4.4 Article

On the structure of rotation sets in hyperbolic surfaces

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WILEY
DOI: 10.1112/jlms.12710

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In this study, we establish the relationship between the genus and possible geometries for homological rotation sets of maps on closed oriented surfaces with a genus g >= 2. We demonstrate that this invariant for Smale diffeomorphisms can be described as the union of at most 2(5g-3) convex sets, all containing zero. By utilizing the theory of hyperbolic dynamics, we extend this bound to a C-0-open and dense set of homeomorphisms, indicating its general validity. Additionally, we provide examples that illustrate the sharpness of this asymptotic order.
Searching for a relation between the genus g >= 2 of a closed oriented surface and the possible geometries for homological rotation sets of its maps, we prove that this invariant for Smale diffeomorphisms is given by a union of at most 2(5g-3) convex sets, all of them containing zero. The classical theory of hyperbolic dynamics allows then to extend this bound to a C-0-open and dense set of homeomorphisms, suggesting this to be a general fact. Examples showing the sharpness for this asymptotic order are provided.

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