Ergodic averages and free $ℤ^2$ actions

• Autorzy: Zoltán Buczolich
• Języki publikacji: EN

Abstrakty

PL

If the ergodic transformations S, T generate a free $ℤ^2$ action on a finite non-atomic measure space (X,S,µ) then for any $c_1,c_2 ∈ ℝ$ there exists a measurable function f on X for which $({N+1})^{-1} ∑_{j=0}^Nf(S^jx) → c_1$ and $(N+1)^{-1} ∑_{j=0}^Nf(T^jx) → c_2 µ$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

Kategorie tematyczne

• 47A35: Ergodic theory
• 28D05: Measure-preserving transformations
• 11K31: Special sequences

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 160
• Numer: 3
• Strony: 247-254

Daty

wydano

1999

otrzymano

1998-04-13

poprawiono

1999-03-12

Twórcy

autor

Zoltán Buczolich

• Department of Analysis, Eötvös Loránd University, Rákóczi út 5, H-1088 Budapest, Hungary

Bibliografia

• [Bu] Z. Buczolich, Arithmetic averages of rotations of measurable functions, Ergodic Theory Dynam. Systems 16 (1996), 1185-1196.
• [M] P. Major, A counterexample in ergodic theory, Acta Sci. Math. (Szeged) 62 (1996), 247-258.
• [OW] D. O. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I: the Rohlin lemma, Bull. Amer. Math. Soc. 2 (1980), 161-164.
• [P] W. F. Pfeffer, The Riemann Approach to Integration, Cambridge Univ. Press, Cambridge, 1993.
• [S] R. Svetic, A function with locally uncountable rotation set, Acta Math. Hungar., to appear.