Cauchy Biorthogonal Polynomials

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408

About the homological discrete Conley index of isolated invariant acyclic continua

This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a locally defined homeomorphism f in R-d and an acyclic continuum X, such as a cellular set or a fixed point, invariant under f and isolated. We prove that the trace of the first discrete homological Conley index of f and X is greater than or equal to -1 and describe its periodical behavior. If equality holds then the traces of the higher homological indices are 0. In the case of orientation-reversing homeomorphisms of R-3, we obtain a characterization of the fixed point index sequence {i(f(n) (,) p}n >= 1 for a fixed point p which is isolated as an invariant set. In particular, we obtain that i(f , p) <= 1. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in R-3