Dynnikov coordinates and pseudo-Anosov braids

• Main Author: Yurttas, Saadet Öykü
• Published: University of Liverpool 2011
• Subjects: 514
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.569517

The aim of this thesis is to study dynamical properties of pseudo -Anosov braids on the n-times punctured disk Dn making use of a particular coordinate system called the Dynnikov coordinate system. The Dynnikov coordinate system gives a homeomorphism from the space of measured foliations MFn on Dn (up to a certain equivalence relation) to Sn = R2n-4\ {O}, and restricts to a bijection from the set of integral laminations (disjoint unions of finitely many essential simple closed curves) on Dn to Cn = Z2n-4 \ {O}. In the first part of the thesis, we introduce a new method for computing the topological entropy of each member of an infinite family of pseudo -Anosov braids making use of Dynnikov's coordinates. The method is developed using the results in Thurston's seminal paper on the geometry and dynamics of surface automorphisms and builds on, more recent work of Moussafir. To be more spe- cific, the method gives a so-called Dynnikov matrix which describes the action of a given pseudo-Anosov braid B near its invariant unstable measured foliation [F, u] on the projective space PSn, and the eigenvalue \ > 1 of this matrix gives the topological entropy of B. In the second part of the thesis, we compare the spectra of Dynnikov matrices with the spectra of the train track transition matrices of a given pseudo-Anosov braid, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions.