Week 11

• 21. Trace formulas / 20 March 2018 /
• If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. The duality is made precise by means of trace formulas. The course is OVER. Trace formulae are beautiful, and there is nothing more to say. Just some mopping up to do.

 Chapter 21 - Trace formulas Dirac delta function Deterministic evolution Classical trace formula for maps Continuous time and Laplace transforms Continuation of the compagnion video (week 9, lecture "Counting" on discrete time and generating functions, watch youtu.be/8DzhYfGNo1U). Discrete-timegenerating functions compared with relating infinitesimal time evolution to the infinite time dynamics via a Laplace transform. The stage is set for the classical trace formula. (A paid video, recorded on a Wacom tablet and edited by Stephen Murphy, GaTech PE Interactive Instructional Media.) Contribution of a periodic orbit, continuous time Intrinsic coordinates Evaluation of the trace along the orbit[27 minutes] Trace formula - an interpretation Global-local duality Don't compute the eigenfunctions

• 22. Spectral determinants / 22 March 2018 /
• We derive the spectral determinants, dynamical zeta functions. While traces and determinants are formally equivalent, determinants are the tool of choice when it comes to computing spectra. Skip sects. 20.5 and 20.6.

 Chapter 22 - Spectral determinants Our zeta functions are exact Compute determinants, not traces From a trace to the determinant Dynamical zeta function as an approximation to the spectral determinant Moral tale: Perils of infinite products
• Homework 11
•  trace formula and spectral determinant  Due 3 April 2018 Discussion forum for week 11

• Optional
•  History of zeta functions, or, Why nobody talks to anybody Gutzwiller before he invented chaos You are blown away, and, There is no "alternative" theory Q: what about equilibria? Sleep deprivation may cause brain damage The leading cause of migranes