4 Nonautonomous Attractors
65
(1) initial condition: ϕ(0, p, x) = x for all p ∈ P and x ∈ R
d ,
(2) cocycle property: ϕ(n + m, p, x) = ϕ(s, θ m ( p), ϕ(m, p, x)) for all n, m ∈
Z
+ , p ∈ P and x ∈ R
d ,
(3) continuity: ( p, x) → ϕ(n, p, x) is continuous for each n ∈Z
+ .
The cocycle property is a generalisation of both the semi-group property and the
2-parameter semi-group property.
Remark 4.1 Note that we have used φ to denote a process and ϕ to denote the cocycle
mapping of a skew product flow. A process φ can be written as a skew product flow
with the driving system given by the left shift operator θ(n 0 ) = n 0 + 1 on the base
space P = Z and the cocycle mapping defined by ϕ(n, n 0 , x 0 ) = φ(n + n 0 , n 0 , x 0 ).
Remark 4.2 The base system θ serves as a driving system which makes the cocycle
mapping nonautonomous. It is like a clock which keeps track of time.
Skew product flows often have very nice properties when the base space P is
compact. This occurs when the driving system is, for example, periodic or almost
periodic. It provides more detailed information about the dynamical behaviour of the
system. George Sell, a pioneering researcher in the area, e.g., see [27], described the
effect of a compact base space as being equivalent to compactifying time.
4.6.1 An Example
Consider a nonautonomous difference equation on R
d
= R given by
x n+1 = f j n (x n ) ,
n ∈ Z,
where the functions f 1 , . . ., f N are continuous and the j n are the components of a biinfinite sequence s = (. . . , j −1 , j 0 , j 1 , j 2 , . . .). Let P = {1, . . . , N }
Z be the totality
of all such bi-infinite sequences. Then
d P
s, s
:=
∞
n=−∞
2
−|n|
j n − j
n
defines a metric on P and(P, d P ) is a compact metric space.
Let θ be the left shift operator on P, i.e., (θs) n = j n+1 for n ∈ Z. Define θ n = θ
n ,
the n-fold composition of θ when n > 0 and of its inverse θ
−1 when n < 0. Then
{θ n } n∈Z is a group under composition on P and the θ n : P → P are continuous in
(P, d P ).
Moreover, the mapping ϕ : Z
+
× P × R
d
→ P defined by
ϕ (n, s, x 0 ) = f j n−1 ◦ · · · ◦ f j 0 (x 0 )
65
(1) initial condition: ϕ(0, p, x) = x for all p ∈ P and x ∈ R
d ,
(2) cocycle property: ϕ(n + m, p, x) = ϕ(s, θ m ( p), ϕ(m, p, x)) for all n, m ∈
Z
+ , p ∈ P and x ∈ R
d ,
(3) continuity: ( p, x) → ϕ(n, p, x) is continuous for each n ∈Z
+ .
The cocycle property is a generalisation of both the semi-group property and the
2-parameter semi-group property.
Remark 4.1 Note that we have used φ to denote a process and ϕ to denote the cocycle
mapping of a skew product flow. A process φ can be written as a skew product flow
with the driving system given by the left shift operator θ(n 0 ) = n 0 + 1 on the base
space P = Z and the cocycle mapping defined by ϕ(n, n 0 , x 0 ) = φ(n + n 0 , n 0 , x 0 ).
Remark 4.2 The base system θ serves as a driving system which makes the cocycle
mapping nonautonomous. It is like a clock which keeps track of time.
Skew product flows often have very nice properties when the base space P is
compact. This occurs when the driving system is, for example, periodic or almost
periodic. It provides more detailed information about the dynamical behaviour of the
system. George Sell, a pioneering researcher in the area, e.g., see [27], described the
effect of a compact base space as being equivalent to compactifying time.
4.6.1 An Example
Consider a nonautonomous difference equation on R
d
= R given by
x n+1 = f j n (x n ) ,
n ∈ Z,
where the functions f 1 , . . ., f N are continuous and the j n are the components of a biinfinite sequence s = (. . . , j −1 , j 0 , j 1 , j 2 , . . .). Let P = {1, . . . , N }
Z be the totality
of all such bi-infinite sequences. Then
d P
s, s
:=
∞
n=−∞
2
−|n|
j n − j
n
defines a metric on P and(P, d P ) is a compact metric space.
Let θ be the left shift operator on P, i.e., (θs) n = j n+1 for n ∈ Z. Define θ n = θ
n ,
the n-fold composition of θ when n > 0 and of its inverse θ
−1 when n < 0. Then
{θ n } n∈Z is a group under composition on P and the θ n : P → P are continuous in
(P, d P ).
Moreover, the mapping ϕ : Z
+
× P × R
d
→ P defined by
ϕ (n, s, x 0 ) = f j n−1 ◦ · · · ◦ f j 0 (x 0 )