Finite number of ergodic random Dirac measures

Question

Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map \begin{align*} \theta\colon T\times \Omega &\to \Omega\\ (t,\omega)&\mapsto \theta_{t}\omega \end{align*} such that $\theta_{t+s}=\theta_{t}\circ \theta_{s}$ for every $s, t\in T$ and $\theta_{t}$ preserves $\mathbb{P}$ for every $t\in T$, that is, $\theta_{t}\mathbb{P}=\mathbb{P}$ for every $t\in T$. In other words, $(\theta_{t})$ is a measure preserving dynamical system on $(\Omega, \mathscr{F}, \mathbb{P})$.

Definition 1 (Random Dynamical Systems and Invariant Measures): Let $(\Omega, \mathscr{F}, \mathbb{P})$ and $(\theta_{t})$ as above and consider a compact metric space $X$ (endowed with its Borel $\sigma$-algebra $\mathscr{B}$).
A measurable map \begin{align*} \varphi\colon T\times \Omega\times X &\to X\\ (t,\omega,x)& \mapsto \varphi(t,\omega)x \end{align*} is called a random dynamical system on $X$ over the measure preserving dynamical system $(\theta_{t})$ if

a) $x\mapsto \varphi(t,\omega)x$ is continuous for $\mathbb{P}$-almost every $\omega$ and every $t\in T$.

b) $\varphi(t+s, \omega)=\varphi(t,\theta_{s}\omega)\circ \varphi(s,\omega)$ for all $t,s\in T$, and $\varphi(0,\omega)=id$, for $\mathbb{P}$-almost every $\omega$.

A random dynamical system $\varphi$ induces a dynamical system over $\Omega\times X$ given by $$ \Theta(t)(\omega, x)=(\theta(t)\omega, \varphi(t,\omega)x),\quad t\in {T} $$ called the skew product associated with $\,\varphi$.

A probability measure $\mu$ on $\Omega\times X$ is said to be a $\varphi$-invariant measure for the random dynamical system $\varphi$, if it satisfies

1. $\Theta(t)\mu=\mu$ for all $t\in T$;

2. $\Pi\mu=\mathbb{P}$, where $\Pi$ is the natural projection on the first factor defined by $\Pi(\omega,x)=\omega$.

Let $\mu$ be a $\varphi$-invariant measure. Since $\Pi\mu=\mathbb{P}$, we can consider the disintegration $d\mu=\mu_{\omega}\, d\mathbb{P}(\omega)$ of $\mu$ with respect to $\mathbb{P}$. A necessary and sufficient condition for $\mu$ to be a $\varphi$-invariant measure is that for every $t\in T$ \begin{equation}\label{inva} \varphi(t,\omega)\mu_{\omega}=\mu_{\theta_{t}\omega} \end{equation} for $\mathbb{P}$-almost every $\omega$.

Definiton 2: We say that $\mu$ is a random Dirac measure if there is measurable map $\Phi\colon \Omega\to X$ such that $$ \mu_{\omega}=\delta_{\Phi(\omega)} $$ for $\mathbb{P}$-almost every $\omega$.

MY QUESTION: I am looking for examples of Random Dynamical Systems (References of Papers) in which there is a finite($\geq$ 2) number of ergodic $\varphi$-invariant measures, which are, additionally, random Dirac measures.

I am most interested in the continuous time cases, but examples (References of Papers)in the discrete time setting is also welcome.

Answer 1

Let $\theta$ be a discrete time dynamical system as in your question. And for each $\omega$, let $A(\cdot)$ be a measurable map from $\Omega$ into $\text{GL}(d,\mathbb R)$ such that $\log \|A(\cdot)\|$ and $\log\|A(\omega)^{-1}\|$ are integrable.

Let me also assume $\mathbb P$ is ergodic. Then the multiplicative etgodic theorem guarantees that there exist $\lambda_1>\ldots>\lambda_k$; multiplicities $d_1,\ldots,d_k$ summing to $d$ and subspaces $V_1(\omega),\ldots,V_k(\omega)$ of dimensions $d_1,\ldots,d_k$ respectively whose direct sum is $\mathbb R^d$ such that:

• $A(\omega)V_i(\omega)=V_i(\sigma\omega)$;

• if $x\in V_i(\omega)\setminus\{0\}$, then $(1/n)\log\|A^{(n)}(\omega)v\|\to\lambda_i$.

The $\lambda_i$ are called Lyapunov exponents and the $V_i(\omega)$ are called Oseledets spaces. A common case is when the multiplicities are all 1. This is called simple Lyapunov spectrum.

Given such a system, one can build a natural map from $\Omega\times P\mathbb R^{d-1}$ to itself by $\bar \Theta(\omega,v)=(\theta\omega, A(\omega)x)$, where matrices act on the projective space in the obvious way.

In the case where the original bicycle has simple Lyapunov spectrum, the delta measures sitting on the directions of the Oseledets spaces are exactly random Dirac measures as you are asking for.

There are also continuous time versions of all of this. These systems arise very naturally in the context of differentiable maps of manifolds or differentiable flows on manifolds.