2019/11/01 by A. M. Vershik, Vershik, Anatoly
Computer Science · Mathematics · #37A05 #94A24 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1911.00509
openalex publication_date 2019/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of combinatorial encoding of continuous dynamical\nsystems and suggest the first examples, which are the most interesting and\nimportant, namely, the combinatorial encoding of a Bernoulli process with\ncontinuous state space, e.g., a sequence of i.i.d. random variables with values\nin the interval with the Lebesgue measure (or a Lebesgue space).\n The main idea is to associate with a random object (a trajectory of the\nrandom process) a path in an N-graded graph and parametrize it with the\nvertices of the graph that belong to this path. This correspondence (encoding)\nis based on the definition of a decreasing sequence of cylinder partitions, and\nthe first problem is to verify whether or not the given combinatorial encoding\nhas the property of distinguishability, which means that our encoding is an\nisomorphism, or, equivalently, the limit of the increasing sequence of finite\npartitions is the partition into singletons bmod ,0. This is a\ngeneralization of the problem of generators in ergodic theory.\n The existence of a suitable N-graded graph is equivalent to the so-called\nstandardness of the orbit partition in the sense of the theory of filtrations\nin measure spaces.\n In the last section, we define the notion of a so-called transfer, a\ntransformation of paths in a graded graph, as a generalization of the shift in\nstationary dynamics.\n