Iterates of maps which are non-expansive in Hilbert's projective metric

Jeremy Gunawardena; Cormac Walsh

Kybernetika (2003)

  • Volume: 39, Issue: 2, page [193]-204
  • ISSN: 0023-5954

Abstract

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The cycle time of an operator on R n gives information about the long term behaviour of its iterates. We generalise this notion to operators on symmetric cones. We show that these cones, endowed with either Hilbert’s projective metric or Thompson’s metric, satisfy Busemann’s definition of a space of non- positive curvature. We then deduce that, on a strictly convex symmetric cone, the cycle time exists for all maps which are non-expansive in both these metrics. We also review an analogue for the Hilbert metric of the Denjoy-Wolff theorem.

How to cite

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Gunawardena, Jeremy, and Walsh, Cormac. "Iterates of maps which are non-expansive in Hilbert's projective metric." Kybernetika 39.2 (2003): [193]-204. <http://eudml.org/doc/33634>.

@article{Gunawardena2003,
abstract = {The cycle time of an operator on $R^n$ gives information about the long term behaviour of its iterates. We generalise this notion to operators on symmetric cones. We show that these cones, endowed with either Hilbert’s projective metric or Thompson’s metric, satisfy Busemann’s definition of a space of non- positive curvature. We then deduce that, on a strictly convex symmetric cone, the cycle time exists for all maps which are non-expansive in both these metrics. We also review an analogue for the Hilbert metric of the Denjoy-Wolff theorem.},
author = {Gunawardena, Jeremy, Walsh, Cormac},
journal = {Kybernetika},
keywords = {Hilbert geometry; Thompson’s part metric; non-expansive map; symmetric cone; cycle time; topical map; iterates; Hilbert geometry; Thompson's part metric; non-expansive map; symmetric cone; cycle time; topical map},
language = {eng},
number = {2},
pages = {[193]-204},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Iterates of maps which are non-expansive in Hilbert's projective metric},
url = {http://eudml.org/doc/33634},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Gunawardena, Jeremy
AU - Walsh, Cormac
TI - Iterates of maps which are non-expansive in Hilbert's projective metric
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 2
SP - [193]
EP - 204
AB - The cycle time of an operator on $R^n$ gives information about the long term behaviour of its iterates. We generalise this notion to operators on symmetric cones. We show that these cones, endowed with either Hilbert’s projective metric or Thompson’s metric, satisfy Busemann’s definition of a space of non- positive curvature. We then deduce that, on a strictly convex symmetric cone, the cycle time exists for all maps which are non-expansive in both these metrics. We also review an analogue for the Hilbert metric of the Denjoy-Wolff theorem.
LA - eng
KW - Hilbert geometry; Thompson’s part metric; non-expansive map; symmetric cone; cycle time; topical map; iterates; Hilbert geometry; Thompson's part metric; non-expansive map; symmetric cone; cycle time; topical map
UR - http://eudml.org/doc/33634
ER -

References

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