Let (X, Σ) be a measurable space and let f be a measurable function from X to itself. A measure μ on (X, Σ) is said to be invariant under f if, for every measurable set A in Σ,
{\displaystyle \mu \left(f^{-1}(A)\right)=\mu (A).} \mu \left(f^{{-1}}(A)\right)=\mu (A).
In terms of the push forward, this states that f∗(μ) = μ.
The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted Mf(X). The collection of ergodic measures, Ef(X), is a subset of Mf(X). Moreover, any convex combination of two invariant measures is also invariant, so Mf(X) is a convex set; Ef(X) consists precisely of the extreme points of Mf(X).
In the case of a dynamical system (X, T, φ), where (X, Σ) is a measurable space as before, T is a monoid and φ : T × X → X is the flow map, a measure μ on (X, Σ) is said to be an invariant measure if it is an invariant measure for each map φt : X → X. Explicitly, μ is invariant if and only if
{\displaystyle \mu \left(\varphi {t}^{-1}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .} \mu \left(\varphi }^{{-1}}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .
Put another way, μ is an invariant measure for a sequence of random variables (Zt)t≥0 (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition Z0 is distributed according to μ, so is Zt for any later time t.
When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1, this being the largest eigenvalue as given by the Frobenius-Perron theorem.
Let (X, Σ) be a measurable space and let f be a measurable function from X to itself. A measure μ on (X, Σ) is said to be invariant under f if, for every measurable set A in Σ,
{\displaystyle \mu \left(f^{-1}(A)\right)=\mu (A).} \mu \left(f^{{-1}}(A)\right)=\mu (A).
In terms of the push forward, this states that f∗(μ) = μ.
The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted Mf(X). The collection of ergodic measures, Ef(X), is a subset of Mf(X). Moreover, any convex combination of two invariant measures is also invariant, so Mf(X) is a convex set; Ef(X) consists precisely of the extreme points of Mf(X).
In the case of a dynamical system (X, T, φ), where (X, Σ) is a measurable space as before, T is a monoid and φ : T × X → X is the flow map, a measure μ on (X, Σ) is said to be an invariant measure if it is an invariant measure for each map φt : X → X. Explicitly, μ is invariant if and only if
{\displaystyle \mu \left(\varphi {t}^{-1}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .} \mu \left(\varphi }^{{-1}}(A)\right)=\mu (A)\qquad \forall t\in T,A\in \Sigma .
Put another way, μ is an invariant measure for a sequence of random variables (Zt)t≥0 (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition Z0 is distributed according to μ, so is Zt for any later time t.
When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1, this being the largest eigenvalue as given by the Frobenius-Perron theorem.