scholarly journals Nilpotency and periodic points in non-uniform cellular automata

2021 ◽  
Vol 58 (4) ◽  
pp. 319-333
Author(s):  
Supreeti Kamilya ◽  
Jarkko Kari
Keyword(s):  
Fixed Point ◽  
Cellular Automata ◽  
Finite Number ◽  
Periodic Point ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
One Dimensional ◽  
Periodic Distribution ◽  
Wang Tile

AbstractNilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

EXTENSIONS OF THE NOTION OF CHAOTIC AREA IN SECOND-ORDER ENDOMORPHISMS

1995 ◽  
Vol 05 (03) ◽  
pp. 751-777 ◽  
Author(s):  
A. BARUGOLA ◽  
J.C. CATHALA ◽  
C. MIRA
Keyword(s):  
Fixed Point ◽  
Finite Number ◽  
Periodic Point ◽  
Critical Points ◽  
Unstable Manifold ◽  
Two Dimensional ◽  
One Dimensional ◽  
Critical Curves ◽  
The One ◽  
Invariant Domains

Properties of chaotic areas (i.e. invariant domains of points positively stable in the Poisson’s sense) of non-invertible maps of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical points in the one-dimensional case). The classical situation is that of a chaotic area bounded by a finite number of critical curves segments. This paper considers another class of chaotic areas bounded by the union of critical curves segments and segments of the unstable manifold of a saddle fixed point, or that of saddle cycle (periodic point). Different configurations are examined, as their bifurcations when a map parameter varies.

scholarly journals Periodic points and transitivity on dendrites

2016 ◽  
Vol 37 (7) ◽  
pp. 2017-2033 ◽  
Author(s):  
GERARDO ACOSTA ◽  
RODRIGO HERNÁNDEZ-GUTIÉRREZ ◽  
ISSAM NAGHMOUCHI ◽  
PIOTR OPROCHA
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
Dense Orbit ◽  
Topological Graphs ◽  
Periodic Decomposition ◽  
Dense Set ◽  
Compact Subsets

We study relations between transitivity, mixing and periodic points on dendrites. We prove that, when there is a point with dense orbit which is a cutpoint, periodic points are dense and there is a terminal periodic decomposition. We also show that it is possible that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It is also possible that a dynamical system is transitive but there is a unique periodic point which, in fact, is the unique fixed point. We also prove that on almost meshed continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all non-empty compact subsets.

Some non-unique fixed point or periodic point results in JS-metric spaces

2019 ◽  
Vol 21 (2) ◽  
Author(s):  
Tanusri Senapati ◽  
Lakshmi Kanta Dey ◽  
Ankush Chanda ◽  
Huaping Huang
Keyword(s):  
Fixed Point ◽  
Periodic Point ◽  
Metric Spaces ◽  
Unique Fixed Point

scholarly journals Non-Unique Fixed Point Theorems in Modular Metric Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 549
Author(s):  
Sharafat Hussain
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Periodic Points ◽  
Modular Metric Spaces ◽  
Modular Spaces

This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive conditions. Our results extend the results of Ćirić, Pachpatte, and Achari in modular metric spaces.

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.

scholarly journals On common fixed and periodic points of commuting functions

1998 ◽  
Vol 21 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Unique Common Fixed Point

It is known that two commuting continuous functions on an interval need not have a common fixed point. It is not known if such two functions have a common periodic point. In this paper we first give some results in this direction. We then define a new contractive condition, under which two continuous functions must have a unique common fixed point.

scholarly journals THE SHADOW-CURVES OF THE ORBIT DIAGRAM PERMEATE THE BIFURCATION DIAGRAM, TOO

2009 ◽  
Vol 19 (09) ◽  
pp. 3017-3031 ◽  
Author(s):  
CHIP ROSS ◽  
MEREDITH ODELL ◽  
SARAH CREMER
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Bifurcation Diagram ◽  
Periodic Point ◽  
Infinite Family ◽  
Periodic Points ◽  
New Phenomena

The "Q-curves [Formula: see text] have long been observed and studied as the shadowy curves which appear illusively — not explicitly drawn — in the familiar orbit diagram of Myrberg's map fc(x) = x2 + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x2 + c as well — by "bifurcation diagram" we mean the collection of all periodic points of fc (attracting, indifferent and repelling) — these collections form what we call "P-curves". We show Q-curves and P-curves intersect in one of two ways: At a superattracting periodic point on a P-curve, the infinite family of Q-curve s which intersect there are all tangent to the P-curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P-curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves. We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P-curves. Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.

scholarly journals ON PERIODIC POINTS OF FREE INVERSE MONOID HOMOMORPHISMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1789-1803 ◽  
Author(s):  
EMANUELE RODARO ◽  
PEDRO V. SILVA
Keyword(s):  
Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Context Sensitive ◽  
Context Free Language ◽  
Context Free ◽  
Free Language ◽  
Free Inverse Monoid

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

scholarly journals A Study on p-Cyclic Orbital Geraghty type Contractions

2018 ◽  
Vol 7 (4.10) ◽  
pp. 883 ◽  
Author(s):  
M. L.Suresh ◽  
T. Gunasekar ◽  
S. Karpagam ◽  
B. Zlatanov ◽  
. .
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Unique Fixed Point ◽  
Best Proximity Point ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Space Setting

Consider a metric space  and the non empty sub sets, of X. A map called p-cyclic orbital Geraghty type of contraction is introduced.  Convergence of a unique fixed point and a best proximity point for this map is obtained in a uniformly convex Banach space setting.  Also, this best proximity point is the unique periodic point of such a map.  

scholarly journals Flexible periodic points

2014 ◽  
Vol 35 (5) ◽  
pp. 1394-1422 ◽  
Author(s):  
CHRISTIAN BONATTI ◽  
KATSUTOSHI SHINOHARA
Keyword(s):  
Periodic Point ◽  
Stable Manifold ◽  
Periodic Points ◽  
Stable Bundle ◽  
Two Dimensional ◽  
General Phenomenon ◽  
One Dimensional ◽  
Stable Direction ◽  
Stable Index

We define the notion of ${\it\varepsilon}$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits ${\it\varepsilon}$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that an ${\it\varepsilon}$-perturbation to an ${\it\varepsilon}$-flexible point allows us to change it to a stable index one periodic point whose (one-dimensional) stable manifold is an arbitrarily chosen $C^{1}$-curve. We also show that the existence of flexible points is a general phenomenon among systems with a robustly non-hyperbolic two-dimensional center-stable bundle.

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