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1998 | 155 | 3 | 215-235
Tytuł artykułu

For almost every tent map, the turning point is typical

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EN
Abstrakty
EN
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, $ʃ g dμ_a = lim_{n → ∞} \frac1n ∑_{i=0}^{n-1} g(T^i_a(c))$. 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.
Słowa kluczowe
Rocznik
Tom
155
Numer
3
Strony
215-235
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-08-12
poprawiono
1997-05-15
Twórcy
autor
Bibliografia
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  • [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.
  • [B] H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), 571-580.
  • [B2] H. Bruin, Combinatorics of the kneading map, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1339-1349.
  • [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.
  • [H] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980), 117-141.
  • [K] G. Keller, Lifting measures to Markov extensions, ibid. 108 (1989), 183-200.
  • [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.
  • [P] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
  • [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.
  • [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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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv155i3p215bwm
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