arXiv subjects
Publications and source records attributed to Katelynn Huneycutt.
Orthogonal polynomials appear naturally in the study of compositions of M\"obius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic M\"obius map. Our results can be viewed as instances of non-autonomous parabolic implosion, including a random perturbative regime in which convergence holds almost surely.
We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urba\'nski. We extend the class of potentials from H\"older to those with the Bowen property on 'good orbits' . We obtain uniqueness of the equilibrium state for potentials satisfying a pressure gap condition which is sharp in the class of potentials we consider. We show that our equilibrium state has the $K$-property, and in particular it has positive entropy. When the potential is H\"older, the theory of equilibrium states is already highly developed. Nevertheless, several interesting results on equilibrium states for H\"older potentials follow readily from our approach. In the family of geometric potentials, we obtain a simple proof of uniqueness of equilibrium states up to the phase transition that occurs at the Hausdorff dimension of the Julia set. For H\"older potentials on parabolic rational maps, we show that hyperbolicity of the potential is equivalent to having a unique equilibrium state which is fully supported. This does not appear to have been stated in the literature before, although it may be considered folklore.
We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms provided through the flow effects. Three different configurations are considered: a clamped panel, a hinged panel, and a flag (a cantilever clamped at the leading edge, free at the trailing edge). After providing the functional framework for the dynamics, recent results on well-posedness and long-time behavior of the associated dynamical system for solutions are presented. Having provided this theoretical context, in-depth numerical stability analyses are provided, focusing both at the onset of flow-induced instability (flutter), and qualitative properties of the post-flutter dynamics across configurations. Modal approximations are utilized, as well as finite difference schemes.