Renormalization in Lorenz maps -- completely invariant sets and periodic orbits

Abstract: The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincar`e maps in geometric models of well known Lorenz attractor. We study connections between periodic points, completely invariant sets and renormalizations. We show that in general, renormalization cannot be fully characterized by a completely invariant set, however there are various situations when such characterization is possible. This way we provide a better insight into the structure of renormalizations in Lorenz maps, correcting some gaps existing in the literature and completing to some extent the description of possible dynamics in this important field of study.

Mathematics Subject Classification ID

Dynamical systems involving maps of the interval (37E05) Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) (37B05) Universality and renormalization of dynamical systems (37E20)

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