Dynamical Asymmetry Is C1-Typical

I will discuss a result with Bonatti and Crovisier from 2009 showing that the C1 generic diffeomorphism f of a closed manifold has trivial centralizer; i.e. fg = gf implies that g is a power of f. I’ll discuss features of the C1 topology that enable our proof (the analogous statement is open in general in the Cr topology, for r greater than 1). I’ll also discuss some features of the proof and some recent work, joint with Danijela Damjanovic and Disheng Xu that attempts to tackle the non-generic case.