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Let F be a pseudo-Anosov homeomorphism of a hyperbolic surface S. In this talk, we’ll describe joint work with Tarik Aougab and Dave Futer that predicts the number of fixed points of F, up to constants that depend only on the surface S. If F satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of F is coarsely equal to its translation length on the Teichmuller space of S. Without this condition, there is still a coarse formula involving subsurface projections of F’s invariant laminations.

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