The recurrence dimension for piecewise monotonic maps of the interval

Franz Hofbauer[1]

  • [1] Fakultät für Mathematik Universität Wien Nordbergstraße 15 A 1090 Wien, Austria

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 3, page 439-449
  • ISSN: 0391-173X

Abstract

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We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [ 0 , 1 ] , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.

How to cite

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Hofbauer, Franz. "The recurrence dimension for piecewise monotonic maps of the interval." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.3 (2005): 439-449. <http://eudml.org/doc/84566>.

@article{Hofbauer2005,
abstract = {We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval $[0,1]$, giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.},
affiliation = {Fakultät für Mathematik Universität Wien Nordbergstraße 15 A 1090 Wien, Austria},
author = {Hofbauer, Franz},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {439-449},
publisher = {Scuola Normale Superiore, Pisa},
title = {The recurrence dimension for piecewise monotonic maps of the interval},
url = {http://eudml.org/doc/84566},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Hofbauer, Franz
TI - The recurrence dimension for piecewise monotonic maps of the interval
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 3
SP - 439
EP - 449
AB - We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval $[0,1]$, giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.
LA - eng
UR - http://eudml.org/doc/84566
ER -

References

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  2. [2] V. Afraimovich and J. Urias, Dimension–like characteristics of invariant sets in dynamical systems, In: “Dynamics and randomness”, A. Maass, S. Martinez and J. San Martin (eds.), Kluwer Academic Publishers, Dordrecht, 2002, pp. 1–30. Zbl1031.37024MR1975573
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  14. [14] Y. Pesin, “Dimension theory in dynamical systems: Contemporary views and applications”, The University of Chicago Press Chicago and London, 1997. Zbl0895.58033MR1489237
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