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Title: Iterates of maps which are non-expansive in Hilbert's projective metric (English)
Author: Gunawardena, Jeremy
Author: Walsh, Cormac
Language: English
Journal: Kybernetika
ISSN: 0023-5954
Volume: 39
Issue: 2
Year: 2003
Pages: [193]-204
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: Hilbert geometry
Keyword: Thompson’s part metric
Keyword: non-expansive map
Keyword: symmetric cone
Keyword: cycle time
Keyword: topical map
Keyword: iterates
MSC: 47H09
MSC: 53C60
idZBL: Zbl 1247.47030
idMR: MR1996557
Date available: 2009-09-24T19:52:44Z
Last updated: 2015-03-23
Stable URL:
Reference: [1] Beardon A. F.: The dynamics of contractions.Ergodic Theory Dynamical Systems 17 (1997), 6, 1257–1266 Zbl 0952.54023, MR 1488316, 10.1017/S0143385797086434
Reference: [2] Bhagwat K. V., Subramanian R.: Inequalities between means of positive operators.Math. Proc. Cambridge Philos. Soc. 83 (1978), 3, 393–401 Zbl 0375.47017, MR 0467372, 10.1017/S0305004100054670
Reference: [3] Busemann H.: The Geometry of Geodesics.Academic Press, New York 1955 Zbl 1141.53001, MR 0075623
Reference: [4] Corach G., Porta, H., Recht L.: A geometric interpretation of Segal’s inequality.Proc. Amer. Math. Soc. 115 (1992), 1, 229–231 Zbl 0749.58010, MR 1075945
Reference: [5] Corach G., Porta, H., Recht L.: Convexity of the geodesic distance on spaces of positive operators.Illinois J. Math. 38 (1994), 1, 87–94 Zbl 0802.53012, MR 1245836
Reference: [6] Donoghue W. F.: Monotone Matrix Functions and Analytic Continuation.Springer–Verlag, Berlin 1974 Zbl 0278.30004, MR 0486556
Reference: [7] Faraut J., Korányi A.: Analysis on Symmetric Cones.Oxford, 1994 Zbl 0841.43002, MR 1446489
Reference: [8] Gaubert S., Gunawardena J.: A non-linear hierarchy for discrete event dynamical systems.In: Proceedings WODES’98, Cagliari 1998
Reference: [9] Gunawardena J.: From max-plus algebra to nonexpansive mappings: a nonlinear theory for discrete event systems.Theoret. Comput. Sci. 293 (2003), 141–167 Zbl 1036.93045, MR 1957616, 10.1016/S0304-3975(02)00235-9
Reference: [10] Gunawardena J., Keane M.: On the Existence of Cycle Times for Some Nonexpansive Maps.HPL-BRIMS-95-03, Hewlett-Packard Labs, 1995
Reference: [11] Kohlberg E., Neyman A.: Asymptotic behavior of nonexpansive mappings in normed linear spaces.Israel J. Math. 38 (1981), 4, 269–275 Zbl 0476.47045, MR 0617673, 10.1007/BF02762772
Reference: [12] Nussbaum R. D.: Hilbert’s Projective Metric and Iterated Nonlinear Maps.Amer. Math. Soc., 1998 Zbl 0666.47028
Reference: [13] Nussbaum R. D.: Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations.Differential and Int. Equations 7 (1994), 6, 1649–1707 Zbl 0844.58010, MR 1269677
Reference: [14] Plant A. T., Reich S.: The asymptotics of nonexpansive iterations.J. Func. Anal. 54 (1983), 3, 308–319 Zbl 0542.47045, MR 0724526, 10.1016/0022-1236(83)90003-4
Reference: [15] Reich S., Shafrir I.: Nonexpansive Iterations in Hyperbolic Spaces.Nonlinear Anal., Theory, Methods & Appls 15 (1990), 6, 537–558 Zbl 0728.47043, MR 1072312, 10.1016/0362-546X(90)90058-O
Reference: [16] Sine R.: Behavior of iterates in the Poincaré metric.Houston J. Math. 15 (1989), 2, 273–289 Zbl 0712.47049, MR 1022069


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