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Flows near compact invariant sets. Part I

Pedro Teixeira (2013)

Fundamenta Mathematicae

It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings...

Generic diffeomorphisms on compact surfaces

Flavio Abdenur, Christian Bonatti, Sylvain Crovisier, Lorenzo J. Díaz (2005)

Fundamenta Mathematicae

We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...

Global attractor of a differentiable autonomous system on the plane

Nguyen Van Chau (1995)

Annales Polonici Mathematici

We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.

Homeomorphisms of inverse limit spaces of one-dimensional maps

Marcy Barge, Beverly Diamond (1995)

Fundamenta Mathematicae

We present a new technique for showing that inverse limit spaces of certain one-dimensional Markov maps are not homeomorphic. In particular, the inverse limit spaces for the three maps from the tent family having periodic kneading sequence of length five are not homeomorphic.

Index filtrations and Morse decompositions for discrete dynamical systems

P. Bartłomiejczyk, Z. Dzedzej (1999)

Annales Polonici Mathematici

On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.

Inverse limit spaces of post-critically finite tent maps

Henk Bruin (2000)

Fundamenta Mathematicae

Let (I,T) be the inverse limit space of a post-critically finite tent map. Conditions are given under which these inverse limit spaces are pairwise nonhomeomorphic. This extends results of Barge & Diamond [2].

Jordan tori and polynomial endomorphisms in ℂ 2

Manfred Denker, Stefan Heinemann (1998)

Fundamenta Mathematicae

For a class of quadratic polynomial endomorphisms f : ℂ 2 → ℂ 2 close to the standard torus map ( x , y ) → ( x 2 , y 2 ) , we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).

Currently displaying 41 – 60 of 130