This is a survey article concerning applications of Hilbert's metric in the analysis and dynamics of linear and nonlinear mappings on cones.
Reading the bibliography…
O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. , 64:1–76, 1907
1907
Earlier work this paper cites.
O. Perron, Zur Theorie der Über Matrizen. Math. Ann. , 64:248–263, 1907
1907
Earlier work this paper cites.
A. Denjoy, Sur l’itération des fonctions analytiques. C. R. Acad. Sc. Paris, Série 1, , 182:255–257, 1926
1926
Earlier work this paper cites.
J. Wolff, Sur l’itération des fonctions bornées. C. R. Acad. Sc. Paris, Série 1 , 182:200–201, 1926
1926
Earlier work this paper cites.
J. Wolff, Sur une généralisation d’un théoreme de Schwartz. C. R. Acad. Sc. Paris, Série 1 , 183:500–502, 1926
1926
Earlier work this paper cites.
K. Löwner, Über monotone Matrixfunktionen. Math. Z
1934
Earlier work this paper cites.
S. Straszewicz, Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math
1935
Earlier work this paper cites.
M.G. Kreĭn and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space. Amer. Math. Soc. Translation 1950 , 26, 1950
1950
Earlier work this paper cites.
L.S. Shapley, Stochastic games. Proc. Natl. Acad. Sci. USA
1953
Earlier work this paper cites.
R. Bellman, Dynamic Programming
1957
Earlier work this paper cites.
G. Birkhoff, Extensions of Jentzsch’s theorem. Trans. Amer. Math. Soc. , 85(1):219–227, 1957
1957
Earlier work this paper cites.
M. Koecher, Positivitätsbereiche im ℝ n \mathbb{R}^{n} . Amer. J. Math
1957
Earlier work this paper cites.
H. Samelson, On the Perron-Frobenius theorem. Michigan Math. J. , 4:57–59, 1957
1957
Earlier work this paper cites.
F.F. Bonsall, Linear operators in complete positive cones. Proc. Lond. Math. Soc. , 8(3):53–75, 1958
1958
Earlier work this paper cites.
E.B. Vinberg, Homogeneous cones. Soviet Math. Dokl
1960
Earlier work this paper cites.
E. Hopf, An inequality for positive integral operators. J. Math. Mech , 12:683–692, 1963
1963
Earlier work this paper cites.
E. Hopf, Remarks on my paper ”An inequality for positive integral operators”. J. Math. Mech. ,12:889–892, 1963
1963
Earlier work this paper cites.
A.C. Thompson, On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. , 14:438–443, 1963
1963
Earlier work this paper cites.
M.A. Krasnosel’skii, Positive Solutions of Operator Equations
1964
Earlier work this paper cites.
F.L. Bauer. An elementary proof of the Hopf inequality for positive operators. Numer. Math. , 7:331–337, 1965
1965
Earlier work this paper cites.
M.V. Menon, Some spectral properties of an operator associated with a pair of nonnegative matrices. Trans. Amer. Math. Soc
1968
Earlier work this paper cites.
U. Hirzebruch, Der Min-Max-Satz von E. Fischer für formal real Jordan-Algebren, Math. Ann
1970
Earlier work this paper cites.
P.P. Zabreiko, M.A. Krasnosel’skii, and Yu.V. Pokornyi, On a class of positive linear operators. Funct. Anal. Appl. , 5:57–70, 1972
1972
Earlier work this paper cites.
P.J. Bushell, Hilbert’s projective metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal. , 52: 330–338, 1973
1973
Earlier work this paper cites.
P.J. Bushell, On the projective contraction ratio for positive linear mappings. J. Lond. Math. Soc. , 6:256–258, 1973
1973
Cited alongside, same era.
B.B. Phadke, A triangular world with hexagonal circles. Geometriae Dedicata
1974
Cited alongside, same era.
T. Bewley and E. Kohlberg, The asymptotic theory of stochastic games. Math. Oper. Res
1976
Cited alongside, same era.
A.J.B. Potter, Applications of Hilbert’s projective metric to certain classes of non- homogeneous operators. Quart. J. Math. Oxford Ser. (2)
1977
Cited alongside, same era.
E. Kohlberg and J.W. Pratt, The contraction mapping approach to the Perron-Frobenius theory: Why Hilbert’s metric?, Math. Oper. Res
1982
Cited alongside, same era.
A. Całka, On a condition under which isometries have bounded orbits. Colloq. Math. , 48:219–227, 1984
1984
Cited alongside, same era.
U. Krause, A nonlinear extension of the Birkhoff-Jentzsch theorem. J. Math. Anal. Appl
1986
Cited alongside, same era.
R.D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps. Mem. Amer. Math. Soc
1988
Cited alongside, same era.
