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The trajectory of the turning point is dense for almost all tent maps

Published online by Cambridge University Press:  14 October 2010

Karen Brucks  and
Michal Misiurewicz
Karen Brucks
Affiliation:
Department of Mathematical Sciences, U-WI-Milwaukee, Milwaukee, WI 53211, USA, (e-mail: [email protected])
Michal Misiurewicz
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Indianapolis, IN 46202, USA, (e-mail: [email protected])
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Abstract

We prove that for almost every (with respect to the Lebesgue measure) a ∈ [√2, 2], the forward trajectory of the turning point of the tent map fa with slope a is dense in the interval of transitivity of fa.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1173 - 1183
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1] Block, L. and Coppel, W. A.. Dynamics in One Dimension (Lecture Notes in Mathematics 1513). Springer, New York, 1992.Google Scholar
[2] Bracks, K. M., Diamond, B., Otero-Espinar, M. V. and Tresser, C.. Dense orbits of critical points for the tent map. Contemp. Math. 117 (1991), 57–61.CrossRefGoogle Scholar
[3] Collet, P. and Eckmann, J.-P.. Iterated maps on the interval as dynamical systems. Progress in Phys. J. Birkhauser, Boston, 1980.Google Scholar
[4] Coven, E. M., Kan, I. and Yorke, J. A.. Pseudo-orbit shadowing in the family of tent maps. Trans. Amer. Math. Soc. 308 (1988), 227–241.Google Scholar
[5] Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics 527). Springer, Berlin, 1976.CrossRefGoogle Scholar
[6] Jakobson, M. V.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[7] Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise montonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
[8] Misiurewicz, M. and Szlenk, W.. Entropy of piecewise monotone mappings. Studia Math. 67 (1980), 45–63.CrossRefGoogle Scholar
[9] Misiurewicz, M. and Visinescu, E.. Kneading sequences of skew tent maps. Ann. Inst. H. Poincare Probab. Stat. 27 (1991), 125–140.Google Scholar
[10] Strien, S. van. Smooth dynamics on the interval. New Directions in Dynamical Systems (London Math. Soc. Lecture Note Ser. 127). Cambridge University Press, 1988, pp. 57–119.Google Scholar