Abstract
For a special class of one-parameter families of unimodal mappings of the form f t(x): [0, 1] → [0, 1], f t = atx/(x + t), 0 ≤ x ≤ 1/2, we establish that, for t ε [0, 1/(a − 2)], a > 2, the topological entropy h(f t) is a function monotonically increasing in the parameter. We prove that there exists a class of one-parameter families of unimodal mappings f t that contains the family indicated above and establish conditions under which the topological entropy h(f t) is a function monotonically increasing in the parameter.
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REFERENCES
L. Alseda, J. Llibre, and M. Misiurewicz, Combinatorial Dynamics and Entropy in One Dimension, World Scientific, Singapore (1993).
H. Bruin, “Non-monotonicity of entropy of interval maps,” Phys. Lett. A, 202, 359–362 (1995).
J. Milnor and W. Thurston, “On iterated maps of the interval,” Lect. Notes Math., 1342, 465–563 (1988).
M. Tsujii, A Simple Proof of Monotonicity of Entropy in the Quadratic Family Using Ruelle Operator, Banach Centre Preprint (1997).
M. Tsujii, “A simple proof for monotonicity of entropy in the quadratic family,” Ergod. Theory Dynam. Syst., 20, 925–933 (2000).
A. Douady and J. H. Hubbard, Etudes Dynamique des Polynômes Complexes, Publ. Math. d'Orsay (1984–1985).
K. Brucks, M. Misiurewicz, and C. Tresser, “Monotonicity properties of the family of trapezoidal maps,” Commun. Math. Phys., 137, 1–12 (1991).
S. F. Kolyada, “One-parameter families of mappings of an interval with negative Schwarzian for which the monotonicity of bifurcations is violated,” Ukr. Mat. Zh., 41, No. 2, 258–261 (1989).
A. Zdunik, “Entropy of transformations of the unit interval,” Fund. Math., 124, 235–241 (1984).
M. Tsujii, Monotonicity of Kneading Sequence for Some Families of Unimodal Maps, Preprint, Hokkaido University (2001).
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston (1980).
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Volkova, O.Y. Monotonicity of Topological Entropy for One-Parameter Families of Unimodal Mappings. Ukrainian Mathematical Journal 55, 1733–1741 (2003). https://doi.org/10.1023/B:UKMA.0000027038.78249.89
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DOI: https://doi.org/10.1023/B:UKMA.0000027038.78249.89