On 𝐶²-stable effects of intermingled basins of attractors in classes of boundary-preserving maps

Kleptsyn, V.; Saltykov, P.

doi:10.1090/S0077-1554-2012-00196-4

On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps
HTML articles powered by AMS MathViewer

by V. A. Kleptsyn and P. S. Saltykov
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2011, 193-217

DOI: https://doi.org/10.1090/S0077-1554-2012-00196-4

Published electronically: January 12, 2012

PDF | Request permission

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.

References

P. Bleher, M. Lyubich, and R. Roeder, Lee–Yang zeros for DHL and $2$D rational dynamics, I. Foliation of the physical cylinder, preprint, arXiv:1009.4691 [math.DS], 2010.
Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
Araceli Bonifant and John Milnor, Schwarzian derivatives and cylinder maps, Holomorphic dynamics and renormalization, Fields Inst. Commun., vol. 53, Amer. Math. Soc., Providence, RI, 2008, pp. 1–21. MR 2477416
Bertrand Deroin and Victor Kleptsyn, Random conformal dynamical systems, Geom. Funct. Anal. 17 (2007), no. 4, 1043–1105. MR 2373011, DOI 10.1007/s00039-007-0606-y
A. Gorodetski and Yu. Ilyashenko, Minimal and strange attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 6, 1177–1183. Nonlinear dynamics, bifurcations and chaotic behavior. MR 1409419, DOI 10.1142/S0218127496000679
A. S. Gorodetskiĭ and Yu. S. Il′yashenko, Some new robust properties of invariant sets and attractors of dynamical systems, Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 16–30, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 33 (1999), no. 2, 95–105. MR 1719330, DOI 10.1007/BF02465190
A. S. Gorodetskiĭ and Yu. S. Il′yashenko, Some properties of skew products over a horseshoe and a solenoid, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 96–118 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(231) (2000), 90–112. MR 1841753
A. S. Gorodetski, Minimal attractors and partially hyperbolic invariant sets of dynamical systems, Cand. Sci. (Phys.–Math.) Dissertation, Faculty of Mechanics and Mathematics, Moscow State University, 2001.
A. S. Gorodetskiĭ, Regularity of central leaves of partially hyperbolic sets and applications, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 6, 19–44 (Russian, with Russian summary); English transl., Izv. Math. 70 (2006), no. 6, 1093–1116. MR 2285025, DOI 10.1070/IM2006v070n06ABEH002340
A. S. Gorodetskiĭ, Yu. S. Il′yashenko, V. A. Kleptsyn, and M. B. Nal′skiĭ, Nonremovability of zero Lyapunov exponents, Funktsional. Anal. i Prilozhen. 39 (2005), no. 1, 27–38, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 1, 21–30. MR 2132437, DOI 10.1007/s10688-005-0014-8
B. M. Gurevich and A. A. Tempel′man, Hausdorff dimension of the set of generic points for Gibbs measures, Funktsional. Anal. i Prilozhen. 36 (2002), no. 3, 68–71 (Russian); English transl., Funct. Anal. Appl. 36 (2002), no. 3, 225–227. MR 1935906, DOI 10.1023/A:1020106306834
Yu. S. Il′yashenko, Diffeomorphisms with intermingled attracting basins, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 60–71, 112 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 42 (2008), no. 4, 298–307. MR 2492427, DOI 10.1007/s10688-008-0043-1
Yuli S. Ilyashenko, Victor A. Kleptsyn, and Petr Saltykov, Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, J. Fixed Point Theory Appl. 3 (2008), no. 2, 449–463. MR 2434457, DOI 10.1007/s11784-008-0088-z
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
Yu. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, preprint, arXiv:1005.0173 [math.DS], 2010.
Ittai Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc. (N.S.) 31 (1994), no. 1, 68–74. MR 1254075, DOI 10.1090/S0273-0979-1994-00507-5
V. Kleptsyn and D. Ryzhov, Special ergodic theorem, a note in preparation.
John Milnor, Fubini foiled: Katok’s paradoxical example in measure theory, Math. Intelligencer 19 (1997), no. 2, 30–32. MR 1457445, DOI 10.1007/BF03024428
A. V. Osipov, Nondensity of the orbital shadowing property in $C^1$-topology, Algebra i Analiz 22 (2010), no. 2, 127–163 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 2, 267–292. MR 2668126, DOI 10.1090/S1061-0022-2011-01140-9
Jacob Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque 261 (2000), xiii–xiv, 335–347 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755446
J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 4, 485–507 (English, with English and French summaries). MR 2145722, DOI 10.1016/j.anihpc.2005.01.001
David Ruelle, Historical behaviour in smooth dynamical systems, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 63–66. MR 1858471
P. S. Saltykov, A special ergodic theorem for Anosov diffeomorphisms on the 2-torus, Funktsional. Anal. i Prilozhen. 45 (2011), no. 1, 69–78 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 45 (2011), no. 1, 56–63. MR 2848741, DOI 10.1007/s10688-011-0006-9
Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057, DOI 10.1007/s002229900035
Floris Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity 21 (2008), no. 3, T33–T36. MR 2396607, DOI 10.1088/0951-7715/21/3/T02
S. R. S. Varadhan, Large deviations, Ann. Probab. 36 (2008), no. 2, 397–419. MR 2393987, DOI 10.1214/07-AOP348
Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525–543. MR 975689, DOI 10.1090/S0002-9947-1990-0975689-7
Lai-Sang Young, Entropy in dynamical systems, Entropy, Princeton Ser. Appl. Math., Princeton Univ. Press, Princeton, NJ, 2003, pp. 313–327. MR 2035829