Structure of Attractor Region for Natural Extension Map Associated with p-adic Continued Fractions

Open Access
Author:
Zydney, Adam Joseph
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Svetlana Katok, Thesis Supervisor
  • Svetlana Katok, Honors Advisor
  • Misha Guysinsky, Thesis Supervisor
Keywords:
  • attractor region
  • natural extension map
  • p-adic numbers
  • analysis
Abstract:
This thesis investigates the behavior of repeated iterations of a map F on two p-adic numbers, ultimately proving the existence of an attractor region for this map and describing that region. A description is also given of the behavior of the map on points inside and outside of the attractor region. First, the p-adic numbers are introduced as an alternative (to the real numbers) completion of the rationals, and definitions are given for canonical expansions, p-adic integers, and p-adic units. Also laid out are the means for arithmetic of p-adic numbers relevant to computing F. With the addition of a non-injective "digit reversing" map from p-adic numbers to real numbers, a computer program (written in Java) produces several graphs of the behavior of F for various initial points. Using these plots as a guideline, analytic proofs are then developed to rigorously show the behavior of F, specifically the existence and description of its attractor region.