Relativity for cyclists

Predrag Cvitanović
Physics, Georgia Tech


Trace formulas relate short time dynamics (unstable periodic orbits) to long time invariant state space densities (natural measure). Higer dimensional dynamics requires inclusion into trace formulas of higher-dimensional compact invariant sets, such as partially hyperbolic invariant tori. A trace formula for partially hyperbolic $(N+1)$-dimensional compact manifolds invariant under a global continuous symmetry is derived here. 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 "quasiperiodic," almost never eventually periodic.

[1] P. Cvitanović, Continuous symmetry reduced trace formulas (in preparation)
[2] P. Cvitanović,, rough draft of chapter "Continuous symmetry reduction"

PS Above "relativity" has nothing to do with Einstein - rather, it is about "relative periodic orbits" which underlie turbulence in Kuramoto-Sivashinsky, Navier-Stokes pipe and plane flows, and higher-dimensional Hamiltonian transport.