Approximate roots of pseudo-Anosov diffeomorphisms

Gendron, T. M.

Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1413-1442.

La Conjecture de la Racine prévoit que chaque difféomorphisme pseudo-Anosov d’une surface fermée a une racine $n$ième approximative de Teichmüller pour tout $n\ge 2$. Dans cet article, on remplace la topologie de Teichmüller par la topologie hauteur-longueur – celle qui est induite par la convergence des différentielles quadratiques tangentes relativement aux fonctionnelles hauteurs et longueurs simultanément – et on prouve que chaque difféomorphisme pseudo-Anosov d’une surface fermée a une racine $n$ième approximative hauteur-longueur pour tout $n\ge 2$.

The Root Conjecture predicts that every pseudo-Anosov diffeomorphism of a closed surface has Teichmüller approximate $n$th roots for all $n\ge 2$. In this paper, we replace the Teichmüller topology by the heights-widths topology – that is induced by convergence of tangent quadratic differentials with respect to both the heights and widths functionals – and show that every pseudo-Anosov diffeomorphism of a closed surface has heights-widths approximate $n$th roots for all $n\ge 2$.

Classification : 30F60, 32G15
Keywords: Teichmuller space, pseudo-Anosov diffeomorphism, root conjecture
Mot clés : espace de Teichmüller, difféomorphisme pseudo-Anosov, conjecture de la racine

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