We briefly present ongoing work about Martin-Löf randomness and the ergodic decomposition theorem. In [ML66], Martin-Löf defined the notion of an algorithmically random infinite binary sequence. The idea was to have an individual notion of a random object, an object that is acceptable as an outcome of a probabilistic process (typically, the tossing of a coin). This notion has proved successful,...

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...

in this paper, a local approach to the concept of hudetz $g$-entropy is presented. the introduced concept is stated in terms of hudetz $g$-entropy. this representation is based on the concept of $g$-ergodic decomposition which is a result of the choquet's representation theorem for compact convex metrizable subsets of locally convex spaces.

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices. 1. Notation and main result 1.1. Let (Ω,F , μ) be a probability space, and let H denote the space of all d × d matrices with entries in the complex space L2(Ω, μ). H is a Hilbert space with the inner product (1) 〈AB〉 = Φ(AB∗), where φ(A) = ∫ Ω tr A(ω)μ(dμ), and tr denotes the normalized trace o...

Measure-preserving systems arise in a variety of contexts, such as probability theory, information theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X → R, the series of values f(x), f(Tx), f(T x)... are the values of a phy...