We study the classical H & eacute;non family fa,b : (x, y) i? (1 - ax(2) + y, bx), 0 < a < 2, 0 < b < 1, and prove that given an integer k = 1, there is a set of parameters Ek of positive two-dimensional Lebesgue measure so that fa,b, for (a, b) ? E-k, has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73-169, 1991). A corresponding statement also holds for the H & eacute;non-like families of Mora and Viana (Acta Math 171:1-71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original H & eacute;non maps. The final main result of the paper is the existence, within the classical H & eacute;non family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

For the family of double standard maps we investigate the structure of the space of parameters a when and when. In the first case the maps have a critical point, but for a set of parameters of positive Lebesgue measure there is an invariant absolutely continuous measure for. In the second case there is an open non-empty set of parameters for which the map is expanding. We show that as, the set accumulates on many points of in a regular way from the measure point of view.

For a many-electron system, whether the particle density rho and the total current density j are sufficient to determine the one-body potential V and vector potential A, is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional E_{V_0,A_0}(rho,j) = <psi(rho,j),H(V_0,\A_0)psi(rho,j)> can be minimal for densities that are not the ground-state densities of the fixed potentials V_0 and A_0. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with rho and j, we show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written (rho,j) -> A(rho,j;x), enters explicitly in E_{V_0,A_0}(rho,j).

We consider families of nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure of , as a function of on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters , and, for each , a set of polynomially recurrent parameters containing as a Lebesgue density point, and constants , , so that, for every -Holder function , In addition, for all , the renormalisation period of satisfies , and there are uniform bounds on the rates of mixing of for all with . If , the set contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter , we find a set of CE parameters accumulating at , a constant , and a function , so that C vertical bar t - t(0)vertical bar(1/2) >= vertical bar integral A(0)d mu(t) - integral A(0)d mu(t0) vertical bar >= C-1 vertical bar t - t(0)vertical bar(1/2), for all t is an element of Delta(MT)'.

In this article, we examine Hohenberg-Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg-Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble-representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble-representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians.