Abstract
We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps.
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L. Alsedà, J. Llibre and M. Misiurewicz,Periodic orbits of maps of Y, Transactions of the American Mathematical Society313 (1989), 475–538.
L. Alsedà, J. Llibre and M. Misiurewicz,Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics5, World Scientific, Singapore, 1993.
L. Alsedà, J. Llibre, F. Mañosas and M. Misiurewicz,Lower bounds of the topological entropy for continuous maps of the circle of degree one, Nonlinearity1 (1988), 463–479.
L. Alsedà, J. Llibre, M. Misiurewicz and C. Simó,Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point, Ergodic Theory and Dynamical Systems5 (1985), 501–517.
L. Alsedà, J. Llibre, M. Misiurewicz and C. Tresser,Periods and entropy for Lorenz-like maps, Annales de l’Institut Fourier39 (1989), 929–952.
S. Baldwin,Generalizations of a theorem of Sharkovskii on orbits of continuous real-valued functions, Discrete Mathematics67 (1987), 111–127.
L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young,Periodic points and topological entropy of one-dimensional maps. Lecture Notes in Mathematics819, Springer-Verlag, Berlin, 1980, pp. 18–34.
A. Blokh,Rotation numbers, twists and a Sharkovskii-Misiurewicz-type ordering for patterns on the interval, Ergodic Theory and Dynamical Systems15 (1995), 1–14.
A. Blokh,Rotation numbers for unimodal maps, MSRI preprint 058-94.
A. Blokh,On rotation intervals for interval maps, Nonlinearity7 (1994), 1395–1417.
A. Blokh,Functional rotation numbers for one-dimensional maps, Transactions of the American Mathematical Society347 (1995), 499–514.
J. Bobok and M. Kuchta,X-minimal orbits for maps on the interval, preprint.
K. Brucks and M. Misiurewicz,Trajectory of the turning point is dense for almost all tent maps, Ergodic Theory and Dynamical Systems16 (1996), 1173–1183.
A. Chenciner, J.-M. Gambaudo and C. Tresser,Une remarque sur la structure des endomorphismes de degré 1du cercle, Comptes Rendus de l’Académie des Sciences, Paris, Série I Math.299 (1984), 145–148.
P. Collet and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems, Progress in Physics1, Birkhäuser, Boston, 1980.
R. Ito,Rotation sets are closed, Mathematical Proceedings of the Cambridge Philosophical Society89 (1981), 107–111.
J.-M. Gambaudo and C. Tresser,A monotonicity property in one dimensional dynamics, Contemporary Mathematics135 (1992), 213–222.
F. Hofbauer,On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, Israel Journal of Mathematics38 (1981), 107–115.
M. Misiurewicz,Periodic points of maps of degree one of a circle, Ergodic Theory and Dynamical Systems2 (1982), 221–227.
M. Misiurewicz,Twist sets for maps of the circle, Ergodic Theory and Dynamical Systems4 (1984), 391–404.
M. Misiurewicz,Minor cycles for interval maps, Fundamenta Mathematicae145 (1994), 281–304.
M. Misiurewicz,Rotation intervals for a class of maps of the real line into itself, Ergodic Theory and Dynamical Systems6 (1986), 117–132.
M. Misiurewicz and Z. Nitecki,Combinatorial patterns for maps of the interval, Memoirs of the American Mathematical Society456 (1990).
M. Misiurewicz and K. Ziemian,Rotation sets for maps of tori, Journal of the London Mathematical Society(2)40 (1989), 490–506.
S. Newhouse, J. Palis and F. Takens,Bifurcations and stability of families of diffemorphisms, Publications Mathématiques de l’Institut des Hautes Études Scientifiques57 (1983), 5–71.
H. Poincaré,Sur les courbes définies par les équations différentielles, Oeuvres completes, Vol. 1, Gauthier-Villars, Paris, 1952, pp. 137–158.
A. N. Sharkovskii,Co-existence of the cycles of a continuous mapping of the line into itself, Ukrainskii Matematicheskii Zhurnal16 (1964), 61–71.
G. Światek,Hyperbolicity is dense in the real quadratic family, SUNY preprint, 1992/10.
K. Ziemian,Rotation sets for subshifts of finite type, Fundamenta Mathematicae146 (1995), 189–201.
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This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.
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Blokh, A., Misiurewicz, M. Entropy of twist interval maps. Isr. J. Math. 102, 61–99 (1997). https://doi.org/10.1007/BF02773795
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DOI: https://doi.org/10.1007/BF02773795