Support overlapping L 1 contractions and exact non-singular transformations

Michael Lin

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 515-520
  • ISSN: 0010-1354

Abstract

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Let T be a positive linear contraction of L 1 of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.

How to cite

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Lin, Michael. "Support overlapping $L_{1}$ contractions and exact non-singular transformations." Colloquium Mathematicae 84/85.2 (2000): 515-520. <http://eudml.org/doc/210830>.

@article{Lin2000,
abstract = {Let T be a positive linear contraction of $L_\{1\}$ of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.},
author = {Lin, Michael},
journal = {Colloquium Mathematicae},
keywords = {completely mixing; asymptotic stability; overlapping supports; Perron-Frobenius operator; Harris recurrent contraction},
language = {eng},
number = {2},
pages = {515-520},
title = {Support overlapping $L_\{1\}$ contractions and exact non-singular transformations},
url = {http://eudml.org/doc/210830},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Lin, Michael
TI - Support overlapping $L_{1}$ contractions and exact non-singular transformations
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 515
EP - 520
AB - Let T be a positive linear contraction of $L_{1}$ of a σ-finite measure space (X,Σ,μ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases: (i) T is the Frobenius-Perron operator of a non-singular transformation ϕ (in which case complete mixing is equivalent to exactness of ϕ). (ii) T is a Harris recurrent operator. (iii) T is a convolution operator on a compact group. (iv) T is a convolution operator on a LCA group.
LA - eng
KW - completely mixing; asymptotic stability; overlapping supports; Perron-Frobenius operator; Harris recurrent contraction
UR - http://eudml.org/doc/210830
ER -

References

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  8. [K] U. Krengel, Ergodic Theorems, de Gruyter, Berlin, 1985. 
  9. [KL] U. Krengel and M. Lin, On the deterministic and asymptotic σ-algebras of a Markov operator, Canad. Math. Bull. 32 (1989), 64-73. Zbl0638.60079
  10. [L] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971), 231-242. Zbl0212.49301
  11. [OS] D. Ornstein and L. Sucheston, An operator theorem on L 1 convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631-1639. Zbl0284.60068
  12. [R] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262. Zbl0838.47040
  13. [Z] R. Zaharopol, Strongly asymptotically stable Frobenius-Perron operators, Proc. Amer. Math. Soc., in press. Zbl0955.47009

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