# multiplicative cocycle

Let $f:M\to M$ be a measurable transformation, and let $\mu $ be an invariant probability measure^{}. Consider $A:M\to GL(d,\text{\mathbf{R}})$, a measurable transformation, where GL(d,R) is the space of invertible^{} square matrices^{} of size $d$. We define ${A}^{-1}:M\to GL(d,\text{\mathbf{R}})$ by ${A}^{-1}(x)={[A(x)]}^{-1}$.
Then we define the sequence^{} of functions:

$${\varphi}^{n}(x)=A({f}^{n-1}(x))\mathrm{\cdots}A(f(x))A(x)$$ |

$${\varphi}^{-n}(x)={[{\varphi}^{n}({f}^{-n}(x))]}^{-1}$$ |

for $n\ge 1$ and $x\in M$.

It is easy to verify that:

$${\varphi}^{m+n}(x)={\varphi}^{n}({f}^{m}(x)){\varphi}^{m}(x)$$ |

for $n,m\in \text{\mathbf{Z}}$ and $x\in M$.

The sequence ${({\varphi}^{n})}_{n}$ is called a multiplicative cocycle, or just cocycle^{} 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 |