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We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K\,{\rightarrow}\, K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by \[ \beta_N = \max_{q+r+s=N}\frac{N!}{q!r!s!}= \frac{N!}{\big\lfloor\frac{N}{3}\big\rfloor!\big\lfloor\frac{N\,{+}\,1}{3}\big\rfloor! \big\lfloor\frac{N\,{+}\,2}{3}\big\rfloor!}\sim \frac{3^{N+1}\sqrt{3}}{2\pi N}, \] where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in $\mathbb{R}^n$, we show that the upper bound is asymptotically sharp

Topics:
QA

Publisher: Cambridge University Press

Year: 2006

OAI identifier:
oai:wrap.warwick.ac.uk:711

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- (2003). A .D .B u r b a n k s ,R .D .N u s s b a u mand
- (1992). a c c e l l i ,G .C o h e n
- (1992). Alternative proof of Sine’s theorem on the size of a regular polygon in R k with the ∞-metric.
- (1989). as. Combinatorics
- (2003). From max-plus algebra to nonexpansive mappings: a nonlinear theory for discrete event systems.
- (1973). Hilbert’s metric and positive contraction mappings in a Banach space.
- (1976). Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics (Springer-Verlag,
- (1995). On the existence of cycle times for some nonexpansive maps.
- Positive equilibria and convergence in subhomogeneous monotone dynamics.
- (1980). Some relations between nonexpansive and order preserving mappings.
- (2003). Spectral theorem for convex monotone homogeneous maps, and ergodic control. Nonlinear Anal.
- (1969). The part metric in convex sets.
- (2004). The Perron–Frobenius theory for homogeneous, monotone functions.
- (1936). Verk¨ urzungen und Isometrien.

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