Hyperbolic crowd

In the theory of dynamic systems, a set that is invariant under a flow is called a hyperbolic set , if the flow along this set contracts in some directions and expands in other directions. This behavior is typical of chaotic dynamic systems.

Definition of discrete dynamic systems

Let be a diffeomorphism of a compact smooth manifold and its differential. An -invariant subset is a hyperbolic set if the restriction of the tangential bundle to itself as the Whitney sum of two -invariant sub-bundles and can be decomposed so that (for a suitable Riemannian metric) the restriction of to a contraction and the restriction of to an expansion is. This means, ${\displaystyle f\colon M\to M}$ ${\displaystyle Df\colon TM\to TM}$ ${\displaystyle f}$ ${\displaystyle \Lambda \subset M}$ ${\displaystyle T_{\Lambda }M:=TM\mid _{\Lambda }}$ ${\displaystyle \Lambda }$ ${\displaystyle Df}$ ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle Df}$ ${\displaystyle E^{s}}$ ${\displaystyle Df}$ ${\displaystyle E^{u}}$

${\displaystyle T_{\Lambda }M=E^{s}\oplus E^{u},}$

${\displaystyle (Df)_{x}E_{x}^{s}=E_{f(x)}^{s}}$and for everyone ${\displaystyle (Df)_{x}E_{x}^{u}=E_{f(x)}^{u}}$ ${\displaystyle x\in \Lambda }$

and there are constants such that ${\displaystyle 0<\lambda <1,c>0}$

${\displaystyle \|Df^{n}v\|\leq c\lambda ^{n}\|v\|}$for everyone and ${\displaystyle v\in E^{s}}$ ${\displaystyle n>0}$

and

${\displaystyle \|Df^{-n}v\|\leq c\lambda ^{n}\|v\|}$for everyone and . ${\displaystyle v\in E^{u}}$ ${\displaystyle n>0}$

Definition of rivers

Be

${\displaystyle \varphi \colon M\times \mathbb {R} \to M}$

a river on a compact smooth manifold . For we denote the figure ${\displaystyle M}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle \varphi _{t}\colon M\to M}$

${\displaystyle \varphi _{t}(x)=\varphi (x,t)}$

and with her differential. We denote the orbit of a point with ${\displaystyle D\varphi _{t}\colon TM\to TM}$ ${\displaystyle x\in M}$

${\displaystyle {\mathcal {O}}(x,\varphi ):=\left\{\varphi (x,t)|t\in \mathbb {R} \right\}=\left\{\varphi _{t}(x)|t\in \mathbb {R} \right\}}$.

A subset that is invariant among all is a hyperbolic set if the restriction of the tangential bundle to itself can be decomposed as the Whitney sum of two -invariant sub-bundles and and of the tangential bundle of the respective orbits, so that (for a suitable Riemannian metric) the restriction of to a contraction and the restriction of on is expansion. This means, ${\displaystyle \varphi _{t}}$ ${\displaystyle \Lambda \subset M}$ ${\displaystyle TM\mid _{\Lambda }}$ ${\displaystyle \Lambda }$ ${\displaystyle Df}$ ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$ ${\displaystyle Df}$ ${\displaystyle E^{s}}$ ${\displaystyle Df}$ ${\displaystyle E^{u}}$

${\displaystyle T_{x}M=T_{x}{\mathcal {O}}(x,\varphi )\oplus E_{x}^{s}\oplus E_{x}^{u}}$for all , ${\displaystyle x\in \Lambda }$

${\displaystyle (D\varphi _{t})_{x}E_{x}^{s}=E_{\varphi _{t}(x)}^{s}}$and for everyone ${\displaystyle (D\varphi _{t})_{x}E_{x}^{u}=E_{\varphi _{t}(x)}^{u}}$ ${\displaystyle x\in \Lambda ,t\in \mathbb {R} }$

and there are constants such that ${\displaystyle 0<\lambda <1,c>0}$

${\displaystyle \|D\varphi _{t}v\|\leq c\lambda ^{t}\|v\|}$for everyone and ${\displaystyle v\in E^{s}}$ ${\displaystyle t>0}$

and

${\displaystyle \|D\varphi _{-t}v\|\leq c\lambda ^{t}\|v\|}$for everyone and . ${\displaystyle v\in E^{u}}$ ${\displaystyle t>0}$

Stable and unstable bundles, stable and unstable manifolds

The bundles and given by the definition of a hyperbolic set are called stable and unstable bundles, their integral manifolds are called stable and unstable manifolds. ${\displaystyle E^{s}}$ ${\displaystyle E^{u}}$

Anosov River, Anosov Diffeomorphism

If is, one speaks of an Anosov flow or Anosov diffeomorphism . More generally, axiom A-flows or axiom A-diffeomorphisms are often considered in the theory of dynamic systems . ${\displaystyle \Lambda =M}$

literature

• Luis Barreira : Ergodic theory, hyperbolic dynamics and dimension theory. University text. Springer, Heidelberg, 2012. ISBN 978-3-642-28089-4
• Eduard Zehnder : Lectures on dynamical systems. Hamiltonian vector fields and symplectic capacities. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zurich, 2010. ISBN 978-3-03719-081-4
• Michael Brin , Garrett Stuck : Introduction to dynamical systems. Cambridge University Press, Cambridge, 2002. ISBN 0-521-80841-3
• Ken Palmer : Shadowing in dynamical systems. Theory and applications. Mathematics and its Applications, 501. Kluwer Academic Publishers, Dordrecht, 2000. ISBN 0-7923-6179-2
• Zbigniew Nitecki : Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms. The MIT Press, Cambridge, Mass.-London, 1971.
• Dmitri Anosov : Dynamical systems in the 1960s: the hyperbolic revolution. Mathematical events of the twentieth century, 1–17, Springer, Berlin, 2006.