Measure-preserving dynamical system
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[edit] A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system with the following structure: is a set, is a σ-algebra over , is a probability measure, so that μ(X) = 1, and μ(∅) = 0, is a measurable transformation which preserves the measure , i.e., . , the identity function on X;, whenever all the terms are well-defined;, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by definingTs = Ts for s ∈ N. The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem. Examples[edit] Examples include: Homomorphisms[edit] The concept of a homomorphism and an isomorphism may be defined. Consider two dynamical systems and . The system is defined as
Interval exchange transformation
Graph of interval exchange transformation (in black) with and . In blue, the orbit generated starting from In mathematics, an interval exchange transformation[1] is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. Formal definition[edit] Let and let be a permutation on of positive real numbers (the widths of the subintervals), satisfying Define a map called the interval exchange transformation associated to the pair as follows. let Then for , define if lies in the subinterval . acts on each subinterval of the form is moved to position Properties[edit] Any interval exchange transformation is a bijection of to itself preserves the Lebesgue measure. The inverse of the interval exchange transformation is again an interval exchange transformation. where for all If is just a circle rotation. is irrational, then is uniquely ergodic. such that
Random dynamical system
evolving according to a succession of maps randomly chosen according to the distribution Q.[1] Motivation: solutions to a stochastic differential equation[edit] Let be a -dimensional vector field, and let . to the stochastic differential equation exists for all positive time and some (small) interval of negative time dependent upon , where denotes a -dimensional Wiener process (Brownian motion). In this context, the Wiener process is the coordinate process. Now define a flow map or (solution operator) by (whenever the right hand side is well-defined). (or, more precisely, the pair ) is a (local, left-sided) random dynamical system. Formal definition[edit] Formally, a random dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. be a probability space, the noise space. as follows: for each "time" , let be a measure-preserving measurable function: for all and Suppose also that That is, . ; in these cases, the maps is ergodic. Now let , the base flow
Base flow (random dynamical systems)
In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system. In the definition of a random dynamical system, one is given a family of maps on a probability space . The measure-preserving dynamical system is known as the base flow of the random dynamical system. are often known as shift maps since they "shift" time. The parameter may be chosen to run over (a two-sided continuous-time dynamical system); (a one-sided continuous-time dynamical system); (a two-sided discrete-time dynamical system); (a one-sided discrete-time dynamical system). Each map is required to be a -measurable function: for all , to preserve the measure : for all , . Furthermore, as a family, the maps satisfy the relations In other words, the maps form a commutative monoid (in the cases and , where would be given by
