On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions

Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoretical and computational machinery beyond modeling and prediction, towards what can be considered as a systematic discovery of "Cole-Hopf-type" transformations for dynamics.Comment: 34 pages, 10 figure

Sets of invariant measures and Cesaro stability

Sets of invariant measures are considered for continuous maps of a metric compact set. We take Kantorovich metric to calculate distance between measures and Hausdorff metrics to calculate distance between compact sets. Consider the function that makes correspondence between a continuous map and the set of all its Borel probability invariant measures. We demonstrate that a typical map is a continuity point of that function. Using approaches of Takens' tolerance stability theory we provide some corollaries that demonstrate that for a typical map points are structurally stable in a statistical sense.Comment: 11 pages, no figure

Full Groups and Orbit Equivalence in Cantor Dynamics

In this note we consider dynamical systems $(X,G)$ on a Cantor set $X$ satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems $(X_1,G_1)$ and $(X_2,G_2)$ are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known Dye's theorem established originally for ergodic measure-preserving actions.Comment: 8 pages, references adde