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Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set ((MIS)) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types -- (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining an MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with polylogarithmic update time. This breaks the natural barrier of $Ω(Δ)$ update time (where $Δ$ is the maximum degree in the graph) for \emph{vertex updates} presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes with approximation factors between 2 and 4 and corresponding running-time trade-offs. We have implemented our algorithms and reported the results of an experimental comparison exploring the trade-off between solution quality and update time for synthetic and real-world map labeling instances.