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In [1], Banks et al. prove that a regular and positively transitive DTDS with
infinite cardinality, has sensitive dependence on initial conditions. In [3] we have
proved the following.
Proposition 3.
Let X, g be a regular and full transitive DTDS with infinite
cardinality. Then it is sensitive to initial conditions.
It is an open problem to find a sensitive, regular, full transitive but non positively
transitive DTDS. Another topological property of dynamical systems related to
the transitivity is the condition of minimality.
Definition 8.
A DTDS X, g is
indecomposable
iff X is not the union of two
nonempty open, disjoint, and positively invariant subsets.
Of course, if two open subsets decompose
X
, then they must be also closed.
This means that for an indecomposable DTDS the phase space
X
cannot be
split into two (nontrivial) clopen sub-dynamical systems; indecomposability is
in a certain sense an
irreducibility
condition [11]. Note that in [10] this property
is also called condition of
invariant connection
and
X
is said to be
invariantly
connected
. It easy to show that, in every DTDS
X, g
, full transitivity implies
indecomposability and the existence of a dense orbit implies indecomposability
too.
The following result can be found in [3].
Theorem 4.
Let X, g be a DTDS. If X is perfect and possesses a dense orbit,
then it is positively transitive.
The above notion of full transitivity holds for any metric space
X
and for
any continuous function
g
:
X → X
. If we consider a compact metric space
and a homeomorphic function, our notion of full transitivity coincides with the
definition of topological transitivity given in [6]. We ask whether this “topo-
logical transitivity” (equal to our full transitivity) coincides with the positive
transitivity. The answer is no, as shown in the following example.
Example 9. A homeomorphic, full transitive and compact DTDS which is not
positively transitive.
Let us consider the subshift of finite type
S, σ
S
on the boolean alphabet gen-
erated by the ECA rule 222. The corresponding set of left forbidden blocks is
F
3
(222) =
{
010
,
100
,
101
,
110
}
. A configuration
x ∈{
0
,
1
}
Z
belongs to
S
iff
x
=
0 =(
... ,
0
,
0
,
0
,...
)or
x
=1=(
... ,
1
,
1
,
1
,...
) or it is of the kind
x
: Z
→ {
0
.
1
}
such that for some
k ∈
Z
x
i
=0if
i<k,x
i
= 1otherwise. The subshift
S, σ
S
is
equipped with the Tychonoff metric:
d
T
(
x,y
)=
+
∞
1
4
|
i
|
|x
i
− y
i
|
. Since
S
is
a closed subset of
{
0
,
1
}
Z
then it is compact. Moreover
σ
S
is a homeomorphism.
A property of this subshift is that for any configuration
x ∈ S \{
0
,
1
}
, the
set
{x}
is open. It is easy to show that
S, σ
S
is full transitive but it is not
positively transitive.
i
=
−∞
