the EF correlators (cf. Theorem \[thm:DL.IID\]).

Strong dynamical localization in finite volumes {#ssec:SimpleDL.finite.L}
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Now we will derive uniform upper bounds on EF correlators in finite, but arbitrarily large balls from the MSA bounds, using a simplified version of the Germinet–Klein argument [@GK01]. Recall that originally the implication “MSA $\Rightarrow$ DL” has been proven by Germinet–De Bièvre [@GD98] and by Damanik–Stollmann [@DS01] (strong dynamical localization). Formally, Germinet and Klein [@GK01] considered differential operators in $\DR^d$, but an adaptation of their technique to lattice models is not difficult. Moreover, it becomes quite elementary when operators in finite balls are considered. Generally speaking, it suffices that finite-volume operators have compact resolvent; on the lattice, the operators $H_{\ball_L(u)}$ are even finite-dimensional and have a finite orthogonal eigenbasis. This allows to avoid an analysis of Hilbert–Schmidt norms of their spectral projections (inevitable in the entire lattice/Euclidean space) and to replace it with an elementary application of Bessel’s inequality.

Denote by $\csB_1(I)$ the set of all Borel functions $\phi:\DR\to\DC$ with ${\rm supp}\,\phi\subset I$ and $\|\phi\|_\infty \le 1$.

\[thm:DL.IID\] Fix an integer $L\in\DN^*$ and assume that the following bound holds for any pair of disjoint balls $\ball_L(x), \ball_L(y)$: $$\pr{ \exists\, E\in I:\, \text{ $\ball_L(x)$ and $\ball_L(y)$ 