Rational Maps and Towers of Finite Fields
We can create sequences from rational maps by iterating a map via composition. As we vary the starting value, this may vary the behavior of the sequence. The sequence could loop back to the starting point, or have a long tail then have a repeating set of numbers, or it can continue unbounded. If we restrict our set of numbers to only live in finite fields, then all points either live on a cycle or live on a tail. So, the question we want to explore is how many points are there that live on a cycle? Does this number vary widely as we travel up a tower of finite fields? In general, this is a difficult question to answer since a random rational function behaves chaotically, but for certain special families we can explore and answer these questions.
This talk is appropriate for faculty, undergraduate students, and a more general audience.
Dr. Bianca Thompson is an assistant professor at Westminster College. She identifies as queer, Latina, and biracial and has traveled from one end of the country to the other for her education (Massachusetts to Hawai'i and back again). Her research falls in the intersection of discrete dynamical systems and number theory in the relatively young field of arithmetic dynamics. She wants to find more ways to create a classroom environment where students know they belong.