On dilatations of surface automorphisms

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.