Approximate Verification of the Symbolic Dynamics of Markov Chains

2015 ◽  
Vol 62 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Manindra Agrawal ◽  
S. Akshay ◽  
Blaise Genest ◽  
P. S. Thiagarajan
Keyword(s):  
Markov Chains ◽  
Symbolic Dynamics

Quasi-stationary distributions for Markov chains on a general state space

1974 ◽  
Vol 11 (4) ◽  
pp. 726-741 ◽  
Author(s):  
Richard. L. Tweedie
Keyword(s):  
Markov Chain ◽  
Invariant Measure ◽  
Markov Chains ◽  
State Space ◽  
Finiteness Condition ◽  
General State ◽  
Stationary Distributions ◽  
General State Space ◽  
Countably Infinite ◽  
Stationary Behaviour

The quasi-stationary behaviour of a Markov chain which is φ-irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspace is finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.

scholarly journals Chains, entropy, coding

1986 ◽  
Vol 6 (3) ◽  
pp. 415-448 ◽  
Author(s):  
Karl Petersen
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Random Walks ◽  
Finite Type ◽  
Periodic Orbits ◽  
Entropy Coding ◽  
Positive Entropy ◽  
Loop Analysis ◽  
Countable State ◽  
Subshifts Of Finite Type

AbstractVarious definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit

2014 ◽  
Vol 24 (07) ◽  
pp. 1450104 ◽  
Author(s):  
Zbigniew Galias
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Third Order ◽  
Return Map ◽  
Topological Sense

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense.

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

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