Unfolding contracting singular cycles. (English) Zbl 0802.58036

The authors’ abstract: “We show that if we start with a Morse-Smale vector field and move through a generic one-parameter family of vector fields to a contracting singular cycle and beyond, we reach a region filled up mostly with hyperbolic flows: the Lebesgue measure of parameter values corresponding to non-Axiom \(A\) flows is zero”.


37D15 Morse-Smale systems
37D99 Dynamical systems with hyperbolic behavior
Full Text: DOI Numdam EuDML


[1] R. BAMÓN , R. LABARCA , R. MAÑ;É and M. J. PACIFICO , The Explosion of Singular Cycles (to appear at Publ. Math. IHES). Numdam | Zbl 0801.58010 · Zbl 0801.58010
[2] M. HIRSCH , C. PUGH and M. SHUB , Invariant Manifolds (Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, 1977 ). MR 58 #18595 | Zbl 0355.58009 · Zbl 0355.58009
[3] R. LABARCA and M. J. PACIFICO , Stability of Singular Horseshoes (Topology, Vol. 25, n^\circ 3, 1986 ). MR 87h:58106 | Zbl 0611.58033 · Zbl 0611.58033
[4] A. ROVELLA , The Dynamics of Perturbations of the Contracting Lorenz Attractor (Thesis, IMPA and to appear at Bol. Soc. Bras. Mat.). Zbl 0797.58051 · Zbl 0797.58051
[5] S. SINGER , Stable Orbits and Bifurcations of Maps of the Interval (SIAM, J. Appl. Math., Vol. 35, 1978 ). MR 58 #13206 | Zbl 0391.58014 · Zbl 0391.58014
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