Transactions of the Moscow Mathematical Society

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ISSN 1547-738X(e) ISSN 0077-1554(p)

On -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
Posted: 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.

References

[BLR] P. Bleher, M. Lyubich, and R. Roeder, Lee-Yang zeros for DHL and D rational dynamics, I. Foliation of the physical cylinder, preprint, arXiv:1009.4691 [math.DS], 2010.
[BDV05] C. Bonatti, L. J. Diaz, and M. Viana, Dynamics beyond uniform hyperbolicity, Springer, Berlin, 2005. MR 2105774 (2005g:37001)
[BM07] A. Bonifant and J. Milnor, Schwarzian derivatives and cylinder maps, preprint, 2007. MR 2477416 (2010f:37041)
[DK07] Bertrand Deroin and Victor Kleptsyn, Random conformal dynamical systems, Geom. Funct. Anal. 17 (2007), no. 4, 1043–1105. MR 2373011 (2010j:37012), http://dx.doi.org/10.1007/s00039-007-0606-y
[GI96] 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 (97e:58151), http://dx.doi.org/10.1142/S0218127496000679
[GI99] 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 (2001c:37031), http://dx.doi.org/10.1007/BF02465190
[GI00] A. S. Gorodetskiĭ and Yu. S. Ilyashenko, Some properties of skew products over a horseshoe and a solenoid, Tr. Mat. Inst. Steklova 231 (2000), 96-118; English transl., Proc. Steklov Inst. Math. 231 (2000), no. 4, 90-112. MR 1841753 (2002i:37040)
[G01] 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.
[G06] 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 (2007k:37033), http://dx.doi.org/10.1070/IM2006v070n06ABEH002340
[GIKN05] 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 (2005m:37062), http://dx.doi.org/10.1007/s10688-005-0014-8
[GT02] 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 (2003j:28026), http://dx.doi.org/10.1023/A:1020106306834
[I08] 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 (2010b:37045), http://dx.doi.org/10.1007/s10688-008-0043-1
[IKS08] 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 (2009h:37043), http://dx.doi.org/10.1007/s11784-008-0088-z
[HPS77] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lect. Notes in Math., vol. 583, Springer, Berlin-New York, 1977. MR 0501173 (58:18595)
[IN10] Yu. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, preprint, arXiv:1005.0173 [math.DS], 2010.
[K94] 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 (94k:58082), /bull/1994-31-01/S0273-0979-1994-00507-5/
[KR] V. Kleptsyn and D. Ryzhov, Special ergodic theorem, a note in preparation.
[Mi97] John Milnor, Fubini foiled: Katok’s paradoxical example in measure theory, Math. Intelligencer 19 (1997), no. 2, 30–32. MR 1457445 (98g:28005), http://dx.doi.org/10.1007/BF03024428
[Os10] A. V. Osipov, Nondensity of the orbital shadowing property in 𝐶¹-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 (2011f:37042), /spmj/2011-22-02/S1061-0022-2011-01140-9/
[Pa00] J. Palis, A global view of dynamics and a conjecture on the dynamics of finitude of attractors, Géometrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque 261 (2000), xiii-xiv; 335-347. MR 1755446 (2001d:37025)
[Pa05] J. Palis, A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 4, 485–507 (English, with English and French summaries). MR 2145722 (2006b:37037), http://dx.doi.org/10.1016/j.anihpc.2005.01.001
[Ru01] D. Ruelle, Historical behaviour in smooth dynamical systems, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 63-66. MR 1858471
[S10] P. S. Saltykov, A special ergodic theorem for Anosov diffeomorphisms on the -torus, Funktsional. Anal. i Prilozhen. 45 (2011), no. 1,69-78; English transl., Funct. Anal. Appl. 45 (2011), no. 1, 56-63. MR 2848741
[SW00] Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057 (2001c:37030), http://dx.doi.org/10.1007/s002229900035
[T08] Floris Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity 21 (2008), no. 3, T33–T36. MR 2396607 (2009a:37043), http://dx.doi.org/10.1088/0951-7715/21/3/T02
[Var08] S. R. S. Varadhan, Large deviations, Ann. Probab. 36 (2008), no. 2, 397–419. MR 2393987 (2009d:60070), http://dx.doi.org/10.1214/07-AOP348
[Y90] Lai-Sang Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 2, 525–543. MR 975689 (90g:58069), /tran/1990-318-02/S0002-9947-1990-0975689-7/
[Y03] Lai-Sang Young, Entropy in dynamical systems, Entropy, Princeton Ser. Appl. Math., Princeton Univ. Press, Princeton, NJ, 2003, pp. 313-327. MR 2035829 (2004k:37010)

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