This paper contains important results about statistical properties of billiards in which the location and shapes of the scatterers may change between collisions. The configuration space is the flat torus \({\mathbb T}^2\) and the scatters are given by an isometric embedding of a finite set of pairwise disjoint closed convex domains \(B_1,B_2,\dots,B_N\) in \({\mathbb R}^2\) with \(C^3\) boundaries \(\Gamma_i, i=1,2,\dots, N\) of strictly positive curvatures. The singularities of the system occur as a moving particle hits a scatterer tangentially. A “configuration” \({\mathcal K}\) is an isometric embedding of a finite union of disjoint scatterers into \({\mathbb T}^2\) and, as usual, the phase space is \(\mathcal M=\cup_{i}{\mathcal M}_{i} \), with \({\mathcal M}_{i}=\Gamma_{i} \times [-\frac{\pi}{2},\frac{\pi}{2}] \).
In order to describe the dynamics, the authors consider disjoint neighborhoods of the obstacles (“buffer zones”) so that as the particle leaves a source obstacle and exits its buffer zone, the target scatterer changes to another nearby obstacle before the particle enters the target buffer zone. Therefore the dynamics is defined in terms of a composition of the elements of sequences of maps \(F_n=F_{{\mathcal K}_{n}, {\mathcal K}_{n-1}}\) associated to sequences of configurations \(({\mathcal K}_n)\) as an iterated systems of transformations \({\mathcal F}_n=F_n \circ F_{n-1} \circ \dots\circ F_1\). Of course it is required to describe conditions under which the billiard maps are defined for all sufficiently close configurations.
The main result proven is the exponential memory loss of initial data for (slowly) moving scatterers.
The exponential decay of correlations for billiards with fixed convex scatterers was proven by L.-S. Young [Ann. Math. (2) 147, No. 3, 585–650 (1998; Zbl 0945.37009)] in the more general context of hyperbolic maps with singularities. This reference is crucial for understanding the strategies of the proof (for instance, the coupling argument) as well as the difficulties one has to overcome in order to extend these arguments to the time dependent case. For that, Section 5 of the paper contains a useful synopsis of the proof. The exposition given on Chapter 7 of the book by N. Chernov and R. Markarian [Chaotic Billiard. Providence, RI: American Mathematical Society (AMS) (2006; doi:10.1090/surv/127)] is also very useful.