Pseudo-Anosov map

Remarks

Example: on the torus $\R^2 / \Z^2$, the map induced by $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, Arnold's cat map, is pseudo-Anosov with no singular points. The eigenvalues of $A$ are $\lambda = \frac{3 + \sqrt 5}{2} > 1$ and $\lambda^{-1}$. The foliations by lines parallel to the two eigenvectors, with transverse measures given by distance across the lines, are contracted by $\lambda^{-1}$ and stretched by $\lambda$.

Non-example: the Dehn twist induced by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ on the torus has only the eigenvalue $1$. It preserves the horizontal foliation without stretching it, so no $\lambda > 1$ fits clauses 2 and 3.

On a closed surface of genus $g \ge 2$, the Euler–Poincaré formula forces singular points, since the foliations cannot be nonsingular. For surfaces with punctures, $1$-prong singularities are also allowed at the punctures. The stretch factor $\lambda$ is an algebraic integer, and $\log \lambda$ is the topological entropy of $f$, the least entropy among all homeomorphisms isotopic to $f$.