Ergodic properties of skew products with Lasota-Yorke type maps in the base

Zbigniew Kowalski

Studia Mathematica (1993)

  • Volume: 106, Issue: 1, page 45-57
  • ISSN: 0039-3223

Abstract

top
We consider skew products T ( x , y ) = ( f ( x ) , T e ( x ) y ) preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and T i , i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these results to random perturbations.

How to cite

top

Kowalski, Zbigniew. "Ergodic properties of skew products with Lasota-Yorke type maps in the base." Studia Mathematica 106.1 (1993): 45-57. <http://eudml.org/doc/216002>.

@article{Kowalski1993,
abstract = {We consider skew products $T(x,y) = (f(x),T_\{e(x)\} y)$ preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and $T_i$, i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these results to random perturbations.},
author = {Kowalski, Zbigniew},
journal = {Studia Mathematica},
keywords = {measure-preserving transformations; skew products; Lasota-Yorke type transformation; eigenfunctions; Frobenius-Perron operator; ergodicity; weak mixing; exactness; random perturbations},
language = {eng},
number = {1},
pages = {45-57},
title = {Ergodic properties of skew products with Lasota-Yorke type maps in the base},
url = {http://eudml.org/doc/216002},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Kowalski, Zbigniew
TI - Ergodic properties of skew products with Lasota-Yorke type maps in the base
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 1
SP - 45
EP - 57
AB - We consider skew products $T(x,y) = (f(x),T_{e(x)} y)$ preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and $T_i$, i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these results to random perturbations.
LA - eng
KW - measure-preserving transformations; skew products; Lasota-Yorke type transformation; eigenfunctions; Frobenius-Perron operator; ergodicity; weak mixing; exactness; random perturbations
UR - http://eudml.org/doc/216002
ER -

References

top
  1. [1] J. C. Alexander and W. Parry, Discerning fat baker's transformations, in: Dynamical Systems, Proc. Special Year, College Park, Lecture Notes in Math. 1342, Springer, 1988, 1-6. Zbl0728.28015
  2. [2] M. Jabłoński, On invariant measures for piecewise C 2 -transformations of the n-dimensional cube, Ann. Polon. Math. 43 (1983), 185-195. Zbl0591.28014
  3. [3] S. A. Kalikow, T, T - 1 transformation is not loosely Bernoulli, Ann. of Math. 115 (1982), 393-409. Zbl0523.28018
  4. [4] Z. S. Kowalski, Bernoulli properties of piecewise monotonic transformations, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 59-61. Zbl0422.28011
  5. [5] Z. S. Kowalski, A generalized skew product, Studia Math. 87 (1987), 215-222. Zbl0651.28013
  6. [6] Z. S. Kowalski, Stationary perturbations based on Bernoulli processes, ibid. 97 (1990), 53-57. Zbl0752.28009
  7. [7] Z. S. Kowalski, The exactness of generalized skew products, Osaka J. Math. 30 (1993), 57-61. Zbl0787.28012
  8. [8] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, 1985. 
  9. [9] I. Meilijson, Mixing properties of a class of skew products, Israel J. Math. 19 (1974), 266-270. Zbl0305.28008
  10. [10] T. Morita, Random iteration of one-dimensional transformations, Osaka J. Math. 22 (1985), 489-518. Zbl0583.60061
  11. [11] T. Morita, Deterministic version lemmas in ergodic theory of random dynamical systems, Hiroshima Math. J. 18 (1988), 15-29. Zbl0698.28009
  12. [12] S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc. 281 (1984), 813-823. Zbl0532.58013
  13. [13] J. P. Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l'un est un schéma de Bernoulli, Israel J. Math. 21 (1975), 177-207. Zbl0329.28008
  14. [14] S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc. 246 (1978), 493-500. Zbl0401.28011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.