TITLE: Pseudo-Anosov maps with same dilatation.
QUESTION [8 upvotes]: Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose $\lambda_f$ denote the stretch factor (or dilatation) of $f$.
Q) Does there exist $f,g\in Mod(S)$ such that $f\neq g$ and $\lambda_f=\lambda_g.$

REPLY [3 votes]: Here is a general source of examples.  Suppose $f$ is a pseudo-Anosov map whose centralizer is not cyclic.  Say $\rho$ is finite order and is in the centralizer of $f$.  Note that $f$ and $\rho f$ have the same dilatation, because they have a common power.  However, $\rho$ can change the way the singularities (and the separatrices) are permuted, and this permutation data is also a conjugacy invariant.  
To be concrete, suppose that $f$ is any pseudo-Anosov map on the twice-punctured torus and that $\tau$ is the hyperelliptic element.  Then $f$ and $\tau f$ are not conjugate. With a bit of care you can play similar games in hyper-elliptic strata in any genus.
There are more subtle obstructions as well -- see the paper
Polynomial invariants of pseudo-Anosov maps by Birman, Brinkmann, and Kawamuro.  Near the end of the paper they say: "It seems to be an open question to describe all the ways to construct all pA maps having a fixed dilatation."
[Edit] Here is another construction.  Suppose that $f$ is an Anosov map on the torus.  Take a power to ensure that $f$ has at least two fixed points $x$ and $y$.  Now form $S_g$ by taking a $g$--fold branched cover over $x$ and $y$.  So $S_g$ is a surface of genus $g$.  We can take a further power of $f$ to ensure that it lifts to a map on $S_g$.  Note that $f$, acting on $S_g$, has exactly two singularities.  Ok.  Now form the surface $T$ by taking a unbranched double cover of $S_2$.  So $T$ has genus three and is homeomorphic to $S_3$.  There is again a power of $f$ that lifts to $T$, and this lift is a pA map with four singularities. Taking the correct powers of $f$ equips $S_3$ and $T$ with pA maps of the same dilatation, but with different numbers of singularities.