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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

On $ C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps


Authors: V. A. Kleptsyn and P. S. Saltykov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2011, 193-217
MSC (2010): Primary 37C70; Secondary 37D25
Published electronically: January 12, 2012
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Abstract: In the spaces of boundary-preserving maps of an annulus and a thickened torus, we construct open sets in which every map has intermingled basins of attraction, as predicted by I. Kan.

Namely, the attraction basins of each of the boundary components are everywhere dense in the phase space for such maps. Moreover, the Hausdorff dimension of the set of points that are not attracted by either of the components proves to be less than the dimension of the phase space itself, which strengthens the result following from the argument due to Bonatti, Diaz, and Viana.


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Additional Information

V. A. Kleptsyn
Affiliation: CNRS, Institut de Recherche Mathématique de Rennes (UMR 6625), France
Email: victor.kleptsyn@univ-rennes1.fr

P. S. Saltykov
Affiliation: Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russian Federation
Email: p_saltykov@pochta.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-2012-00196-4
Keywords: Dynamical system, attractor, stability, partially hyperbolic skew product, Hölder rectifying map
Published electronically: January 12, 2012
Additional Notes: Supported in part by RFBR grant no. 10-01-00739-a and joint RFBR–CNRS grant no. 10-01-93115-NTsNI_a.
Article copyright: © Copyright 2012 American Mathematical Society