Dynamical system concepts

The idea behind dynamical systems theory is studying, understanding, and estimating the long place behavior of a system that changes in time. The characterization of this behavior consists in knowing which are the conditions of a system such that it has a particular comportment; some examples of these comportments are the following:

• The system has a periodic behavior
• The system recurrently returns to a given set
• The system goes to all the possible sets that cover the space of the system
• The system never leaves a given set

Using some useful topological concepts we define a dynamical systems in the following way: ds

Definition 1   A dynamical system $({{X}_{}},\Psi)$ consists of a metric, compact space ${{X}_{}}$ and a continuous mapping $\Psi:{{X}_{}}\rightarrow {{X}_{}}$ that maps elements of the space ${{X}_{}}$ to the space ${{X}_{}}$.

A consequence of definition 3 is the concept of an orbit of a given point in the space ${{X}_{}}$: or

Definition 2   In a dynamical system $({{X}_{}},\Psi)$, an orbit is the trajectory that a given point ${x_{}}\in {{X}_{}}$ has in the space ${{X}_{}}$ with the successive application of the mapping $\Psi$ over the point ${x_{}}$

The reason for the compactness of the space ${{X}_{}}$ is that such spaces can be covered and thereby represented by a finite number of sets. With this, the orbit of a point ${x_{}}$ in the space ${{X}_{}}$ can be described by the finite number of sets that it reaches, and this feature provides an easier analysis. In the study of the dynamical behavior in $({k^{}},1/2)$ reversible one dimensional cellular automata, we are interested in characterizing the orbits of the configurations ${{c}_{}}$ in the configuration space ${({{C}_{}},{\mathfrak{C}_{}})}$ applying the global mapping ${\Phi^{-1}}$. In particular, we want to describe the periodic, recurrent and the transitive behavior, for this motive, based on the works of J. de Vries [dV93] and Clark Robinson [Rob95] we present some definitions of these kinds of behavior. pp

Definition 3   A point ${x_{}}$ in a dynamical system $({{X}_{}},\Psi)$ is called a periodic point with minimum period $n$ if $\Psi^n({x_{}})={x_{}}$ and $\Psi^j({x_{}}) \neq {x_{}}$ for $0 \leq j \leq n$

fp Definition 5 says that after $n$ iterations of the mapping $\Psi$, the point ${x_{}}$ comes back to the same place. If a point ${x_{}}$ in a dynamical system $({{X}_{}},\Psi)$ has period equal $1$, then it is a fixed point. The analysis of periodic points and periodic orbits is the beginning in the study of a dynamical system. Now, we can ask if there are orbits that come back not to the same point but to the same open set of the initial point. If this happens for every open set in the space ${{X}_{}}$, then we have the following definition: nw

Definition 4   A dynamical system $({{X}_{}},\Psi)$ is non-wandering if for every open set ${\mathcal{O}_{}}$ of the space ${{X}_{}}$ there exists an integer $n>0$ such that $\Psi^n({\mathcal{O}_{}}) {\underset{}{\cap}}{\mathcal{O}_{}}\neq \varnothing$, that is, there exists a point ${x_{}}\in {\mathcal{O}_{}}$ such that $\Psi^n({x_{}}) \in {\mathcal{O}_{}}$

In the previous definition, we are using the topologycal nature of the space ${{X}_{}}$ utilizing the open sets of this space for characterizing the non-wandering orbits that mapping $\Psi$ can generate. Until now, we have defined orbits with a recurrent behavior, but another question is if the dynamical behavior of the system is such that an orbit can reach each one of the neighborhoods that cover the space ${{X}_{}}$. This idea gives the following definition: tp

Definition 5   A dynamical system $({{X}_{}},\Psi)$ is topologically transitive if there exists a point ${x_{}}$ in the space ${{X}_{}}$ such that for all open set ${\mathcal{O}_{}}$ in the space ${{X}_{}}$ there is an integer $n$ such that $\Psi^n({x_{}}) {\underset{}{\cap}}{\mathcal{O}_{}}\neq \varnothing$

Definition 7 establishes a point that cross all the neghbborhoods in a finite covering of a dynamical system $({{X}_{}},\Psi)$. Another transitive behaviors are also possible: te

Definition 6   A dynamical system $({{X}_{}},\Psi)$ is topologically ergodic if for all pair ${\mathcal{O}_{1}}$ and ${\mathcal{O}_{2}}$ of open sets of the space ${{X}_{}}$, we have that $\Psi^n({\mathcal{O}_{1}}) {\underset{}{\cap}}{\mathcal{O}_{2}} \neq \varnothing$

This definition looking for the existence of a topologically transitive point in every open set of the space ${{X}_{}}$. Periodicity and transitivity of points and open sets are the dynamical behaviors that we shall analyse in $({k^{}},1/2)$ reversible one dimensional cellular automata. In addition, we will define one more concept, if there exists an orbit such that it reaches a given open set and it remains there forever. tom

Definition 7   A dynamical system $({{X}_{}},\Psi)$ is topologically mixing if for all pair ${\mathcal{O}_{1}}$ and ${\mathcal{O}_{2}}$ of open sets in the space ${{X}_{}}$, there exists an integer $n_0$ such that $\Psi^n({\mathcal{O}_{1}}) {\underset{}{\cap}}{\mathcal{O}_{2}} \neq \varnothing$ for all $n \geq n_0$

In section 5 we will develop some matrix methods for detecting the existence of the different points and sets described in the previous definitions in $({k^{}},1/2)$ reversible one dimensional cellular automata.