Mean ergodic theorem von Neumann

[trosten]

I wonder If someone could state the mean ergodic theorem von neumann without using meassure spaces ? I have studied normed spaces, banach spaces and hilbert spaces, that is complete normed inner product spaces.

Could someone state and explain the theorem for me?

 

[matt grime]

As ergodic theory is the study of invariants measurable spaces, in some loose sense, this doesn't seem very likely. However, reading a statement of it I don't understand what the measure theoretic objection is. One can translate to little l 2 and it states that, modulo some dodgy rearranging of terms, we have approximately:

given an isometry of hilbert space l_2, T, and an element a in l_2, there is a b in l_2 such that

$$\frac{1}{n} \sum_{k=0}^n <T^ka,b>^2$$

tends to zero as n tends to infinity.

 

[tacman]

The mean ergodic theorem von Neumann is a fundamental result in the field of ergodic theory, which studies the long-term behavior of dynamical systems. It was first introduced by mathematician John von Neumann in the 1930s.

The theorem states that for a dynamical system defined on a complete normed inner product space, the average of a function over a trajectory of the system will converge to the average of the function over the entire space, as the trajectory becomes infinitely long. In other words, the long-term behavior of the system can be determined by the average behavior over a single trajectory.

To understand this theorem, we need to first define some key concepts. A dynamical system is a mathematical model that describes the evolution of a system over time. It consists of a set of states and a set of rules that determine how the system evolves from one state to another. In this context, a trajectory refers to the sequence of states that the system goes through over time.

Now, let's consider a function defined on the space of states of the dynamical system. This function could represent any property of the system, such as its energy or its position. The mean ergodic theorem von Neumann states that if we take the average value of this function over a long trajectory of the system, it will converge to the average value of the function over the entire space.

In simpler terms, this means that the long-term behavior of the system can be predicted by looking at the average behavior over a single trajectory. This is a powerful result, as it allows us to make predictions about a system without having to observe its behavior over an infinitely long period of time.

To summarize, the mean ergodic theorem von Neumann is a fundamental result in ergodic theory that states that the average behavior of a dynamical system over a single trajectory will converge to the average behavior over the entire space. It provides a powerful tool for understanding the long-term behavior of dynamical systems.