A consequence of birkhoff ergodic theorem tells us that ergodicity is equivalent to:

$\forall A,B \in \mathcal{B} \ \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\longrightarrow} \mu(A)\mu(B)$

In other words, a system is ergodic if every pair of measurable sets are asymptotically independent in Cesàro sum. With this definition of ergodicity is natural to define the mixing systems that are the systems in which every pair of measurable sets are asymptotically independent.

I was wandering if exists a class of system that for every pair of measurable sets $A,B$ exists $N > 0$ such that $\forall n \geq N$ $\mu(A \cap T^{-n})=\mu(A)\mu(B)$.

That class of systems are clearly smaller than the mixing class, but I don't know if there are examples or anything about them. I think that the Bernoulli and Markov shifts presents that kind of behavior.

Any help or reference will help :)

A consequence of Birkhoff ergodic theorem tells us that ergodicity is equivalent to:

$\forall A,B \in \mathcal{B} \quad \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\longrightarrow} \mu(A)\mu(B)$.

In other words, a system is ergodic if every pair of measurable sets are asymptotically independent in Cesàro sum. With this definition of ergodicity is natural to define the mixing systems that are the systems in which every pair of measurable sets are asymptotically independent.

I was wondering if exists a class of system that for every pair of measurable sets $A,B$ exists $N > 0$ such that $\forall n \geq N$ $\mu(A \cap T^{-n}B)=\mu(A)\mu(B)$.

That class of systems are clearly smaller than the mixing class, but I don't know if there are examples or anything about them. I think that the Bernoulli and Markov shifts presents that kind of behavior.

Any help or reference will help :)

A consequence of birkhoff ergodic theorem tells us that ergodicity is equivalent to:

$\forall A,B \in \mathcal{B} \ \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\longrightarrow} \mu(A)\mu(B)$

In other words, a system is ergodic if every pair of measurable sets are asymptotically independent in Cesàro sum. With this definition of ergodicity is natural to define the mixing systems that are the systems in which every pair of measurable sets are asymptotically independent.

I was wandering if exists a class of system that for every pair of measurable sets $A,B$ exists $N > 0$ such that $\forall n \geq N$ $\mu(A \cap T^{-n})=\mu(A)\mu(B)$.

That class of system are clearly more small that the mixing class, but I don't know if there are examples or anything about them. I think that the Bernoulli and Markov shifts present that kind of behavior.

Any help or reference will help :)

A consequence of birkhoff ergodic theorem tells us that ergodicity is equivalent to:

$\forall A,B \in \mathcal{B} \ \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to \infty}{\longrightarrow} \mu(A)\mu(B)$

In other words, a system is ergodic if every pair of measurable sets are asymptotically independent in Cesàro sum. With this definition of ergodicity is natural to define the mixing systems that are the systems in which every pair of measurable sets are asymptotically independent.

I was wandering if exists a class of system that for every pair of measurable sets $A,B$ exists $N > 0$ such that $\forall n \geq N$ $\mu(A \cap T^{-n})=\mu(A)\mu(B)$.

That class of systems are clearly smaller than the mixing class, but I don't know if there are examples or anything about them. I think that the Bernoulli and Markov shifts presents that kind of behavior.

Any help or reference will help :)