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Iterations of concave maps, the Perron-Frobenius theory, and applications to circle packings. (English) Zbl 1022.05012
An \(\overline{a}\) pseudocircle packing is a collection of (1) an plane embedding of a finite graph with vertex set \(V\), (2) an angle parameter vector \(\overline{a}=(a_0,\dots,a_{|V|-1}) \in R^{|V|}\), and (3) a sequence of \(|V|\) circles with radii \(\overline{r}=(r_0,\dots,r_{|V|-1})\in R^{+|V|}\). In the nice paper under review, the author develops the theory of finite pseudocircle packings as a generalization of the finite cicrle packings. A mapping \(f_{\overline{a}}: R^{+|V|} \to R^{+|V|}\) is a key tool in the investigations as isotonity, superadditivity and existence and uniqueness of a positive eigenvalue are proved. The eigenvalue and the corresponding eigenpoint are determined by two normalizations of \(f_{\overline{a}}\) which behave as contractions. Moreover, the eigenpoint provides the radii of a corresponding pseudocircle packing. Among the machinery used are the Sperner lemma, Perron-Frobenius theory for nonnegative matrices, geometric inequalities (which are interesting themselves) and an algorithm from algebraic geometry.
05B40 Combinatorial aspects of packing and covering
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
11C20 Matrices, determinants in number theory
15B48 Positive matrices and their generalizations; cones of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
35P15 Estimates of eigenvalues in context of PDEs
47H10 Fixed-point theorems
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
12Y05 Computational aspects of field theory and polynomials (MSC2010)
52C26 Circle packings and discrete conformal geometry