# For almost every tent map, the turning point is typical

Fundamenta Mathematicae (1998)

- Volume: 155, Issue: 3, page 215-235
- ISSN: 0016-2736

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topBruin, Henk. "For almost every tent map, the turning point is typical." Fundamenta Mathematicae 155.3 (1998): 215-235. <http://eudml.org/doc/212253>.

@article{Bruin1998,

abstract = {Let $T_a$ be the tent map with slope a. Let c be its turning point, and $μ_a$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃ g dμ_a = lim_\{n → ∞\} \frac\{1\}\{n\} ∑_\{i=0\}^\{n-1\} g(T^i_a(c))$. As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.},

author = {Bruin, Henk},

journal = {Fundamenta Mathematicae},

keywords = {absolutely continuous invariant measure; Birkhoff ergodic theorem; tent map},

language = {eng},

number = {3},

pages = {215-235},

title = {For almost every tent map, the turning point is typical},

url = {http://eudml.org/doc/212253},

volume = {155},

year = {1998},

}

TY - JOUR

AU - Bruin, Henk

TI - For almost every tent map, the turning point is typical

JO - Fundamenta Mathematicae

PY - 1998

VL - 155

IS - 3

SP - 215

EP - 235

AB - Let $T_a$ be the tent map with slope a. Let c be its turning point, and $μ_a$ the absolutely continuous invariant probability measure. For an arbitrary, bounded, almost everywhere continuous function g, it is shown that for almost every a, $ʃ g dμ_a = lim_{n → ∞} \frac{1}{n} ∑_{i=0}^{n-1} g(T^i_a(c))$. As a corollary, we deduce that the critical point of a quadratic map is generically not typical for its absolutely continuous invariant probability measure, if it exists.

LA - eng

KW - absolutely continuous invariant measure; Birkhoff ergodic theorem; tent map

UR - http://eudml.org/doc/212253

ER -

## References

top- [BC] M. Benedicks and L. Carleson, On iterations of $1-a{x}^{2}$ on (-1,1), Ann. of Math. (2) 122 (1985), 1-25. Zbl0597.58016
- [BGMY] L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one dimensional maps, in: Lecture Notes in Math. 819, Springer, 1980, 18-34.
- [BM] K. Brucks and M. Misiurewicz, Trajectory of the turning point is dense for almost all tent maps, Ergodic Theory Dynam. Systems 16 (1996), 1173-1183. Zbl0874.58014
- [B] H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), 571-580. Zbl0827.58015
- [B2] H. Bruin, Combinatorics of the kneading map, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1339-1349. Zbl0886.58023
- [DGP] B. Derrida, A. Gervois and Y. Pomeau, Iteration of endomorphisms on the real axis and representations of numbers, Ann. Inst. H. Poincaré Phys. Théor. 29 (1978), 305-356. Zbl0416.28012
- [H] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980), 117-141. Zbl0433.54009
- [K] G. Keller, Lifting measures to Markov extensions, ibid. 108 (1989), 183-200. Zbl0712.28008
- [MS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. (3) 25, Springer, Berlin, 1993.
- [M] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 17-51. Zbl0477.58020
- [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. Zbl0153.19101
- [Sa] D. Sands, Topological conditions for positive Lyapunov exponents in unimodal maps, Ph.D. thesis, Cambridge, 1994.
- [Sc] J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dynam. Systems 17 (1997), 675-694. Zbl0908.58017
- [T] H. Thunberg, Absolutely continuous invariant measures and superstable periodic orbits: weak*-convergence of natural measures, Ph.D. thesis, Stockholm, 1996.

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