I was reading this seminal paper
on logarithmic law of geodesics by Sullivan. I have trouble in understanding some of the details in the paper. It would be very helpful if someone could draw some light on these.
The section $6$ Disjoint spheres and mixing of geodesic flow he draws the picture in the universal cover. I am not getting how this statement follows ** The integral counts $e^{-dt}$ ( number of $\Gamma$ orbit balls approximately $t$ away from a fixed lift of $B$). So the orbit points is fixed width spherical shells is caught between two constant times $e^{dt}$.I am not sure how this $e^{dt}$ terms are coming in integral count of number of orbit points.
In the next page the term $e^{-d(x,\gamma x)}$ over one horosphere is commensurable to the largest term each being comparable to solid angle of the horosphere viewed from fixed lift of $B$. I am not sure how this is coming i.e why the angle is $e^{-d(x,\gamma x)}$
In section 9 the main result i.e theorem 6 he has taken dist(v(t))=max{1, distance from a fixed point in $V$ to the point achieved after traveling time $t$ along the geodesic} . I am just curious why we need to take maximum of 1 and distance from fixed pont, can't we work only with distance from fixed point to the point achieved after time $t$. I am not sure what is the need of $1$.
In theorem $6$ he also used volume of the part $V$ where $dist\geq T$ bounded above and below by constant times $e^{-dT}$. Does this anyway follows from question 1 understanding?
I am sorry for the long post. But it would be very helpful if I can get a better understanding of this paper. Any help is very much appreciated.
