Searching for Targets of Nonuniform Size Using Mixing Transformations: Constructive Upper Bounds and Limit Laws

Abstract

We investigate searching for closed balls $B$ in the $n$-dimensional torus ${\mathbb {T}}^n$, where the ball’s center is located uniformly at random in ${\mathbb {T}}^n$ and its volume is uniformly distributed in $(\delta , V], 0<\delta \ll V \ll 1$. For exponentially mixing maps and multi-valued maps, it is shown that hitting time of a small ball and the expected hitting time are of ${\fancyscript{O}}(1/\hbox {vol}(B))$ and ${\fancyscript{O}}(-\ln \delta )$, respectively, for every ball center and almost every initial condition. Along the way, an ergodic theory framework is developed for ${\mathfrak {B}}$-regular, multi-valued maps. Finally, discrete-time maps are used to generate continuous-time dynamics on ${\mathbb {T}}^n$, and it is shown that the asymptotic behavior for the continuous case is the same as the discrete case.