Dynamical system | AdamátorWiki

Definition

A dynamical system is a pair

(X, F)

, where

X

is a compact metric space and

F : X \to X

is a continuous function. Hereafter, assume that

(X, F)

is a dynamical system.

Definition

A subset

Y \subseteq X

is invariant if

F (Y) \subseteq Y

Definition

The orbit of a point

x \in X

𝒪 (x) ≔ {F^{n} (x) | n \in ℕ} .

Definition

A morphism between two dynamical systems

(X, F), (Y, G)

is a continuous function

π : X \to Y

such that

π \circ F = G \circ π

A bijective morphism is a conjugacy and the systems are conjugate.
A surjective morphism is a factor map, $(Y, G)$ is a factor of $(X, F)$ and $(X, F)$ is an extension of $(Y, G)$ .
If $π$ is injective, $(X, F)$ is a subsystem of $(Y, G)$ .

Definition

A subset

Z \subseteq X

is invariant if

F (Z) \subseteq Z

Definition

The orbit relation

𝒪 \subseteq X^{2}

is defined as

(x, y) \in 𝒪 ⟺ \exists n > 0, y = F^{n} (x) .

Definition

The recurrence relation

ℛ \subseteq X \times X

is defined as

(x, y) \in ℛ ⟺ \forall B (y), \exists n > 0, F^{n} (x) \in B (y) .

Definition

The nonwandering relation

𝒩 \subseteq X \times X

is defined as

(x, y) \in 𝒩 ⟺ \forall B (x), B (y), \exists n > 0, \exists z \in B (x), F^{n} (z) \in B (y) .

Definition

The chain relation

𝒞 \subseteq X \times X

is defined as

(x, y) \in 𝒞 ⟺ \forall ε > 0, \exists (x = x_{0}, x_{1}, \dots, x_{n - 1}, x_{n} = y), \forall i \in ⟦ 0, n ⦈, d (F (x_{i}), x_{i + 1}) < ε .

Theorem

𝒪 \subseteq ℛ \subseteq 𝒩 = \bar{𝒪} \subseteq 𝒞

Theorem

𝒪

and

𝒞

are transitive.

Theorem

𝒩

and

𝒞

are closed.

Definition

(X, F)

minimal if $ℛ = X^{2}$ ;
transitive if $𝒩 = X^{2}$ ;
chain-transitive if $𝒞 = X^{2}$ .

Definition

A point

x \in X

is transitive if its orbit is dense, i.e.

\bar{𝒪 (x)} = X

. The set of all transitive points is denoted

𝒯

Theorem

(X, F)

is minimal iff:

every $x \in X$ is transitive;
$(X, F)$ does not have a proper subsystem (a set $\emptyset \neq Y \neq X$ that is closed and invariant).

Theorem

(X, F)

is transitive iff:

$𝒯$ is residual;
$𝒯$ is non-empty.

Definition

Denote by

| R |

the diagonal of a relation

R

. Then

$| 𝒪 |$ is the set of periodic points, i.e. a point $x$ is periodic if $\exists n > 0, F^{n} (x) = x$ ;
$| ℛ |$ is the set of recurrent points, i.e. a point $x$ is recurrent if $\forall B (x), \exists n > 0, F^{n} (x) \in B (x)$ ;
$| 𝒩 |$ is the set of nonwandering points, i.e. a point $x$ is nonwandering if $\forall B (x), \exists n > 0, \exists z \in B (x), F^{n} (z) \in B (x)$ ;
$| ℛ |$ is the set of chain-recurrent points, i.e. a point $x$ is chain-recurrent if $\forall ε > 0, \exists (x = x_{0}, x_{1}, \dots, x_{n - 1}, x_{n} = x), \forall i \in ⟦ 0, n ⦈, d (F (x_{i}), x_{i + 1}) < ε$ .

Theorem

| 𝒪 | \subseteq | ℛ | \subseteq | 𝒩 | \subseteq | 𝒞 |

Theorem

| 𝒪 |, | ℛ |, | 𝒩 |, | 𝒞 |

are invariant.

Theorem

| 𝒩 |, | 𝒞 |

are closed.

Definition

(X, F)

is chaotic if

𝒩 = X^{2}

\bar{| 𝒪 |} = X

and

X

is infinite.

Definition

A point

x \in X

is fixed if

F (x) = x

Definition

A point

x \in X

is preperiodic if

F^{m} (x)

is periodic for some

m \in ℕ

Definition

A point

x \in X

is quasi-periodic if

\forall B (x), \exists p > 0, \forall n \geq 0, F^{n p} (x) \in B (x)

Definition

A point

x \in X

is almost periodic if

\forall B (x), \exists p > 0, \forall n \geq 0, \exists k < p, F^{n + k} (x) \in B (x)

Theorem

For a given point

x \in X

, the following series of implications holds:

fixed ⟹ periodic ⟹ quasi-periodic ⟹ almost periodic ⟹ recurrent ⟹ nonwandering ⟹ chain-recurrent

Theorem Birkhoff

A point

x \in X

is almost periodic iff

\bar{𝒪 (x)}

is a minimal system.

Definition

A point

x \in X

is equicontinuous if

\forall ε > 0, \exists B (x), \forall y \in B (x), \forall n \geq 0, d (F^{n} (y), F^{n} (x)) < ε .

The set of all equicontinuous points is denoted

ℰ

. If

ℰ = X

(X, F)

is equicontinuous. If

ℰ

is residual,

(X, F)

is almost equicontinuous.

Definition

(X, F)

is sensitive if

\exists ε > 0, \forall x \in X, \forall B (x), \exists y \in B (x), \exists n \geq 0, d (F^{n} (y), F^{n} (x)) \geq ε .

(X, F)

Bibliography