Entropy rigidity near real and complex hyperbolic metrics.

The HADES seminar on Tuesday, October 1st, will be at 2:00pm in Room 740.

Speaker: Tristan Humbert

Abstract: Topological entropy is a measure of the complexity of a
dynamical system. The variational principle states that topological
entropy is the supremum over all invariant probability measures of the
metric entropies. For an Anosov flow, the supremum is uniquely attained
at a measure called the measure of maximal entropy (or Bowen-Margulis
measure).

An important example of Anosov flow is given by the geodesic flow on a
negatively curved closed manifold. For these systems, another important
invariant measure is given by the Liouville measure : the smooth volume
associated to the metric.

A natural question, first raised by Katok is to characterize for which
negatively curved metrics the two measures introduced above coincide.
The Katok’s entropy conjecture states that it is the case if and only if
g is a locally symmetric metric. The conjecture was proven by Katok for
surfaces but remains open in higher dimensions.

In this talk, I will explain how one can combine microlocal techniques
introduced by Guillarmou-Lefeuvre for the study of the marked length
spectrum with geometrical methods of Flaminio to obtain Katok’s entropy
conjecture in neighborhoods of real and complex hyperbolic metrics (in
all dimensions).

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