The goal of this project is to explore surprising connections between two seemingly unrelated problems, one belonging to analysis, and the other belonging to number theory: The behavior of chaotic maps on the one hand, and the representation of numbers in "non-standard", or "exotic", number systems on the other hand. A chaotic map is, roughly speaking, a function that when iterated a large number of times behaves in unpredictable ways. We will focus on a particular class of such maps, namely piecewise linear functions from [0,1] to [0,1]. Examples include the "tent map", which consists of an up-slope followed by a down-slope, and maps of the form f(x)= {ax+b}, where the braces denote the fractional part. These maps lead in a natural way to representations, or "encodings", of real numbers with respect to generalized, or "exotic", number systems, a connection that has been used in recent years to design encryption and compression schemes. Despite the apparent simplicity of these maps, their behavior under iteration is still not well understood. In this project we will investigate these maps - from both the "chaotic map" and the "generalized number systems" angle - using computer experimentation and visualization, and hopefully also gain new theoretical insights.

**Faculty Member: ** A.J. Hildebrand and Ken Stolarsky

**Difficulty: **Intermediate

**Team Meetings: **Twice a week

**Prerequisites: **Completion of Calculus 3. Exposure to an upper level proof-based course such as elementary number theory, real or complex analysis, or differential equations, would be desirable, though specific knowledge in any of these areas is not required. Prior experience with Mathematica (preferred) or other appropriate software such as Matlab or Python would be helpful, but the necessary skills can also be acquired at the beginning of the project.

More information on Chaotic maps, exotic number systems, and arithmetic coding schemes