## Departmental PhD Thesis Exam – Pablo Carrasco

Monday, June 28, 2010, 11:00 a.m.,
in BA 6183, 40 St. George St.

PhD Candidate:  Pablo Carrasco

PhD Advisors:  Charles Pugh and Michael Shub

Thesis Title:  Compact Dynamical Foliations
(http://www.math.toronto.edu/pablo/Thesis%20-%20Pablo%20Carrasco.pdf)

Thesis Abstract:

According to the work of Dennis Sullivan, there exists a smooth flow on the
5-sphere all of whose orbits are periodic although there is no uniform bound
on their periods.  The question addressed in this thesis is whether such an
example can occur in the partially hyperbolic context.  That is, does there
exist a  partially hyperbolic diffeomorphism of a compact manifold such that
all the leaves of its center foliation are compact although there is no
uniform bound for their volumes. We will show that the answer to the
previous question under the very mild hypothesis of local product structure
is no.

The thesis is organized as follows. In the first chapter we give the
necessary background and results in partially hyperbolic dynamics needed for
the rest of the work, studying in particular the geometry of the center
foliation. Chapter two is devoted to a general discussion of compact
foliations. We give proof or sketches of all the relevant results used.
Chapter three is the core of the thesis, where we establish the non
existence of Sullivan's type of examples in the partially hyperbolic domain,
and generalize to maps where the center foliation has arbitrary dimension.
The last chapter is devoted to applications of the results of chapter three,
where in particular it is proved that if the center foliation of a
dynamically coherent transitive partially hyperbolic map is compact, then it
is plaque expansive.