On a one-dimensional analogue of the Smale horseshoe

• Autorzy: Ryszard Rudnicki
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991
• Tom: 54
• Numer: 2
• Strony: 147-153

Daty

wydano

1991

otrzymano

1989-10-18

Twórcy

autor

Ryszard Rudnicki

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

Bibliografia

• [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.
• [3] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181-202.
• [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.
• [6] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971), 231-242.
• [7] L.-S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc. 287 (1985), 41-48.

Uwagi

1985 Mathematics Subject Classification: Primary 58F13.