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 $n\ge 1$); or $f\#$ fixes a finite union of disjoint circles $\gamma i:S1\hookrightarrow S$; 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 $\lambda f\ge 1$ (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 $f\ast$ on the homology group $H1(S;Z)$. The numbers $\lambda f$ thus satisfy monic polynomials with integer coefficients. Fried \cite{Fried} showed that $\lambda f$ is a unit in $Z[\lambda f]$ and that the Galois conjugates of $\lambda f$ (excepts perhaps one of $\pm \lambda f-1$) lie in the open annulus . Numbers satisfying these properties are called \emph{biPerron}. If the Galois conjugates of an algebraic integer $\lambda \ge 1$ are only in the disk , 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 $An$ has only positive entries. If only the sum $A+A2+...$ 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 $Bg(R)$ be the set of bi-Perron units no larger than $R$ whose minimal polynomial has degree at most $2g$, and say $Dg(R)$ is the set of dilatations no larger than $R$ of pseudo-Anosov maps with orientable invariant foliations on $Sg$.\ 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 $Sg$ grows like $R4g-3/\log (R)$ as $R\to \infty$. This is an upper bound for the number of dilatations, $|Dg(R)|$. We showed (with H. Baik and C. Wu) that $|Bg(R)|$ grows like $Rg(g+1)/2$ as $R\to \infty$. Since dilatations $Dg(R)$ form a subset of biPerron units $Bg(R)$, we get that for $g\ge 6$ the proportion of dilatations $Dg(R)$ inside biPerron units $Bg(R)$ 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.