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Restrictions on rotation sets for commuting torus homeomorphisms. (English) Zbl 1366.37103

Summary: Let \(K_1,\: K_2\subset \mathbb{R}^2\) be two convex, compact sets. We would like to know if there are commuting torus homeomorphisms \(f\) and \(h\) homotopic to the identity, with lifts \(\tilde f\) and \(\tilde h\) such that \(K_1\) and \(K_2\) are their rotation sets respectively. In this work, we prove some cases where it cannot happen, assuming some restrictions on rotation sets.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E45 Rotation numbers and vectors
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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