Cellular automata (CA), formally introduced by von Neumann in the late 1940s and early 1950s, are a class of spatially and temporally discrete deterministic systems, characterized by local interactions and an inherently parallel form of evolution [
rules
, 30 topologically-distinct Bernoulli shift rules, 10 complex Bernoulli shift rules, and 8 hyper Bernoulli shift rules. Recently, the dynamical properties of Chua’s periodic rules and robust Bernoulli-shift rules with distinct parameters have been investigated from the viewpoint of symbolic dynamics [11-17].
By a theorem of Hedlund [
is an one-dimensional cellular automata (1-D CA) if and only if it is continuous and commutes with the shift map
. For any 1-D CA, there exists a radius 
and a local map 
such that
where notetions will be precisely defined below. In particular, 
is an ECA global map when 
and 
. Each ECA can be expressed by a 3-bit Boolean function and coded by an integer
, which is the decimal notation of the output binary sequence of the Boolean function [5,7,18].
Let 
be a metric space and
: 
be a continuous map. 
is said to be topologically transitive or simply transitive if, for any non-empty open subsets 
and 
of 
there exists a natural number 
such that
; 
is topologically mixing or simply mixing if there exists a natural number 
such that 
for all
; 
is sensitive to initial conditions (or simply sensitive) if there exists a 
such that, for 
and for any neighborhood 
of
, there exists a 
and a natural number 
such that
, where 
is a distance defined on
.
Let 
be the set of periodic points of
. 
is said to be a dense subset of 
if, for any 
and any constant
, there exists a 
such that
.
Definition 1. 
is chaotic on 
in the sense of Devaney if (1)
is transitive, (2) 
is a dense subset of
, (3)
is sensitive [
It has been proved that additive 1-D CA are chaotic [
For a finite alphabet
, a word over 
is a finite sequence 
of elements of
. Denote by 
the set of all words of length
. If 
is a finite or infinite word and 
is an interval of integers on which 
is defined, then denote
. Moreover, 
is said to be a subword of
, denoted as
, if 
for some interval
, where 
is the set of all integers. The set of bi-infinite configurations is denoted by
, and a distance 
on
is defined by
where
,
, and 
if 
, or 
otherwise. It is known that 
is a compact, perfect and totally disconnected metric space [
For a map
, a set 
is said to be 
if
. If 
is closed and
then 
is called a subsystem of the dynamical system
. For example, let 
be a set of some words of length 
over
, and 
be the set of the bi-infinite configurations consisting of all the words in
. Then, 
is a subsystem of
, where 
is the left shift 
(or the right shift
) defined on
, and 
is called the determinative block system of
. The subsystem
, or simply 
is called a subshift of finite type (SFT) with respect to
.
Furthermore, 
can be described by a finite directed graph, 
, where each vertex is labeled by a word in
, and 
is the set of edges connecting the vertices in
. Two vertices 
and 
are connected by an edge 
if and only if
. One can think of each element of 
as a bi-infinite path on the graph
. Whereas a directed graph corresponds to a square transition matrix 
with 
if and only if there is an edge from vertex 
to vertex
, where 
is the number of elements in
, and 
(or
) is the code of the corresponding vertex in
,
. Thus, 
is precisely defined by the transition matrix
.
Remarkably, a 
square matrix 
is irreducible if, for any
, there exists an 
such that
; aperiodic if there exists an 
such that 
for all
, where 
is the 
entry of the power matrix
. If 
is an SFT of
, then it is transitive if and only if 
is irreducible; it is mixing if and only if 
is aperiodic. Equivalently, 
is irreducible if and only if for every ordered pair of vertices 
and 
there is a path in 
starting at 
and ending at 
[23,24].
of a 1-D CA 
is called a Bernoulli-shift subsystem if there exists an integer pair 
with 
such that
, where 
is the radius of the local map 
of the CA 
and 
is the left (or right) shift map.
as the left shift in the following discussion.
is a Bernoulli-shift subsystem of a 1-D CA 
with
then there exists a 
-sequence set 
such that
.
is the local map of
, then one can get the 
times iteration of 
Thus, 
, if and only if
for all
. Let
where 
is a finite set since 
Then, it follows that
.
in Proposition 1 is called the Bernoulli subshift of finite type (BSFT), and 
is called the determinative block system of
. If BSFT is an infinite set, then it is said to be non-trivial.
and
, of a 1-D CA 
with
.
if and only if
.
be a BSFT of a 1-D CA 
with 
If the shift 
is transitive, then 
is also transitive.
on 
is equivalent to the existence of a 
such that
where
starting from 
and 
is its closure [14,15]. It can be verified that for any 
, there exists at least an 
such that
.
, the 
.
above, consider the orbit
. It is clear that 
is closed and
. Because 
is 
-invariant and closed, one has 
and
. Obviously, 
.
denote the determinative block system of
. On one hand, based on Proposition 2 and
, one has
. On the other hand, since
, it follows that
, but the 
-sequence set consisting of 
is
. This implies
and
. Thus, 
, i.e., 
is transitive on
.
is transitive on a BSFT, since 
is transitive on SFT if and only if the transition matrix corresponding to the SFT is irreducible [23,24].
. One has
, where 
, 
, so 
contains two points. It is clear that 
is transitive but 
is not. In case BSFT is an infinite set, whether the CA transitivity implies the shift transitivity on the BSFT is still an open problem.
is a finite set on which 
is transitive, then it is a set of 
points, 
for some 
and 
, it is said that 
is trivial.
is transitive but not sensitive on a SFT 
if and only if 
for some 
and
. It is easy to be verified that 
and they are common multiples of 
and
.
be a BSFT of a 1-D CA 
with
. If the shift 
is transitive, then the set of periodic points of 
is dense in
.
be the BSFT, and 
be its determinative block system. For any 
and
, there exists a positive integer 
such that
and for
it follows that
.
is transitive on 
there exists a path from
to 
in the graph
. Let
-subsequences belong to
.
where
.
and
, where 
. Thus, 
and
, i.e., 
and
.
is dense in
.
is transitive on a non-trivial BSFT of a 1-D CA
, then 
is sensitive on the BSFT.
is chaotic in the sense of Devaney on its transitive non-trivial BSFT.
and
for all other triples
, and the corresponding global map is denoted by 
,
, where 
and
is a closed set because 
is an open set. Then, it can be easily verified that 
for any 
and 
for any
. This implies that
for
.
is
is irreducible, so 
is transitive on
.
, where 
is the identity matrix, and let 
denote the elements of the power matrix
. It can be easily verified that 
for all
, so 
is aperiodic. Recall that a matrix 
is irreducible if and only if 
is aperiodic [23,24]. Hence, 
is irreducible and so 
is transitive on
.
is chaotic in the sense of Devaney on
.