# On a one-dimensional analogue of the Smale horseshoe

Annales Polonici Mathematici (1991)

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

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topRyszard 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

top- [1] P. Billingsley, Probability and Measure, Wiley, New York 1979.
- [2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 47, Springer, Berlin 1975. Zbl0308.28010
- [3] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181-202. Zbl0311.58010
- [4] A. Lasota, Thoughts and conjectures on chaos, preprint.
- [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] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971), 231-242. Zbl0212.49301
- [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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