Week 15
Have: evolution law. Want: invariant predictions 
 29. Discrete symmetry factorization / 26 April 2022 /
 30. Continuous symmetry factorization / 28 April 2022 bonus lecture /
 Homework 15
 Optional
3 disk pinball symmetries suffice to illustrate all that is needed to factorized spectral determinants for any system with a discrete symmetry: character. Discrete symmetry tiles the state space, and dynamics can be reduced to dynamics on the fundamental domain, together with a finite matrix that keeps track of the tile the full state space trajectory lands on. We need some group theory (one needs to underatand the projection to irreducible representations) and illustrate how different classes of periodic orbits contribute to different invariant subspaces for the 3disk pinball. Read sects. 25.2  25.6.
Trace formulas relate short time dynamics (unstable periodic orbits) to long time invariant state space densities (natural measure). A trace formula for a partially hyperbolic (N+ 1)dimensional compact manifold invariant under a global continuous symmetry is derived. In this extension of “periodic orbit” theory there are no or very few periodic orbits  the relative periodic orbits that the trace formula has support on are almost never eventually periodic.
Chapter 26 Continuous symmetry factorization This chapter needs a major revision, the videos are more uptodate than the chapter itself  
Group averaging  
The Great Orthonormality Theorem  
Average over space and time  
Discrete time averaging  
Continuous time averaging  
If discrete symmetry, average over it  
If continuous symmetry, average over it 
discrete factorization Due 28 April 2015 (note!) 

Discussion forum for week 15 
John F Gibson solves the NavierStokes 