For almost every tent map, the turning point is typical

Henk Bruin

Fundamenta Mathematicae (1998)

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

Abstract

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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 d μ a = l i m n → ∞ 1 n ∑ i = 0 n - 1 g ( T a i ( 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.

How to cite

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Bruin, 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

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  11. [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. Zbl0153.19101
  12. [Sa] D. Sands, Topological conditions for positive Lyapunov exponents in unimodal maps, Ph.D. thesis, Cambridge, 1994. 
  13. [Sc] J. Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergodic Theory Dynam. Systems 17 (1997), 675-694. Zbl0908.58017
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