Ergodic properties of skew products withfibre maps of Lasota-Yorke type

Zbigniew Kowalski

Applicationes Mathematicae (1994)

  • Volume: 22, Issue: 2, page 155-163
  • ISSN: 1233-7234

Abstract

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We consider the skew product transformation T(x,y)= (f(x), T e ( x ) ) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and T s s ∈ S is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling of hard rock with high rotational speed.

How to cite

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Kowalski, Zbigniew. "Ergodic properties of skew products withfibre maps of Lasota-Yorke type." Applicationes Mathematicae 22.2 (1994): 155-163. <http://eudml.org/doc/219088>.

@article{Kowalski1994,
abstract = {We consider the skew product transformation T(x,y)= (f(x), $T_\{e(x)\}$) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and $\{T_s\}_\{s \in S\}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling of hard rock with high rotational speed.},
author = {Kowalski, Zbigniew},
journal = {Applicationes Mathematicae},
keywords = {Frobenius-Perron operator; invariant measure; motion of cogged bits; fibre maps; skew product transformation; Lasota-Yorke type maps; ergodic properties},
language = {eng},
number = {2},
pages = {155-163},
title = {Ergodic properties of skew products withfibre maps of Lasota-Yorke type},
url = {http://eudml.org/doc/219088},
volume = {22},
year = {1994},
}

TY - JOUR
AU - Kowalski, Zbigniew
TI - Ergodic properties of skew products withfibre maps of Lasota-Yorke type
JO - Applicationes Mathematicae
PY - 1994
VL - 22
IS - 2
SP - 155
EP - 163
AB - We consider the skew product transformation T(x,y)= (f(x), $T_{e(x)}$) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and ${T_s}_{s \in S}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling of hard rock with high rotational speed.
LA - eng
KW - Frobenius-Perron operator; invariant measure; motion of cogged bits; fibre maps; skew product transformation; Lasota-Yorke type maps; ergodic properties
UR - http://eudml.org/doc/219088
ER -

References

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  1. [1] N. Dunford and J. Schwartz, Linear Operators I, Interscience, New York, 1958. 
  2. [2] P. Góra and A. Boyarsky, Compactness of invariant densities for families of expanding, piecewise monotonic transformations, Canad. J. Math. 61 (1989), 855-869. Zbl0689.28007
  3. [3] K. Horbacz, Statistical properties of the Ejgielies model of a cogged bit, Zastos. Mat. 21 (1991), 15-26. Zbl0759.47005
  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, Stationary perturbations based on Bernoulli processes, Studia Math. 97 (1990), 53-57. Zbl0752.28009
  6. [6] Z. S. Kowalski, Ergodic properties of skew products with Lasota-Yorke type maps in the base, ibid. 106 (1993), 45-57. Zbl0815.28013
  7. [7] A. Lasota and P. Rusek, An application of ergodic theory to the determination of the efficiency of cogged drilling bits, Archiwum Górnictwa 3 (1974), 281-295 (in Polish). 
  8. [8] A. Lasota and J. A. Yorke, On the existence of invariant measure for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. Zbl0298.28015
  9. [9] T. Morita, Asymptotic behavior of one-dimensional random dynamical systems, J. Math. Soc. Japan 37 (1985), 651-663. Zbl0587.58027
  10. [10] T. Morita, Deterministic version lemmas in ergodic theory of random dynamical systems, Hiroshima Math. J. 18 (1988), 15-29. Zbl0698.28009

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