Limit set

${\displaystyle \omega (x,f)=\bigcap _{n=1}^{\infty }{\overline {\bigcup _{k=n}^{\infty }\{f^{k}(x)\}}}.}$

If ${\displaystyle f}$ is a homeomorphism (that is, a bicontinuous bijection), then the ${\displaystyle \alpha }$-limit set is defined in a similar fashion, but for the backward orbit; i.e. ${\displaystyle \alpha (x,f)=\omega (x,f^{-1})}$.

Both sets are ${\displaystyle f}$-invariant, and if ${\displaystyle X}$ is compact, they are compact and nonempty.

Definition for flows

Given a

${\displaystyle (T,X,\varphi )}$

${\displaystyle \varphi :\mathbb {R} \times X\to X}$

${\displaystyle x}$

${\displaystyle \omega }$

${\displaystyle x}$

${\displaystyle (t_{n})_{n\in \mathbb {N} }}$

${\displaystyle \mathbb {R} }$

${\displaystyle \lim _{n\to \infty }t_{n}=\infty }$

${\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}$.

For an orbit ${\displaystyle \gamma }$ of ${\displaystyle (T,X,\varphi )}$, we say that ${\displaystyle y}$ is an ${\displaystyle \omega }$-limit point of ${\displaystyle \gamma }$, if it is an ${\displaystyle \omega }$-limit point of some point on the orbit.

Analogously we call ${\displaystyle y}$ an ${\displaystyle \alpha }$-limit point of ${\displaystyle x}$ if there exists a sequence ${\displaystyle (t_{n})_{n\in \mathbb {N} }}$ in ${\displaystyle \mathbb {R} }$ so that

${\displaystyle \lim _{n\to \infty }t_{n}=-\infty }$

${\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}$.

For an orbit ${\displaystyle \gamma }$ of ${\displaystyle (T,X,\varphi )}$, we say that ${\displaystyle y}$ is an ${\displaystyle \alpha }$-limit point of ${\displaystyle \gamma }$, if it is an ${\displaystyle \alpha }$-limit point of some point on the orbit.

The set of all ${\displaystyle \omega }$-limit points (${\displaystyle \alpha }$-limit points) for a given orbit ${\displaystyle \gamma }$ is called ${\displaystyle \omega }$-limit set (${\displaystyle \alpha }$-limit set) for ${\displaystyle \gamma }$ and denoted ${\displaystyle \lim _{\omega }\gamma }$ (${\displaystyle \lim _{\alpha }\gamma }$).

If the ${\displaystyle \omega }$-limit set (${\displaystyle \alpha }$-limit set) is disjoint from the orbit ${\displaystyle \gamma }$, that is ${\displaystyle \lim _{\omega }\gamma \cap \gamma =\varnothing }$ (${\displaystyle \lim _{\alpha }\gamma \cap \gamma =\varnothing }$), we call ${\displaystyle \lim _{\omega }\gamma }$ (${\displaystyle \lim _{\alpha }\gamma }$) a

Alternatively the limit sets can be defined as

${\displaystyle \lim _{\omega }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t>s\}}}}$

and

${\displaystyle \lim _{\alpha }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t<s\}}}.}$

Examples

• For any
  periodic orbit
  ${\displaystyle \gamma }$ of a dynamical system, ${\displaystyle \lim _{\omega }\gamma =\lim _{\alpha }\gamma =\gamma }$
• For any fixed point ${\displaystyle x_{0}}$ of a dynamical system, ${\displaystyle \lim _{\omega }x_{0}=\lim _{\alpha }x_{0}=x_{0}}$

Properties

• ${\displaystyle \lim _{\omega }\gamma }$ and ${\displaystyle \lim _{\alpha }\gamma }$ are closed
• if ${\displaystyle X}$ is compact then ${\displaystyle \lim _{\omega }\gamma }$ and ${\displaystyle \lim _{\alpha }\gamma }$ are
  compact and connected
• ${\displaystyle \lim _{\omega }\gamma }$ and ${\displaystyle \lim _{\alpha }\gamma }$ are ${\displaystyle \varphi }$-invariant, that is ${\displaystyle \varphi (\mathbb {R} \times \lim _{\omega }\gamma )=\lim _{\omega }\gamma }$ and ${\displaystyle \varphi (\mathbb {R} \times \lim _{\alpha }\gamma )=\lim _{\alpha }\gamma }$

• Julia set
• Stable set
• Limit cycle
• Periodic point
• Non-wandering set
• Kleinian group

References

Further reading

• ISBN 978-0-8218-8328-0
  .