James, J. D. James

We explore a novel method of approximating contractible invariant circles in maps. The process begins by leveraging improvements on Birkhoff's Ergodic Theorem via Weighted Birkhoff Averages to compute high precision estimates on several Fourier modes. We then set up a Newton-like iteration scheme to further improve the estimation and extend the approximation out to a sufficient number of modes to yield a significant decay in the magnitude of the coefficients of high order. With this approximation in hand, we explore the phase space near the approximate invariant circle with a form numerical continuation where the rotation number is perturbed and the process is repeated. Then, we turn our attention to a completely different problem which can be approached in a similar way to the numerical continuation, finding a Siegel disk boundary in a holomorphic map. Given a holomorphic map which leads to a formally solvable cohomological equation near the origin, we use a numerical continuation style process to approximate an invariant circle in the Siegel disk near the origin. Using an iterative scheme, we then enlarge the invariant circle so that it approximates the boundary of the Siegel disk.