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C¹ stability of endomorphisms on two-dimensional manifolds

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Iglesias J., Portela A., Rovella A.
Fundamenta Mathematicae
|
2012
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tom 219
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nr 1
37-58
EN
A set of necessary conditions for C¹ stability of noninvertible maps is presented. It is proved that the conditions are sufficient for C¹ stability in compact oriented manifolds of dimension two. An example given by F. Przytycki in 1977 is shown to satisfy these conditions. It is the first example known of a C¹ stable map (noninvertible and nonexpanding) in a manifold of dimension two, while a wide class of examples are already known in every other dimension.
2
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Stability modulo singular sets

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Iglesias J., Portela A., Rovella A.
Fundamenta Mathematicae
|
2009
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tom 204
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nr 2
155-175
EN
A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.
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C¹ stable maps: examples without saddles

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Iglesias J., Portela A., Rovella A.
Fundamenta Mathematicae
|
2010
|
tom 208
|
nr 1
23-33
EN
We give here the first examples of C¹ structurally stable maps on manifolds of dimension greater than two that are neither diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C¹ stable. A necessary condition for the existence of such examples is also given.
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