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Let $\phi$ be a pseudo-Anosov on a surface with punctures with asymptotic translation length $L(\phi)$. If under the forgetful map $\phi$ restricts to a pseudo-Anosov $\hat{\phi}$ on a connected subsurface with asymptotic translation length $L(\hat{\phi})$ is $L(\phi)\geq L(\hat{\phi})$?

This would be the analogy for dilatations of these mapping classes.

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    $\begingroup$ A mapping class fixing a subsurface (up to isotopy) preserves the boundary curves, and is therefore reducible and not pseudo-Anosov. $\endgroup$
    – Ian Agol
    Aug 16, 2013 at 12:44

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