M.A. Krasnosel’skii, J. A. Lifshits, and A.V. Sobolev. Postive Linear Systems: the Method of Positive Operators. Sigma Ser. Appl. Math. 5, Heldermann Verlag, 1989
1989
Cited alongside, same era.
A.F. Beardon, Iteration of contractions and analytic maps. J. Lond. Math. Soc. (2) , 41(1):141–150, 1990
1990
Cited alongside, same era.
R.D. Nussbaum, Omega limit sets of nonexpansive maps: finiteness and cardinality estimates. Differential Integral Equations , 3(3):523–540, 1990
1990
Cited alongside, same era.
R. Lyons and R.D. Nussbaum, On transitive and commutative finite groups of isometries, in Fixed Point Theory and Applications (K-K. Tan ed.), World Scientific: Singapore, pp. 189–228, (1992)
1992
Cited alongside, same era.
A. Neyman, Stochastic games and nonexpansive maps. In A. Neyman and S. Sorin, editors, Stochastic Games and Applications (Stony Brook, NY, 1999)
2003
Later among the works it cites.
S. Gaubert and J. Gunawardena, The Perron-Frobenius theory for homogeneous, monotone functions. Trans. Amer. Math. Soc. , 356(12): 4931–4950, 2004
2004
Later among the works it cites.
T. Foertsch and A. Karlsson, Hilbert metrics and Minkowski norms. J. Geom. , 83(1-2): 22–31, 2005
2005
Later among the works it cites.
B. Lemmens and M. Scheutzow, On the dynamics of sup-norm nonexpansive maps. Ergodic Theory & Dynam. Systems , 25(3):861–871, 2005
2005
Later among the works it cites.
M. Akian, S. Gaubert, B. Lemmens, and R.D. Nussbaum, Iteration of order preserving subhomogeneous maps on a cone. Math. Proc. Cambridge Philos. Soc. , 140(1):157–176, 2006
2006
Later among the works it cites.
K. Koufany, Application of Hilbert’s projective metric on symmetric cones. Acta Math. Sin. (Engl. Ser.)
2006
Later among the works it cites.
B. Lemmens and C. Sparrow, A note on periodic points of order preserving subhomogeneous maps. Proc. Amer. Math. Soc. , 134(5):1513–1517, 2006
2006
Later among the works it cites.
O. Bletz-Siebert and T. Foertsch, The Euclidean rank of Hilbert geometries. Pacific J. Math
2007
Later among the works it cites.
Y. Lim, Hilbert’s projective metric on Lorentz cones and Birkhoff formula for Lorentzian compressions. Linear Algebra Appl
2007
Later among the works it cites.
B. Lins, Asymptotic Behavior and Denjoy-Wolff Theorems for Hilbert Metric Nonexpansive Maps
2007
Later among the works it cites.
B. Lins, A Denjoy-Wolff theorem for Hilbert metric nonexpansive maps on a polyhedral domain. Math. Proc. Cambridge. Philos. Soc. , 143(1):157–164, 2007
2007
Later among the works it cites.
R.D. Nussbaum, fixed point theorems and Denjoy-Wolff theorems for Hilbert’s projective metric in infinite dimensions. Topol. Methods Nonlinear Anal. , 29(2):199–250, 2007
2007
Later among the works it cites.
B. Lins and R.D. Nussbaum, Denjoy-Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators. J. Funct. Anal
2008
Later among the works it cites.
C. Walsh, The horofunction boundary of the Hilbert geometry. Adv. Geom
2008
Later among the works it cites.
A. Bernig, Hilbert geometry of polytopes. Arch. Math. (Basel)
2009
Later among the works it cites.
A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric. Math. Proc. Cambridge Philos. Soc
2009
Later among the works it cites.
B. Colbois and P. Verovic, Hilbert domains that admit a quasi-isometric embedding into Euclidean space. Adv. Geom
2011
Later among the works it cites.
B. Colbois, C. Vernicos, and P. Verovic, Hilbert geometry for convex polygonal domains. J. Geom
2011
Later among the works it cites.
B. Lemmens, Nonexpansive mappings on Hilbert’s metric spaces. Topol. Methods Nonlinear Anal. , 38(1):45–58, 2011
2011
Later among the works it cites.
B. Simon, Convexity. An analytic viewpoint
2011
Later among the works it cites.
S. Gaubert and G. Vigeral, A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces. Math. Proc. Cambridge Philos. Soc
2012
Later among the works it cites.
B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobebius theory
2012
Later among the works it cites.
R.D. Nussbaum, A. Priyadarshi, and S. Verduyn Lunel, Positive operators and Hausdorff dimension of invariant sets. Trans. Amer. Math. Soc
2012
Later among the works it cites.
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