scholarly journals On One-Sided, D-Chaotic CA Without Fixed Points, Having Continuum of Periodic Points With Period 2 and Topological Entropy log(p) for Any Prime p

Entropy ◽  
2014 ◽  
Vol 16 (11) ◽  
pp. 5601-5617
Author(s):  
Wit Forys ◽  
Janusz Matyja
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Periodic Points ◽  
Log P ◽  
Period 2

scholarly journals A note on periodic points of expanding maps of the interval

1986 ◽  
Vol 33 (3) ◽  
pp. 435-447 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Periodic Points ◽  
Expanding Maps ◽  
Continuous Maps

We sharpen a result of Byers on the existence of periodic points for some continuous expanding maps of the interval and generalize it to some classes of continuous maps of the interval which are not necessarily expanding. We then use these results to construct one-parameter families of continuous maps of the interval which have a bifurcation form fixed points directly to period 3 points together with a series of reverse bifurcations from period 3 points back to fixed points. Consequently, our results also provide examples of one-parameter families of continuous maps of the interval whose topological entropy jumps form zero to some positive number and then changes back to zero as the parameter varies.

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).

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

INFINITE DIMENSIONAL KRAWCZYK OPERATOR FOR FINDING PERIODIC ORBITS OF DISCRETE DYNAMICAL SYSTEMS

2007 ◽  
Vol 17 (12) ◽  
pp. 4261-4272 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete Dynamical Systems ◽  
Krawczyk Operator ◽  
Infinite Dimensional ◽  
Dispersal Model ◽  
Period 2 ◽  
Existence Of Zeros

In this work, we introduce the Krawczyk operator for infinite dimensional maps. We prove two properties of this operator related to the existence of zeros of the map. We also show how the Krawczyk operator can be used to prove the existence of periodic orbits of infinite dimensional discrete dynamical systems and for finding all periodic orbits with a given period enclosed in a specified region. As an example, we consider the Kot–Schaffer growth-dispersal model, for which we find all fixed points and period-2 orbits enclosed in the region containing the attractor observed numerically.

scholarly journals Chain recurrent points of a tree map

1999 ◽  
Vol 59 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Tao Li ◽  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Limit Set ◽  
Recurrent Point ◽  
Continuous Map ◽  
Tree Map

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if thew-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

scholarly journals Fixed points and periodic points of semiflows of holomorphic maps

2003 ◽  
Vol 2003 (4) ◽  
pp. 217-260
Author(s):  
Edoardo Vesentini
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Complex Banach Space ◽  
Periodic Points ◽  
General Question ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
Spin Factor ◽  
Cartan Factor

Letϕbe a semiflow of holomorphic maps of a bounded domainDin a complex Banach space. The general question arises under which conditions the existence of a periodic orbit ofϕimplies thatϕitself is periodic. An answer is provided, in the first part of this paper, in the case in whichDis the open unit ball of aJ∗-algebra andϕacts isometrically. More precise results are provided when theJ∗-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflowϕgenerated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.

Tilings of amenable groups

2019 ◽  
Vol 2019 (747) ◽  
pp. 277-298 ◽  
Author(s):  
Tomasz Downarowicz ◽  
Dawid Huczek ◽  
Guohua Zhang
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Finite Subset ◽  
Dimensional Space ◽  
Amenable Group ◽  
Free Action ◽  
Amenable Groups ◽  
Entropy Zero

Abstract We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

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