On a one-dimensional analogue of the Smale horseshoe

Ryszard Rudnicki

Annales Polonici Mathematici (1991)

  • Volume: 54, Issue: 2, page 147-153
  • ISSN: 0066-2216

Abstract

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We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have ∫ φ ( T n x ) f ( x ) d x → ∫ φ d μ , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then n - 1 ∑ i = 0 n - 1 φ ( T i x ) → ∫ φ d μ for Lebesgue-a.e. x.

How to cite

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Ryszard Rudnicki. "On a one-dimensional analogue of the Smale horseshoe." Annales Polonici Mathematici 54.2 (1991): 147-153. <http://eudml.org/doc/262334>.

@article{RyszardRudnicki1991,
abstract = {We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have $∫φ(T^\{n\}x)f(x)dx → ∫φdμ$, where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then $n^\{-1\}∑_\{i=0\}^\{n-1\} φ(T^\{i\}x) → ∫φdμ$ for Lebesgue-a.e. x.},
author = {Ryszard Rudnicki},
journal = {Annales Polonici Mathematici},
keywords = {Smale horseshoe; chaos; completely mixing; cylinder measure; invariant measure},
language = {eng},
number = {2},
pages = {147-153},
title = {On a one-dimensional analogue of the Smale horseshoe},
url = {http://eudml.org/doc/262334},
volume = {54},
year = {1991},
}

TY - JOUR
AU - Ryszard Rudnicki
TI - On a one-dimensional analogue of the Smale horseshoe
JO - Annales Polonici Mathematici
PY - 1991
VL - 54
IS - 2
SP - 147
EP - 153
AB - We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have $∫φ(T^{n}x)f(x)dx → ∫φdμ$, where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then $n^{-1}∑_{i=0}^{n-1} φ(T^{i}x) → ∫φdμ$ for Lebesgue-a.e. x.
LA - eng
KW - Smale horseshoe; chaos; completely mixing; cylinder measure; invariant measure
UR - http://eudml.org/doc/262334
ER -

References

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  1. [1] P. Billingsley, Probability and Measure, Wiley, New York 1979. 
  2. [2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 47, Springer, Berlin 1975. Zbl0308.28010
  3. [3] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181-202. Zbl0311.58010
  4. [4] A. Lasota, Thoughts and conjectures on chaos, preprint. 
  5. [5] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. Zbl0298.28015
  6. [6] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971), 231-242. Zbl0212.49301
  7. [7] L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc. 287 (1985), 41-48. Zbl0552.58022

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