Infinitesimally stable endomorphisms

Ikeda, Hiroshi

doi:10.1090/S0002-9947-1994-1250821-2

Infinitesimally stable endomorphisms
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Trans. Amer. Math. Soc. 344 (1994), 823-833 Request permission

Abstract:

It is well known that infinitesimal stability of diffeomorphisms is an open property. However, infinitesimal stability of endomorphisms is not an open property. So we consider the interior of the set of all infinitesimally stable endomorphisms. We prove that if f belongs to the interior of the set of all infinitesimally stable endomorphisms, then f is $\Omega$-stable. This means a generalization of Smale’s $\Omega$-stability theorem for diffeomorphisms. Moreover, it is proved that for Anosov endomorphisms structural stability is equivalent to lying in the interior of the set of infinitesimally stable endomorphisms.

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