3
$\begingroup$

Let $V$ be a smooth Anosov vector field on a compact $n$ dimensional manifold $X$. Let $D(X)$ denote the set of distance functions $d$ on $X$ that are equivalent to fixed Riemannian distance. For each $d\in D(X)$ consider the log Lipschitz constant of the time one flow $$L_d (V) := \sup_{x\neq y} \log \frac{ d(e^Vx,e^Vy)}{d(x,y)}.$$ Then I think it is well known (eg. Thm 7.15 in Walters's ergodic theory book) that the topological entropy of $V$ satisfies $$h_{\mathrm{top}} (V) \leq n L_d (V).$$ But the question is whether the infimum of the right-hand side over all such equivalent distances $d\in D(X)$ equals the entropy

$$h_{\mathrm{top}} (V) = \inf_{d\in D(X)} \ nL_d (V) ? $$

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.