Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems

Polyomorphisms Conjugate to Dilations

Consider polynomial maps f: Cn → Cn and their dilations sf(x) by complex scalars s. That is, maps f whose components fi are polynomials with complex coefficients in the n variables (x1, x2,..., xn) = x ∈ Cn. The question, first raised by O.-H. Keller in 1939 [10] for polynomials over the integers but now also raised for complex polynomials and, as such, known as The Jacobian Conjecture (JC), asks whether a polynomial map f with nonzero constant Jacobian determinant det f ′(x) need be a polyomorphism: I.e., bijective with polynomial inverse. It suffices to prove injectivity because in 1960–62 it was proved, first in dimension 2 by Newman [19] and then in all dimensions by Bia lynicki-Birula and Rosenlicht [4], that, for polynomial maps, surjectivity follows from injectivity; and furthermore, under Keller’s hypothesis, the inverse f−1(x) will be polynomial, at least in the complex case, if the polynomial map is bijective. The group of all polyomorphisms of Cn is denoted GAn(C). It is isomorphic to the group AutC[x] of automorphisms σ of the polynomial ring C[x] by means of the correspondence φ(f) = σ where σ(xi) = fi(x). Polynomial maps f(x) satisfying det f ′(x) = const 6 = 0 are called Keller maps. We can and do assume that f(0) = 0 and f ′(0) = I. Five main problems arise: Problem#1. Classify all Keller maps f: Cn → Cn. Open for n ≥ 2

Winter biology of Culex pipiens quinquefasciatus say, (Diptera: Culicidae) from Córdoba, Argentina

Adult cohorts and immature stages were kept under field conditions during the autumn and winter of three consecutive years. Survival, oviposition and development time from egg to adult were considered. The adult cohorts were studied under three experimental conditions: unfed cohorts, cohorts fed with sugar solution ansd cohorts fed with both sugar solution and blood (chicken). Female longevity showed significant differences among the three treatments. Females of unfed cohorts lived up to three weeks; females fed with sugar solution survived until six weeks, while those fed both with sugar and blood lived at most fourteen weeks; after the blood intake eggs were laid. In the immature stages, the highest relative mortality rates occurred during the egg and larval stages. Total pre-adult mortality varied between 59.09 and 89.71%. The developmental duration from egg to adult was between 43-62 days; there were no differences among results obtained for the three years