  The classic technique of Baker [J. ACM '94] is the most fundamental approach
for designing approximation schemes on planar, or more generally
topologically-constrained graphs, and it has been applied in a myriad of
different variants and settings throughout the last 30 years. In this work we
propose a dynamic variant of Baker's technique, where instead of finding an
approximate solution in a given static graph, the task is to design a data
structure for maintaining an approximate solution in a fully dynamic graph,
that is, a graph that is changing over time by edge deletions and edge
insertions. Specifically, we address the two most basic problems -- Maximum
Weight Independent Set and Minimum Weight Dominating Set -- and we prove the
following: for a fully dynamic $n$-vertex planar graph $G$, one can:
  * maintain a $(1-\varepsilon)$-approximation of the maximum weight of an
independent set in $G$ with amortized update time $f(\varepsilon)\cdot
n^{o(1)}$; and,
  * under the additional assumption that the maximum degree of the graph is
bounded at all times by a constant, also maintain a
$(1+\varepsilon)$-approximation of the minimum weight of a dominating set in
$G$ with amortized update time $f(\varepsilon)\cdot n^{o(1)}$.
  In both cases, $f(\varepsilon)$ is doubly-exponential in
$\mathrm{poly}(1/\varepsilon)$ and the data structure can be initialized in
time $f(\varepsilon)\cdot n^{1+o(1)}$. All our results in fact hold in the
larger generality of any graph class that excludes a fixed apex-graph as a
minor.
