dc.contributor.author Rafiqi, Ahmad dc.date.accessioned 2018-10-23T13:36:24Z dc.date.available 2018-10-23T13:36:24Z dc.date.issued 2018-08-30 dc.identifier.other Rafiqi_cornellgrad_0058F_10931 dc.identifier.other http://dissertations.umi.com/cornellgrad:10931 dc.identifier.other bibid: 10489873 dc.identifier.uri https://hdl.handle.net/1813/59775 dc.description.abstract Suppose $S$ is a compact topological surface without boundary, oriented and connected. In \S$1$ we go over Thurston's classification of \emph{automorphisms} of $S$: continuous functions $f:S\to S$ with continuous inverses. The classification provides an $f_\#$ homotopic to a given automorphism $f$, which is either: periodic ($f_\#^n$ is the identity for some $n\ge1$); or $f_\#$ fixes a finite union of disjoint circles $\gamma_i:S^1\hookrightarrow S$; or is what Thurston calls pseudo-Anosov (pA). In the pA case, there is a finite number of points on $S$ whose complement carries two transverse foliations by $1$-submanifolds each with a holonomy invariant measure on transverse arcs. Such a structure on $S$ can be described in terms of a quadratic differential on a Riemann surface structure on it which we go over in \S$1.2$. Further, there is a number $\lambda_f\ge1$ (called the \emph{dilatation} or \emph{stretch factor}) such that the two measured foliations are stretched and shrunk respectively by $\lambda_f$ under $f_\#$.\\ When the foliations of a pA map $f$ can be oriented consistently in neighborhoods on $S$, the dilatation $\lambda_f$ is an eigenvalue of the induced action $f_*$ on the homology group $H_1(S;\mathbb{Z})$. The numbers $\lambda_f$ thus satisfy monic polynomials with integer coefficients. Fried \cite{Fried} showed that $\lambda_f$ is a unit in $\mathbb{Z}[\lambda_f]$ and that the Galois conjugates of $\lambda_f$ (excepts perhaps one of $\pm\lambda_f^{-1}$) lie in the open annulus $\{z\in\mathbb{C}:1/\lambda_f<|z|<\lambda_f\}$. Numbers satisfying these properties are called \emph{biPerron}. If the Galois conjugates of an algebraic integer $\lambda\ge1$ are only in the disk $\{|z|<\lambda\}$, it is called a \emph{Perron} number. We describe Fried's proof and some properties of dilatations in \S$1.3$. In \cite[Problem 2]{Fried}, Fried asked whether some power of a biPerron unit is always a surface automorphism stretch factor. This thesis is an attempt to solve this problem. \\ In \S$2$ we go over a well-known rectangular decomposition of a Riemann surface with a quadratic differential. When the associated foliations are orientable, the quadratic differential is the square of an Abelian differential and we study this latter structure on a Riemann surface in some detail.\\ A square matrix $A$ of non-negative entries is called \emph{mixing} if some power $A^n$ has only positive entries. If only the sum $A+A^2+...$ is positive, $A$ is \emph{ergodic}. By the Perron-Frobenius theorem, an ergodic matrix has a real eigenvalue $\lambda>0$ (called the \emph{Perron root}) bigger than all its other eigenvalues in absolute value, and whose eigenspace is $1$-dimensional. If the matrix has integer entries, the Perron root is a Perron number. Lind \cite{Lind84} showed the converse that all Perron numbers are eigenvalues of ergodic integer matrices. In Chapter 3 (see \cite{BRW16}), under additional hypotheses on an erdogic matrix $A$ with entries in $\{0, 1\}$ we construct a closed orientable surface $S$ of genus $g\le\dim{A}/2$ with a self-homeomorphism $f$ such that the dilatation $\lambda_f$ is the Perron root of $A$ (which is therefore biPerron).\\ On the other hand, Hamenst\"adt \cite{Hamen13} showed that out of dilatations smaller than $R>0$ on a surface of fixed genus $g$, the proportion of those that have only real Galois conjugates approaches $1$ as $R\to\infty$. This suggests Fried's conjecture may be false and motivated our study of the asymptotic behavior of biPerron units which we present in chapter 4. Let $B_g(R)$ be the set of bi-Perron units no larger than $R$ whose minimal polynomial has degree at most $2g$, and say $D_g(R)$ is the set of dilatations no larger than $R$ of pseudo-Anosov maps with orientable invariant foliations on $S_g$.\\ Eskin-Mirzakhani-Rafi (2016) and Hamenst\"adt (2016) independently showed that the number of periodic orbits of length less than $\log(R)$ for the Teichm\"uller flow on the moduli space of area one Abelian differentials on $S_g$ grows like $R^{4g-3}/{\log(R)}$ as $R\to\infty$. This is an upper bound for the number of dilatations, $|D_g(R)|$. We showed (with H. Baik and C. Wu) that $|B_g(R)|$ grows like $R^{g(g+1)/2}$ as $R \to\infty$. Since dilatations $D_g(R)$ form a subset of biPerron units $B_g(R)$, we get that for $g\ge6$ the proportion of dilatations $D_g(R)$ inside biPerron units $B_g(R)$ approaches $0$ as $R\to\infty$. Our result does not disprove Fried's conjecture.\\ For the lower genera (2, 3, 4 and 5) we get an interesting result as well. If there are exactly $n$ closed geodesics in Moduli space of the same length $\log(\lambda)$, we say $n$ is the \emph{multiplicity} of each of these geodesics. For geodesics bounded in length, the proportion of those closed geodesics that have multiplicity greater than some positive integer $k$ approaches $1$ as the bound in length approaches $\infty$. That is, asymptotically almost all geodesics in Moduli spaces for these genera have arbitrarily high multiplicities. dc.language.iso en_US dc.rights Attribution 4.0 International * dc.rights.uri https://creativecommons.org/licenses/by/4.0/ * dc.subject Mathematics dc.subject BiPerron dc.subject Dilatation dc.subject Pseudo-Anosov dc.title On dilatations of surface automorphisms dc.type dissertation or thesis thesis.degree.discipline Mathematics thesis.degree.grantor Cornell University thesis.degree.level Doctor of Philosophy thesis.degree.name Ph. D., Mathematics dc.contributor.chair Hubbard, John Hamal dc.contributor.committeeMember Manning, Jason F. dc.contributor.committeeMember Hatcher, Allen E dcterms.license https://hdl.handle.net/1813/59810 dc.identifier.doi https://doi.org/10.7298/X4QJ7FJH
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