On stretch-factors of pseudo-Anosov homeomorphisms.

Abstract

RESEARCH PROJECT SUMMARYThe project is in the general fields of topology, topological dynamics and hyperbolic geometry. More specifically I've been working on problems related to surface homeomorphisms and their `stretch-factors'.Past researchFried [F] proved that if » is the dilatation associated to a pseudo-Anosov homeomorphism of a compact surface then both » and 1/» are roots of monic polynomials over the integers (so » is an algebraic unit) and that all its Galois conjugates (excepts perhaps one of ±»^(-1 )) lie in the open annulus with inner out outer radii 1/» and ». Numbers satisfying these properties are called biPerron units. In his paper, Fried conjectured that all biPerron units are dilatations of surface homeomorphisms and not just vice-versa as he proved. There are many related works. Thurston [Th1] gave a partial answer to the Out(Fn)-version of this question (see [Th1] Theorem 1.8).Thurston [Th1] gave a few examples of constructions of pseudo-Anosov maps for given biPerron algebraic units. He constructed post-critically finite piecewise-linear maps of the unit intervalwith the dilatation as the slopes, and surfaces were constructed by computing the É-limit set of extensions of the interval maps to the plane that stretched horizontally and shrunk vertically by ». Theseexamples motivated my previous work ([BRW], with H. Baik and C. Wu) to construct pseudo-Anosovmaps with prescribed biPerron units as dilatations. We considered those biPerron units that werethe leading eigenvalues of matrices consisting of 0s and 1s (with some additional properties, see [BRW]).For these we were able to construct closed orientable surfaces (with quadratic differentials comingfrom the shape of the matrix) and define homeomorphisms on them such that the dilatation was thefixed biPerron unit. Even though this construction gave us many new examples of pseudo-Anosovmaps, it does not produce all pseudo-Anosov maps. Whether such a construction works for allbiPerron units is unknown.Future directionsIn [BRW] we gave a sufficient condition for the leading eigenvalue of a matrix to be a pseudo-Anosovdilatation on a closed surface. One direction would be to investigate whether surfaces of finite typecould be obtained for a bigger class of biPerron units. However, if we do not require the resultingsurface to be of finite-type, we can significantly weaken the conditions, and obtain pseudo-Anosov like maps. More precisely, we still have two invariant and transverse, measured foliations on thesurface - except perhaps with infinitely many singularities - one still expanded and the other contracted by the dilatation. If the singularities of the foliation accumulate at a finite number of points weget a generalized pseudo-Anosov map (in the sense of [dCH]).References:[BRW] Baik, H., Rafiqi, A. and Wu, C. (2016). Constructing pseudo-Anosov maps with given dilatations, Geom. Dedicata, Volume 180, Issue 1, 39{48.[BRW2] Baik, H., Rafiqi, A., &Wu, C. (2016), Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation? Accepted: Ergodic Theory and Dynamical Systems.[dCH] de Carvalho, A. and Hall, T. (2004). Unimodal generalized pseudo-Anosov maps. Geom. Topol., 8, 1127{1188.[EMR] Eskin, A., Mirzakhani, M., Ra_, K. Counting closed geodesics in strata, arXiv:1206.5574.[F] Fried, D. (1985). Growth rate of surface homeomorphisms and ow equivalence. Ergod. Theory Dyn. Syst., 5(04), 539{563.[H13] Hamenstadt, U. (2013). Bowen's construction for the Teichmuller ow, J. Mod. Dynamics 7, 489{526.[H2] Hamenstadt, U. Typical properties of periodic Teichmuller geodesics, arXiv:1409.5978.[Mc] McMullen, C. T. Polynomial invariants for fibered 3-manifolds and Teichmuller geodesics for foliations. Annales scientifiques de l'Ecole normale suprieure. Vol. 33. No. 4. 2000.[Th1] Thurston, W. (2014). Entropy in dimension one. arXiv:1402.2008.[Th2] Thurston, W. P. A norm for the homology of 3-manifolds, M