Abstract
A well-known conjecture about the transformationT a :x ↦ax (1−x) on [0, 1], where 2≤a≤4, says that the mapa ↦h top (T a ) is monotone. In this paper we show that this is connected with a property of the polynomialsP k (t) (4≤t≤8) given byP 0 (t)=0 andP k+1 (t)=(t−P k t)2)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a∈[2,4]:h top (T c ), ifc<a andd>a}.
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References
Hofbauer, F.: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math.34, 213–237 (1979).
Hofbauer, F.: On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II., to appear in Israel J. Math.
Denker, M., Ch. Grillenberger, andK. Sigmund: Ergodic Theory on Compact Spaces. Lecture Notes Math. 527. Berlin-Heidelberg-New York: Springer. 1976.
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Hofbauer, F. The topological entropy of the transformationx ↦ax (1−x). Monatshefte für Mathematik 90, 117–141 (1980). https://doi.org/10.1007/BF01303262
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DOI: https://doi.org/10.1007/BF01303262