For a certain diagonal flow on SL(d, Z)\ SL(d, R) where
d ≥ 2, we show that any bounded subset (with measure
zero boundary) of the horosphere or a translated horosphere
equidistributes, under a suitable normalization, on a target
shrinking into the cusp. This type of equidistribution is
shrinking target horospherical equidistribution ...
For a certain diagonal flow on SL(d, Z)\ SL(d, R) where
d ≥ 2, we show that any bounded subset (with measure
zero boundary) of the horosphere or a translated horosphere
equidistributes, under a suitable normalization, on a target
shrinking into the cusp. This type of equidistribution is
shrinking target horospherical equidistribution (STHE), and
we show STHE for several types of shrinking targets. Our
STHE results extend known results for d=2 and L\ PSL(2, R)
where L is any cofinite Fuchsian group with at least one cusp.
The two key tools needed to prove our STHE results for the
horosphere are a renormalization technique and Marklof’s
result on the equidistribution of the Farey sequence on
distinguished sections. For our STHE results for translated
horospheres, we introduce translated Farey sequences, develop
some of their geometric and dynamical properties, generalize
Marklof’s result by proving the equidistribution of translated
Farey sequences for the same distinguished sections, and use
this equidistribution of translated Farey sequences along with
the renormalization technique to prove our STHE results for
translated horospheres.
