Dilatation of surface diffeomorphisms

Question

Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism. Define dilatation of $f_*$ as the largest absolute value of eigenvalues of $f_*$. Fix a negatively curved metric $g$ on $S$. (Say, $g$ is a hyperbolic metric.)

Question: Is it true that for any $\varepsilon>0$ there exists a diffeomorphism $h$ homotopic to $f$ such that $$ dil(h,g)<dil(f_*)+\varepsilon? $$ If not, what can one say about the $inf_h(dil(h,g)-dil(f_*))$?

Here the dilatation of $h$ with respect to $g$ is defined as $$ dil(h,g)=max_{v\in TS}\frac{\|Dhv\|_g}{\|v\|_g}. $$

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Misha points out that in general answer is no. So, let me ask follow up questions.

Q2: Assume that $f$ is p-Anosov with dilatation $\lambda$. Is it true that for any $\varepsilon>0$ there exists a diffeomorphism $h$ homotopic to homeomorphism $f$ such that $$ dil(h,g)<\lambda+\varepsilon? $$ If not, what can one say about the $inf_h(dil(h,g)-\lambda)$?

Q3: A more vague question is: how does $dil(h,g)$ varies with $h$ and $g$?

Answer 1

The answer to question two is also "no". The right way to minimize the dilatation is vary both the metric and the representative $f'$ of the mapping class $[f]$. There is a large family of pairs $(f',g)$ that exactly realize the dilatation. However, for generic $g$, no map $f'$ in the mapping class $[f]$ can realize the dilatation.

Question three is answered by thinking about the "Lipschitz distance" between the metric $g$ and the "Lipschitz geodesics" stabilized by $[f]$. (At least, this will give a lower bound.)

As Misha says, you should read Thurston's paper "Minimal stretch maps between hyperbolic surfaces". This is an interesting area of mathematics - for example Bill Goldman gives a connection between minimal stretch maps and the problem of building Lorentz manifolds.

Answer 2

The canonical reference is actually:

\bib{MR3134416}{article}{
   author={McMullen, Curtis T.},
   title={Entropy on Riemann surfaces and the Jacobians of finite covers},
  journal={Comment. Math. Helv.},
  volume={88},
   date={2013},
  number={4},
   pages={953--964},
   issn={0010-2571},
   review={\MR{3134416}},
   doi={10.4171/CMH/308},
}