Trajectory of the turning point is dense for a co-σ-porous set of tent maps

Karen Brucks; Zoltán Buczolich

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 2, page 95-123
  • ISSN: 0016-2736

Abstract

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It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map T a with slope a is dense in the interval of transitivity of T a . We prove that the complement of this set of parameters of full measure is σ-porous.

How to cite

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Brucks, Karen, and Buczolich, Zoltán. "Trajectory of the turning point is dense for a co-σ-porous set of tent maps." Fundamenta Mathematicae 165.2 (2000): 95-123. <http://eudml.org/doc/212465>.

@article{Brucks2000,
abstract = {It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map $T_a$ with slope a is dense in the interval of transitivity of $T_a$. We prove that the complement of this set of parameters of full measure is σ-porous.},
author = {Brucks, Karen, Buczolich, Zoltán},
journal = {Fundamenta Mathematicae},
keywords = {tent map; dense trajectory; -porous},
language = {eng},
number = {2},
pages = {95-123},
title = {Trajectory of the turning point is dense for a co-σ-porous set of tent maps},
url = {http://eudml.org/doc/212465},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Brucks, Karen
AU - Buczolich, Zoltán
TI - Trajectory of the turning point is dense for a co-σ-porous set of tent maps
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 2
SP - 95
EP - 123
AB - It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map $T_a$ with slope a is dense in the interval of transitivity of $T_a$. We prove that the complement of this set of parameters of full measure is σ-porous.
LA - eng
KW - tent map; dense trajectory; -porous
UR - http://eudml.org/doc/212465
ER -

References

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  1. [1] L. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, New York, 1992. 
  2. [2] K. Brucks, B. Diamond, M. V. Otero-Espinar and C. Tresser, Dense orbits of critical points for the tent map, in: Contemp. Math. 117, Amer. Math. Soc., 1991, 57-61. Zbl0746.34029
  3. [3] K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems 16 (1996), 1173-1183. Zbl0874.58014
  4. [4] H. Bruin, Invariant measures of interval maps, Ph.D. thesis, Delft, 1994. 
  5. [5] H. Bruin, Combinatorics of the kneading map, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1339-1349. Zbl0886.58023
  6. [6] H. Bruin, Quasi-symmetry of conjugacies between interval maps, Nonlinearity 9 (1996), 1191-1207. Zbl0895.58018
  7. [7] H. Bruin, Topological conditions for the existence of absorbing Cantor sets, Trans. Amer. Math. Soc. 350 (1998), 2229-2263. Zbl0901.58029
  8. [8] H. Bruin, For almost every tent map, the turning point is typical, Fund. Math. 155 (1998), 215-235. Zbl0962.37015
  9. [9] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980), 117-141. Zbl0433.54009
  10. [10] F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319-337. Zbl0702.58034
  11. [11] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993. Zbl0791.58003
  12. [12] D. Preiss and L. Zajíček, Fréchet differentiation of convex functions in a Banach space with a separable dual, Proc. Amer. Math. Soc. 91 (1984), 202-204. Zbl0521.46034
  13. [13] D. L. Renfro, On some various porosity notions, preprint, 1995. 
  14. [14] D. Sands, Topological conditions for positive Lyapunov exponent in unimodal maps, Ph.D. thesis, Cambridge, 1994. 
  15. [15] S. van Strien, Smooth dynamics on the interval, in: New Directions in Dynamical Systems, London Math. Soc. Lecture Note Ser. 127, Cambridge Univ. Press, Cambridge, 1988, 57-119. 
  16. [16] B. S. Thomson, Real Functions, Lecture Notes in Math. 1170, Springer, New York, 1985. Zbl0581.26001
  17. [17] L. Zajíček, Porosity and σ-porosity, Real Anal. Exchange 13 (1987-88), 314-347. 

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