2、Classical Dynamics and Ergodic Theory: An In - Depth Exploration

Classical Dynamics and Ergodic Theory: An In - Depth Exploration

1. Introduction to Classical Dynamical Systems

The choice of appropriate observables to describe a dynamical system is closely related to the structure of the chosen phase - space (X). For statistical descriptions, (X) needs a measure - structure, and measurable functions serve as suitable observables. If (X) has a topology, continuous functions are typical observables.

1.1 Definition of Classical Dynamical Systems

Classical dynamical systems are represented as triplets ((X, T, \mu)), where:
1. (X) is a measure space with an assigned (\sigma) - algebra (\Sigma) of measurable sets.
2. (T) is measurable, i.e., if (A\in\Sigma), then (T^{-1}(A)\in\Sigma).
3. (X) has