On the inverse limit stability of endomorphisms

الأصل · EN

We present several results suggesting that the concept of C¹-inverse limit stability is free of singularity theory. We describe an example of a C¹-inverse stable endomorphism which is robustly transitive with persistent critical set. We show that every (weak) axiom A, C¹-inverse limit stable endomorphism satisfies a certain strong transversality condition (T). We prove that every attractor-repellor endomorphism satisfying axiom A and Condition (T) is C¹-inverse limit stable. The latter is applied to Hénon maps, rational functions of the sphere and others. This leads us to conjecture that C¹-inverse stable endomorphisms are those which satisfy axiom A and the strong transversality condition (T).