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Example 5. A non positively transitive DTDS which possesses a dense orbit.
Let
N
,g
s
be the DTDS of the example 4. The orbit
{g
s
(0)
}
t∈
N
of initial state
0
∈
N is dense since it coincides with the whole phase space N.
This behavior holds also in the case of compact DTDS.
Example 6. A compact and non positively transitive DTDS with a dense orbit.
Let us consider the phase space
X
=
{
0
}∪{
2
n
:
n ∈
N
}
equipped with the
metric
d
(
x, y
)=
|x − y|
induced by the usual metric of R; in this metric every
singleton
{x}
[
x
= 0] is a clopen. The topology induced from
d
besides the trivial
open sets
{X, ∅}
contains
P
(
{
2
n
:
n ∈
N
}
), the power set of
{
2
n
:
n ∈
N
}
, and
the open balls centered in 0. In particular this topological space is
compact
.
Let
g
:
X → X
be the map defined as:
∀x ∈ X
,
g
(
x
)=
2
x
. This map is
continuous in the topology induced from
d
. The unique positive orbit dense in
X
is the sequence of initial state
x
=1:
γ
1
=
{
1
,
(1
/
2)
,
(1
/
2
2
)
,... ,
(1
/
2
t
)
,...}
.
But this dynamical system is not positively transitive. Indeed, if we consider the
two nonempty open sets
A
=
{
1
/
2
}
and
B
=
{
1
}
, then we have that
∀n ∈
N:
g
n
(
A
)
∩ B
=
∅
.
Note that there exist DTDS's which are positively transitive but without
dense orbits.
Example 7. A positively transitive DTDS which does not possess any dense orbit.
Let
Per
(
σ
)
,σ
the dynamical subsystem constituted by all periodic points of
the full shift
A
Z
,σ
. It is a positively transitive DTDS which has no dense orbit.
We now introduce a weaker notion of transitivity named
Full Transitivity
.
Definition 7.
A DTDS X, g is said to be
topologically full transitive
iff for
every pair of nonempty open subsets A and B of the phase space, there exists an
integer t
0
∈
Z
such that A ∩ g
−t
0
(
B
)
=
∅.
As stressed in the introduction, this is the notion of transitivity one can find
in [6] in the case of a homeomorphic DTDS on a compact space. We want to
remark that for any pair
A, B
of nonempty open subset of
X
the following three
conditions are mutually equivalent:
∃t
0
∈
Z :
A ∩ g
−t
0
(
B
)
=
∅
Full Transitivity)
∃t
1
∈
Z :
g
t
1
(
A
)
∩ B
=
∅
FT1)
∃t
2
∈
Z :
g
t
2
(
B
)
∩ A
=
∅
FT2)
Note that the existence of a orbit dense is a su
@
cient condition to guarantee
the full transitivity. It is easy to show that a positively transitive DTDS is full
transitive. The converse does not hold, see the example 4 and the following
example where a homeomorphic DTDS is involved.
Example 8.
Let
X
= Z be the set of integer numbers equipped with the trivial
metric and
g
p
: Z
→
Z
∀x ∈
Z,
g
p
(
x
)=
x −
1. It is easy to show that the system
Z
,g
p
is homeomorphic and full transitive, but it is not positively transitive.
Indeed, if
A
=
{
3
}
and
B
=
{
5
}
, there is no integer
n
0
∈
N s.t.
g
n
p
(
A
)
∩ B
=
∅
.
