Colloquium: "Hyperbolicity in Dynamics and Geometry"

Abstract: A  dynamical  system,  such  as a flow  or  transformation,  is called  hyperbolic if  it  tends  to stretch and contract  the  underlying space  in different  directions. This  behavior  often appears  for  systems  that are  sufficiently mixing and/or  volume  preserving  --  think of  the  striations  that  appear  when  mixing a  drop of milk into coffee  --  and it  lends  these  systems  a  kind of  rigidity that  can be  useful  in understanding  their long-term  behavior.   We  will  look  at  a  number  of  hyperbolic  dynamical  systems, including Anosov and pseudo-Anosov flows  and transformations,  and  illustrate some of  the uses  and  consequences of their  hyperbolic  behavior. In addition, we  will  see  that  the  dynamics  of  a  hyperbolic  system  can often be understood  in terms  of  a  simpler, lower-dimensional  system  that  lies "at  infinity."  This is  an  important part of  the  proof  of  Calegari's  Conjecture,  which relates  the  dynamical  hyperbolicity of  a  flow  with the geometric  hyperbolicity  of  its  underlying space:  It  says  that  any flow  on a  closed hyperbolic  3-manifold whose orbits  are coarsely  comparable  to geodesics  is  equivalent, on the  large  scale, to  a  hyperbolic  flow.

Tea @ 3:45 in Cupples I, Room 200

Host: Rachel Roberts