The problem of combinatorial encoding of a continuous dynamics and the notion of transfer of paths in graphs

TL;DR

This paper introduces a novel combinatorial encoding of continuous dynamical systems using paths in graded graphs, exploring properties like distinguishability and standardness, and generalizes the concept of shift transformations.

Contribution

It proposes the first examples of combinatorial encoding for continuous dynamical systems, linking graph paths with measure-theoretic properties and introducing the notion of transfer of paths.

Findings

Encoding isomorphic to the original process under certain conditions

Existence of suitable graded graphs relates to standardness of orbit partitions

Transfer of paths generalizes shift transformations in stationary dynamics

Abstract

We introduce the notion of combinatorial encoding of continuous dynamical systems and suggest the first examples, which are the most interesting and important, namely, the combinatorial encoding of a Bernoulli process with continuous state space, e.g., a sequence of i.i.d. random variables with values in the interval with the Lebesgue measure (or a Lebesgue space). The main idea is to associate with a random object (a trajectory of the random process) a path in an $\N$-graded graph and parametrize it with the vertices of the graph that belong to this path. This correspondence (encoding) is based on the definition of a decreasing sequence of cylinder partitions, and the first problem is to verify whether or not the given combinatorial encoding has the property of distinguishability, which means that our encoding is an isomorphism, or, equivalently, the limit of the increasing sequence of finite partitions is the partition into singletons $\bmod\,0$. This is a generalization of the problem of generators in ergodic theory. The existence of a suitable $\N$-graded graph is equivalent to the so-called standardness of the orbit partition in the sense of the theory of filtrations in measure spaces. In the last section, we define the notion of a so-called transfer, a transformation of paths in a graded graph, as a generalization of the shift in stationary dynamics.