Abstract
We study fractional quadratic transformationsT of the sphere and try to determine their topological entropy. In the case whereT is a constant mapping or a homeomorphism, the topological entropy is of course zero. In the other cases, we have the following results. IfT has only one fixed point, its entropy is log 2. IfT has exactly two fixed points, it can be written asT z =z−z −1 +v, and ifv is real, then the entropy ofT is again log 2. A general result ofMisiurewicz andPrzytycki shows that the entropy ofT is at least log2, and we conjecture that this entropy is always equal to log2 in the remaining cases, i. e. two fixed points andv not real, and three fixed points.
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Literatur
Denker, M., C. Grillenberger, andK. Sigmund: Ergodic Theory on Compact Spaces. Lecture Notes Math. 527. Berlin-Heidelberg-New York: Springer. 1976.
Misiurewicz, M., andF. Przytycki: Topological entropy and degree of smooth mappings. Preprint.
Shub, M.: Dynamical systems, filtrations and entropy. Bull. Amer. Math. Soc.80, 27–41 (1974).
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Denker, M., Keane, M.S. Eine bemerkung zur topologischen entropie. Monatshefte für Mathematik 85, 177–183 (1978). https://doi.org/10.1007/BF01534860
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DOI: https://doi.org/10.1007/BF01534860