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Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the Riemannian exponential map.

For any measurable subset $C\subset S$ and any $R\in\mathbb{R}_+$, we put $$ [0,R]\cdot C=\{tv\in T_{x_0}X\mid t\in[0,R],\,v\in C\} $$ and define the volume entropy of $C$ as $$ h(C):=\lim_{R\rightarrow+\infty}\frac{1}{R}\log \mathrm{vol}\big(\exp\big([0,R]\cdot C\big)\big). $$ So $h(S)$ is the usual volume entropy of $X$.

I want to know whether the following proposition on continuity of $h(C)$ is true.

Proposition. For any $\epsilon>0$, there exists $\delta>0$, such that for any measurable subset $C$ of $S$ whose complement has measure less than $\delta$, we have $h(C)>h(S)-\epsilon$.

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    $\begingroup$ Just a suggestion: have a look to Roblin's book (Mémoires SMF), where he proves that the critical exponent of a group $\Gamma$ is equal to the exponential growth rate of any orbit in any cone, and not in the full space. You could use it for the statement that you wish, which is, to my knowledge, unknown. Barbara $\endgroup$
    – Barbara Schapira
    Commented Jun 24, 2014 at 21:19

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