A Horseshoe with Positive Measure.
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Rufus Bowen (1975)
Inventiones mathematicae
Anna Cima, Armengol Gasull, Francesc Mañosas (2001)
Annales Polonici Mathematici
In LaSalle's book "The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on ℝ is a global attractor, and proposes to study the natural extensions of these conditions in ℝⁿ. Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor...
Guido Gentile (1995)
Forum mathematicum
Ricardo Mañé (1987)
Publications Mathématiques de l'IHÉS
Carlos Gutierrez (1995)
Annales de l'I.H.P. Analyse non linéaire
Janina Kotus, Fopke Klok (1988)
Compositio Mathematica
И.У. Бронштейн, В.П. Бурдаев (1982)
Matematiceskie issledovanija
Feliks Przytycki (1976)
Studia Mathematica
A. Ambrosetti, D. Arcoya, J. L. Gámez (1998)
Rendiconti del Seminario Matematico della Università di Padova
Wu, Jinn-Wen (2003)
Applied Mathematics E-Notes [electronic only]
Patrick Bonckaert, Vincent Naudot (2001)
Annales de la Faculté des sciences de Toulouse : Mathématiques
Giovanni Leoni (1996)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Katarzyna Horbacz (1989)
Annales Polonici Mathematici
Charles Pugh, Michael Shub (1975)
Inventiones mathematicae
M. J. Dias Carneiro, Jacob Palis (1989)
Publications Mathématiques de l'IHÉS
Sheldon E. Newhouse, Jacob Palis, Floris Takens (1983)
Publications Mathématiques de l'IHÉS
I.P. Malta, M.J. Pacifico (1983)
Inventiones mathematicae
J. Iglesias, A. Portela, A. Rovella (2012)
Fundamenta Mathematicae
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.
J. Iglesias, A. Portela, A. Rovella (2010)
Fundamenta Mathematicae
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.
Kazuhiro Sakai (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
The notion of C¹-stably positively expansive differentiable maps on closed manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.
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