Attractors (Mathematics)

A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the
standard horseshoe, but where the set of points that escape are now reinjected and contribute to
richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing
an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from
classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic
topology to show that for suitable parameters the flow contains a strange attractor.

In this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this allowed us to obtain a maximum principle for solutions of the dissipative two-dimensional quasi-geostrophic equation the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek.