A global perspective for non-conservative dynamics
Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 4, p. 485-507
@article{AIHPC_2005__22_4_485_0,
     author = {Palis, Jacob},
     title = {A global perspective for non-conservative dynamics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {22},
     number = {4},
     year = {2005},
     pages = {485-507},
     doi = {10.1016/j.anihpc.2005.01.001},
     zbl = {02191851},
     mrnumber = {2145722},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2005__22_4_485_0}
}
Palis, J. A global perspective for non-conservative dynamics. Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 4, pp. 485-507. doi : 10.1016/j.anihpc.2005.01.001. http://www.numdam.org/item/AIHPC_2005__22_4_485_0/

[1] Abraham R., Smale S., Nongenericity of Ω-stability, in: Global Analysis, Berkeley 1968, Proc. Sympos. Pure Math., vol. XIV, Amer. Math. Soc., 1970. | Zbl 0215.25102

[2] Afraimovich V.S., Bykov V.V., Shil'Nikov L.P., On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR 234 (1977) 336-339. | MR 462175 | Zbl 0451.76052

[3] Afraimovich V.S., Shil'Nikov L.P., The accessible transitions from Morse-Smale systems to systems with several periodic motions, Izv. Akad. Nauk SSSR 38 (1974) 1248-1288. | MR 423422 | Zbl 0322.58007

[4] Afraimovich V.S., Shil'Nikov L.P., On bifurcations of codimension 1, leading to the appearance of fixed points of a countable set of tori, Dokl. Akad. Nauk SSSR 262 (1982) 777-780. | MR 644867 | Zbl 0509.58033

[5] Alves J.F., Bonatti C., Viana M., SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000) 351-398. | MR 1757000 | Zbl 0962.37012

[6] Andronov A., Leontovich E.A., Some cases of dependence of limit cycles on parameters, Uchen. Zap. Gor'kov. Univ. 6 (1939) 3-24.

[7] Andronov A., Leontovich E.A., Gordon I.I., Majer A.G., Qualitative Theory of Dynamical Systems of Second Order, Nauka, Moscow, 1966, p. 569. | MR 199506

[8] Andronov A., Leontovich E.A., Gordon I.I., Majer A.G., The Theory of Bifurcations of Dynamical Systems on the Plane, Nauka, Moscow, 1967, p. 487. | MR 235228 | Zbl 0257.34001

[9] Andronov A., Pontryagin L., Systèmes grossiers, Dokl. Akad. Nauk USSR 14 (1937) 247-251. | Zbl 0016.11301

[10] Anosov D.V., Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov Math. Inst. 90 (1967) 1-235. | MR 224110 | Zbl 0176.19101

[11] Arnéodo A., Coullet P., Tresser C., Possible new strange attractors with spiral structure, Comm. Math. Phys. 79 (1981) 673-679. | MR 623967 | Zbl 0485.58013

[12] Arnold V.I., Small denominators I: On the mapping of a circle to itself, Izv. Akad. Nauk Math. Ser. 25 (1961) 21-86. | MR 140699 | Zbl 0152.41905

[13] Arnold V.I., Lectures on bifurcations and versal families, Uspekhi Math. Nauk 27 (1972) 119-184. | MR 413191 | Zbl 0264.58005

[14] Arnold V.I., Kolmogorov's hydrodynamics attractors. turbulence and stochastic processes: Kolmogorov's ideas 50 years on, Proc. Roy. Soc. London Ser. A 434 (1991) 19-22. | MR 1124924 | Zbl 0726.76045

[15] Arnold V.I., Afraimovich V.S., Ilyasheko Yu., Shilnikov L.P., Bifurcation Theory and Catastrophe Theory, Springer, 1999. | MR 1733750 | Zbl 1038.37500

[16] Arroyo A., Rodriguez Hertz F., Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. H. Poincaré 20 (2003) 805-841. | Numdam | MR 1995503 | Zbl 1045.37006

[17] Avila A., Lyubich M., De Melo W., Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math. 154 (2003) 451-550. | MR 2018784 | Zbl 1050.37018

[18] A. Avila, C.G. Moreira, Phase-parameter relation sharp statistical properties of unimodal maps, Contemp. Math., in press.

[19] A. Avila, C.G. Moreira, Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations, Publ. Math. IHES, in press. | Numdam | MR 2217046 | Zbl 1078.37030

[20] A. Avila, C.G. Moreira, Statistical properties of unimodal maps: the quadratic family, Ann. of Math., in press. | MR 2153401 | Zbl 1078.37029

[21] Avila A., Moreira C.G., Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative, Astérisque 286 (2003) 81-118. | MR 2052298 | Zbl 1046.37021

[22] Bamón R., Labarca R., Mañé R., Pacifico M.J., The explosion of singular cycles, Publ. Math. IHES 78 (1993) 207-232. | Numdam | MR 1259432 | Zbl 0801.58010

[23] Benedicks M., Carleson L., The dynamics of the Hénon map, Ann. of Math. 133 (1991) 73-169. | MR 1087346 | Zbl 0724.58042

[24] M. Benedicks, M. Viana, Random perturbations and statistical properties of Hénon-like maps, Ann. Inst. H. Poincaré Anal. Non Linéaire. | Numdam | MR 2259614 | Zbl 1131.37033 | Zbl 05072659

[25] Benedicks M., Viana M., Solution of the basin problem for Hénon-like attractors, Invent. Math. 143 (2001) 375-434. | MR 1835392 | Zbl 0967.37023

[26] Benedicks M., Young L.-S., SBR-measures for certain Hénon maps, Invent. Math. 112 (1993) 541-576. | MR 1218323 | Zbl 0796.58025

[27] Birkhoff G.D., Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi. Lyncaei 1 (1935) 85-216. | JFM 60.1340.02 | Zbl 0016.23401

[28] Bonatti C., Diaz L., Viana M., Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., vol. 102, Springer, 2004. | MR 2105774 | Zbl 1060.37020

[29] Bonatti C., Díaz L.J., Pujals E., A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003) 355-418. | MR 2018925 | Zbl 1049.37011

[30] Bonatti C., Díaz L.J., Pujals E., Rocha J., Robust transitivity and heterodimensional cycles, Astérisque 286 (2003) 187-222. | MR 2052302 | Zbl 1056.37024

[31] Bonatti C., Pumariño A., Viana M., Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris, