Carolina Dynamics Symposium 2026

The observation of pairs of moons around Saturn, which display differing, but stable, orbital properties, led to the development of the coorbital model in celestial mechanics. The model assumes a central mass with two small moons with nearly equal radii from the central body. Cors and Hall (2003) identified a parameter that determines if the moons simply pass each other or if they switch orbits (also called a horseshoe orbit). In early solar system formation, protoplanets pass through a phase of coorbital configuration where the Cors-Hall parameter permits collisions. We will develop coordinates for near-collision dynamics and consider outcomes of these interactions.

A bounded topological speedup of a Cantor minimal system (X,T) is another system (X,T^p) where p : X → ℕ is bounded (equivalently, continuous). Many classes of Cantor minimal systems are known to be closed under the taking of bounded speedups (odometers, expansive maps, substitutions, etc.), and in a 2023 paper, Ash, Dykstra and LeMasurier conjecture that any minimal bounded speedup of a Toeplitz system must itself be Toeplitz. In this talk, I'll discuss joint work with Aimee S.A. Johnson (Swarthmore) that shows this conjecture to be false. In particular, we prove that any proper, primitive and aperiodic substitution of constant length admits a minimal bounded speedup that is not Toeplitz.

How do poles and critical points of complex valued functions behave under iteration? Interestingly, we are able to visualize the orders of critical points and poles by graphing contour plots of the real part of a given complex function. As we follow the orbit of a critical point or pole, we can continue to generate contour plots to observe the order. In this talk, we look at the patterns created and summarize what causes orders to increase under iteration.

Given our recent results (with David Aulicino, Aaron Calderon, Nick Salter and Carlos Matheus) on Siegel-Veech constants for cyclic covers we investigate, if analogous results hold for more general classes of groups, such as solvable groups for example. For simplicity we restrict our presentation to torus-covers. We will state an alternative version to our previous formula for Siegel-Veech constants, one that serves as potential formula for more general deck transformation groups. We will present steps needed in a proof of that formula for solvable groups, and particularly investigate the structure of the Hurwitz spaces of covers. The geometric properties of these spaces explain certain structural properties displayed in the formula. After the necessary algebraic background on solvable groups, our presentation will be geometric and intuitive.

Two major discoveries of the last century demonstrated that there is a natural bridge between random and deterministic systems (Kolmogorov, 1958): stochasticity of deterministic systems, often called chaotic dynamics. Another discovery demonstrated that under small perturbations of integrable dynamical systems there are remnants of integrability which allow to call such systems quasi-integrable (Kolmogorov-Arnold-Moser (KAM) theory). I will explain all possible types of evolution (dynamics) of deterministic (dynamical) systems and will demonstrate them on the simplest (visual) examples. Mostly, the presentation will concentrate on dynamics (evolution) of conservative dynamical systems, although (if time permits) dissipative systems will be also discussed. At least the simplest strange attractors in dissipative systems will be presented. Some simplest but important open problems in the area will be also formulated.