Week Six: More Discrete 1D Maps and Chaos
This week we continued to look at discrete 1D maps and started to read a key paper in chaos theory.
1 In Class
Photos of work on the decimal shift map and tent map (from HW5) are in this Google Drive. You can also look at Desmos graphs for these explorations (thanks Aidan!).
We started discussing Li and Yorke’s paper “Period Three Implies Chaos”. The goal of looking at this paper will be to understand the statement of the main result (Theorem 1), the ideas of the proof of the first part of Theorem 1 (proof of T1), and the proof in Appendix 1 that a point of period 5 does not guarantee a point of period 3.
In class, we talked through the introduction and what to expect to see in the main result. That list included:
- having a point of period 3 guarantees that for all natural numbers \(p\), a periodic point exists with that period
- having a point of period 3 guarantees that there are uncountably many aperiodic points
- those aperiodic points have orbits/trajectories that behave in ways that feel “chaotic”
2 Homework 6 – Due Friday, Mar. 6 by the start of class
Submit your work to the HW 6 assignment on Blackboard!