In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that :


$a(p)=\lim\_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at each point $p$ in manifolds of negative curvature which it is the main result implies purely exponential growth of volume of geodesic spheres.Really I'am curiouse to know if this limit does exist for Manifolds with positive curvature ,we may look to an example for application this in unit tangent bundle of $S^4$ and probably cohomology $CP^3$ which they admit Riemannian metrics with positive sectional curvature almost everywhere ? and what about continuity of $a(p)$ in this case ?



Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?
When $M$ has positive curvature, the limit should always be 0. Indeed if $M$ has non-negative Ricci curvature, Bishop-Gromov tells you that $\frac{Vol S(p, r)}{n\omega\_nr^{n-1}}\leq 1$, so the volumes grow at most polynomially. 


