Measure-preserving dynamical system Totally Explained

Everything about Measure-preserving Dynamical System totally explained

In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.

Definition

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it's a system ť

(X, mathcal h_mu(T,Q)

where the supremum is taken over all finite measurable partitions. A theorem of Ya. Sinai in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process is

log 2

, since every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals

[0,1/2)

and

[1/2,1]

. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of

2^nx

.
If the space X is endowed with a metric, then the topological entropy may also be defined.

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