multiplicative cocycle

Let f:MM be a measurable transformation, and let μ be an invariant probability measureMathworldPlanetmath. Consider A:MGL(d,𝐑), a measurable transformation, where GL(d,R) is the space of invertiblePlanetmathPlanetmath square matricesMathworldPlanetmath of size d. We define A-1:MGL(d,𝐑) by A-1(x)=[A(x)]-1. Then we define the sequenceMathworldPlanetmath of functions:


for n1 and xM.

It is easy to verify that:


for n,m𝐙 and xM.

The sequence (ϕn)n is called a multiplicative cocycle, or just cocycleMathworldPlanetmath defined by the transformation A.

Title multiplicative cocycle
Canonical name MultiplicativeCocycle
Date of creation 2014-03-19 22:13:54
Last modified on 2014-03-19 22:13:54
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 4
Author Filipe (28191)
Entry type Definition
Synonym cocycle; multiplicative linear cocycle
Related topic Furstenberg-Kesten theorem
Defines multiplicative cocycle