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We extend the recently developed discrete geometric singular perturbationtheory to the non-normally hyperbolic regime. Our primary tool is the Takensembedding theorem, which provides a means of approximating the dynamics ofparticular maps with the time-1 map of a formal vector field. First, we showthat the so-called reduced map, which governs the slow dynamics near slowmanifolds in the normally hyperbolic regime, can be locally approximated by thetime-one map of the reduced vector field which appears in continuous-timegeometric singular perturbation theory. In the non-normally hyperbolic regime,we show that the dynamics of fast-slow maps with a unipotent linear part can belocally approximated by the time-1 map induced by a fast-slow vector field inthe same dimension, which has a nilpotent singularity of the correspondingtype. The latter result is used to describe (i) the local dynamics oftwo-dimensional fast-slow maps with non-normally singularities of regular fold,transcritical and pitchfork type, and (ii) dynamics on a (potentially highdimensional) local center manifold in $n$-dimensional fast-slow maps withregular contact or fold submanifolds of the critical manifold. In general, ourresults show that the dynamics near a large and important class ofsingularities in fast-slow maps can be described via the use of formalembedding theorems which allow for their approximation by the time-1 map of afast-slow vector field featuring a loss of normal hyperbolicity.

Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic Points