Metric ergodicity

In mathematics , metric ergodicity is a reinforcement of the concept of ergodicity .

Metric ergodicity

A dimensionally maintaining effect of a group on a dimensional space is called metric ergodic if for every isometric effect of the group on a separable metric space every - equivariate figure is almost everywhere constant. ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle G}$

From metric ergodicity, ergodicity follows by applying the condition . ${\displaystyle M=\left\{0,1\right\}}$

Relative metric ergodicity

definition

An equivariate mapping between Lebesgue G spaces is relatively metrically ergodic if there is an equivariate mapping with for every equivariate Borel map with a fiber-wise isometric G effect and for all equivariate maps with . ${\displaystyle p\colon A\to B}$ ${\displaystyle q\colon M\to V}$ ${\displaystyle f\colon A\to M,f_{0}\colon B\to V}$ ${\displaystyle f_{0}p=qf}$ ${\displaystyle f_{1}\colon B\to M}$ ${\displaystyle f_{1}p=f,qf_{1}=f_{0}}$

properties

• The linking of relatively metric ergodic G-maps is again relatively metric ergodic.
• If relative metric is ergodic then this also applies , but not necessarily applies . ${\displaystyle gf}$ ${\displaystyle g}$ ${\displaystyle f}$
• If the projection is relatively metric ergodic, then metric is ergodic. ${\displaystyle A\times B\to B}$ ${\displaystyle A}$
• If a grid is in a Lie group , then a relatively metric ergodic map is also a relatively metric ergodic map. ${\displaystyle \Gamma }$ ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle G}$

literature

• U. Bader, A. Furman: Boundaries, rigidity of representations, and Lyapunov exponents , Proceedings of ICM 2014, Invited Lectures, (2014), 71-96.
• U. Bader, A. Furman: Boundaries, Weyl groups, and Superrigidity , Electron. Res. Announc. Math. Sci., Vol 19 (2012), 41-48.
• U. Bader, B. Duchesne, J. Lcureux (2014). Furstenberg Maps for CAT (0) Targets of Finite Telescopic Dimension.