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Standard deviation of recurrence times for piecewise linear transformations

Journal Article


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Abstract


  • This paper is concerned with dynamical systems of the form (X,f) where X is a bounded interval and f

    comes from a class of measure-preserving, piecewise linear transformations on X. If A⊆X is a Borel set and x∈A, the Poincaré recurrence time of x relative to A is defined to be the minimum of {n:n∈Nandfn(x)∈A}, if the minimum exists, and ∞ otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.

Authors


  •   Ismael, Mimoon (external author)
  •   Nillsen, Rodney
  •   Williams, Graham H. (external author)

Publication Date


  • 2017

Citation


  • Ismael, M., Nillsen, R. & Williams, G. H. (2017). Standard deviation of recurrence times for piecewise linear transformations. Asian-European Journal of Mathematics, 10 (1), 1750009-1-1750009-10.

Scopus Eid


  • 2-s2.0-84973652226

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1321&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/320

Start Page


  • 1750009-1

End Page


  • 1750009-10

Volume


  • 10

Issue


  • 1

Place Of Publication


  • Singapore

Abstract


  • This paper is concerned with dynamical systems of the form (X,f) where X is a bounded interval and f

    comes from a class of measure-preserving, piecewise linear transformations on X. If A⊆X is a Borel set and x∈A, the Poincaré recurrence time of x relative to A is defined to be the minimum of {n:n∈Nandfn(x)∈A}, if the minimum exists, and ∞ otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.

Authors


  •   Ismael, Mimoon (external author)
  •   Nillsen, Rodney
  •   Williams, Graham H. (external author)

Publication Date


  • 2017

Citation


  • Ismael, M., Nillsen, R. & Williams, G. H. (2017). Standard deviation of recurrence times for piecewise linear transformations. Asian-European Journal of Mathematics, 10 (1), 1750009-1-1750009-10.

Scopus Eid


  • 2-s2.0-84973652226

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1321&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/320

Start Page


  • 1750009-1

End Page


  • 1750009-10

Volume


  • 10

Issue


  • 1

Place Of Publication


  • Singapore