scholarly journals The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Risong Li ◽  
Tianxiu Lu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Real Line ◽  
Cyclic Permutation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Compact Subinterval ◽  
Dimensional Dynamical System ◽  
Necessary And Sufficient

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s, s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (01) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds under P 0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (1) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

Some results on the classification of expansive automorphisms of compact abelian groups

1996 ◽  
Vol 16 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Fabio Fagnani
Keyword(s):  
Finite Group ◽  
Topological Entropy ◽  
Abelian Groups ◽  
Topological Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Abelian Groups ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

scholarly journals Constant slope, entropy, and horseshoes for a map on a tame graph

2019 ◽  
Vol 40 (11) ◽  
pp. 2970-2994
Author(s):  
ADAM BARTOŠ ◽  
JOZEF BOBOK ◽  
PAVEL PYRIH ◽  
SAMUEL ROTH ◽  
BENJAMIN VEJNAR
Keyword(s):  
Topological Entropy ◽  
Transition Matrix ◽  
Sufficient Condition ◽  
Branch Points ◽  
Necessary And Sufficient Condition ◽  
Peano Continuum ◽  
Monotone Maps ◽  
Markov Map ◽  
Necessary And Sufficient ◽  
Constant Slope

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

scholarly journals A criterion for differentiability

1956 ◽  
Vol 40 ◽  
pp. 8-11
Author(s):  
A. M. Macbeath
Keyword(s):  
Real Number ◽  
Number System ◽  
The Other ◽  
Alternative Methods ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Explicit Mention ◽  
Necessary And Sufficient ◽  
Real Number System

There are two alternative methods of defining the concept of “convergence” of a sequence, one involving explicit mention of the limit, the other (Cauchy's condition) giving a necessary and sufficient condition in terms of the elements of the sequence only. The two definitions are equivalent, because of the property of completeness of the real number system.

scholarly journals Entropy and semi-conjugacy in dimension two

1988 ◽  
Vol 8 (4) ◽  
pp. 585-596 ◽  
Author(s):  
Michael Handel
Keyword(s):  
Topological Entropy ◽  
Closed Surface ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

AbstractWe prove that if a diffeomorphism f of a closed surface is homotopic to and has the same topological entropy as a pseudo-Anosov homeomorphism g, then f is semi-conjugate to g. As part of the proof, a necessary and sufficient condition is given for a pseudo-orbit of a pseudo-Anosov homeomorphism g to be shadowed by an actual orbit of g.

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