Seminar

Symbolic dynamics, and the associated topological entropy, are well

developed tools for analyzing two-dimensional area-preserving

dynamics, such as arise in 2D symplectic maps and the chaotic mixing

of 2D fluids.  For example, topological entropy has been useful in

quantifying the mixing of fluids stirred by periodically braiding

rods.  However, at present no analogous symbolic techniques exist for

extracting topological dynamics from symplectic maps in higher

dimensions.  Here, we address chaotic, volume-preserving maps in

three-dimensions, which is a stepping stone to 4D symplectic maps and

a system of intrinsic interest for mixing of 3D fluids.  We address

this challenge using the topology of intersecting codimension-one

stable and unstable manifolds.  This leads to a symbolic dynamics of

2D surfaces based on homotopy theory.  This symbolic dynamics can be

understood as resulting from stirring by loops that undergo a kind of

3D braiding.  The resulting theory provides a rigorous lower bound on

the growth rates of both two-dimensional surfaces and one-dimensional

curves.  We illustrate our theory with a mathematical model of a

chaotic ring vortex.  Finally, we will present results that hint at

the presence of a subtle duality in the topological dynamics.