Hausdorff Dimension of the Julia Set of a Hyperbolic Expanding Map from Cantor Set Correspondence

Dynamical systems in chemistry and physics are often modeled using iterative maps, whose behavior for a given initial condition can be categorized as either predictable or chaotic. The Julia set of an iterative map consists of all initial conditions that exhibit chaotic behavior upon iteration. The properties of the Julia set reflect important features of the original system itself. One particular property — the Hausdorff dimension — tells us how the chaotic initial conditions are distributed throughout space. For a specific parameterized iterative quadratic map, we constructed the Julia set in a manner similar to the Cantor set. Using this new construction and numerical computation, we tightly bounded the dimension as the parameter approaches -2. With this, we can capture the moment the Julia set transitions from being disconnected to connected. We can use this characterization to address more difficult problems in dynamical system research, including questions in scattering theory and counting Pollicott-Ruelle resonances.