2024/04/02 by Fernando A. Oliveira, Oliveira, Fernando
Computer Science · #B20 #B40 #C29 #Cognitive Science and Mapping #Constraint Satisfaction and Optimization #Data Management and Algorithms #Dynamical Systems (math.DS) #E30 #FOS: Mathematics #Primary: 37 #Secondary: A10
paper · pdf · doi:10.48550/arxiv.2404.02209
openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that for the standard map family, for all values of the parameter, except one, the mapping has positive topological entropy. The main tool is the following result. Let S be a compact connected orientable surface and f:S → S an area preserving orientation preserving C \e 1 diffeomorphism of S. Assume that U is an invariant domain of S such that frSU has a finite number of connected components. Let b be a regular ideal boundary point of U which is fixed under the induced action by f on the ideal boundary of U, and let f:C(b) → C(b) the homeomorphism on the corresponding circle of prime ends. Let Z(b) be the impression of b in S and assume that all fixed points of f in Z(b) are non degenerate. If there exists a fixed prime end e ∈ C(b) then we know the following. (1) If p is the principal point of e then p is also a fixed point of Z(b) and p is a saddle. (2) C(b) has a finite number of fixed prime ends and there exists a finite singular covering ϕ:C(b) → Z(b), which is a semiconjugacy between the mapping of prime ends on C(b) and the restriction of f to Z(b). In particular, Z(b) is the connected union of finitely many saddle connections and the corresponding saddles. This can be seen as a two dimensional generalization of the dynamics of homeomorphisms of the circle with fixed points.