Pseudo-Anosov mappings and toral automorphisms

Abstract

For every irreducible automorphism $\varphi \in {\text{SL}}_3(\mathbb{Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $\varphi$, whose stretch factor is the product of the expanding eigenvalues of $\varphi$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $\varphi$ has exactly two eigenvalues outside the unit circle (and whose product is positive). Furthermore for any irreducible hyperbolic automorphism $\varphi \in {\text{SL}}_n(\mathbb{Z})$ of the $n$-torus, $n\ge 4$, we construct a pseudo-Anosov mapping semiconjugate and almost-isomorphic to any sufficiently large power of $\varphi$.

There is a mistake in the proof

References in corpus (1)

• Tree substitutions and Rauzy fractals

Cited by in corpus (1)

• The geometry of bi-Perron numbers with real or unimodular Galois conjugates