Contractive local IFSs are combinatorially stable under concordant shadowing and topologically stable under the open set condition on manifolds of dimension at least three, but can be combinatorially unstable as shown by beta-transformation constructions.
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Stability Theory for Local Iterated Function Systems
Contractive local IFSs are combinatorially stable under concordant shadowing and topologically stable under the open set condition on manifolds of dimension at least three, but can be combinatorially unstable as shown by beta-transformation constructions.