Ergodic river

Ergodic flows are a term from the theory of dynamic systems . Clearly, ergodicity of a river means that almost all points belong to a single flow line.

definition

Let be a probability measure on a probability space and a flow that receives the measure , i.e. H. for all and all measurable quantities applies , wherein . ${\displaystyle \mu }$ ${\displaystyle \Omega }$ ${\displaystyle \phi \colon \Omega \times \mathbb {R} \to \Omega }$ ${\displaystyle \mu }$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle A\subset \Omega }$ ${\displaystyle \mu (\phi _{t}(A))=\mu (A)}$ ${\displaystyle \phi _{t}(A)=\left\{\phi (x,t)\colon x\in A\right\}}$

Then an ergodic flow is called if for every -invariant set : ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle A\subset \Omega }$

${\displaystyle \mu (A)=0}$or . ${\displaystyle \mu (A)=1}$

(A set is called -invariant if it holds for all .) ${\displaystyle A}$ ${\displaystyle \phi }$ ${\displaystyle \phi _{t}(A)=A}$ ${\displaystyle t\in \mathbb {R} }$

An equivalent definition says that it is ergodic if and only if the only -invariant functions are the constant functions . (A function is called -invariant if the equation holds for all for- almost all .) ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle f\in L^{1}(\Omega ,\mu )}$ ${\displaystyle \phi }$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle \mu }$ ${\displaystyle x\in \Omega }$ ${\displaystyle f(\phi (x,t))=f(x)}$

properties

• Because all orbits

${\displaystyle \left\{\phi (x,t),t\in \mathbb {R} \right\}}$

are (with ) -invariant, in particular exactly one orbit must have dimension 1 and all other orbits must have dimension 0. In particular, an ergodic flow defines an ergodic action of the group of real numbers . ${\displaystyle x\in \Omega }$ ${\displaystyle \phi }$ ${\displaystyle \mathbb {R} }$

• The ergodic law applies to ergodic flows :

${\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}f(\phi (x,t))dt=\int _{\Omega }fd\mu }$

for almost everyone and every function . ${\displaystyle \mu }$ ${\displaystyle x\in \Omega }$ ${\displaystyle f\in L^{1}(\Omega ,\mu )}$

Examples

• The geodetic flow of a compact Riemannian manifold with negative sectional curvature is ergodic.
• The horocyclic flow of a compact hyperbolic surface is ergodic.

literature

• A. Katok and B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995, ISBN 0-521-34187-6 .