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Let $l_{g,n}$ be the logarithm of minimum dilatation for pseudo-Anosov homeomorphisms on surface of genus $g$ with $n$ punctures. Let $n$ be fixed and $g$ varies. Is the asymptotic behavior of $l_{g,n}$ known? Does it behave like $\frac{1}{g}$?

Added: Penner proved the lower bound $l_{g,n} \geq \frac{\log2}{12g-12+4n}$ for surfaces of negative Euler characteristic. He also shows an upper bound $l_{g,0} \leq \frac{\log11}{g}$ by constructing explicit examples of low dilatation pseudo-Anosov maps on closed surfaces. This shows that for closed surfaces the asymptotic behavior of $l_{g,0}$ is like $\frac{1}{g}$ as genus grows. Since the lower bound works in general no matter how many punctures we have, if one can construct examples of low dilatation pseudo-Anosov maps on surfaces with arbitrary fixed number of punctures, then the same asymptotic behavior follows.

PS: Chia Yen-Tsai states in her paper, the asymptotic behavior of least pseudo-Anosov dilatations, that the above question has a positive answer for $n = 0,1,2,3$ or $4$.

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    $\begingroup$ If there is an upper bound comparable to Penner's lower bound (up to a constant), then a theorem of Farb-Leininger-Margalit implies that the mapping tori of manifolds realizing $l_{g,n}$ for fixed $n$ will be obtained by Dehn filling on finitely many fibered 3-manifolds. Thus, to construct upper bounds, one could try to find a 3-manifold which fibers with $n$ cusps, and which has non-peripheral homology, so that the fibered faces which have the same boundary slopes give all possible genus. $\endgroup$
    – Ian Agol
    Commented May 22, 2016 at 5:26
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    $\begingroup$ See also the work of Valdivia: arxiv.org/pdf/1006.4409v3.pdf $\endgroup$
    – Dan Margalit
    Commented Jun 28, 2016 at 2:01

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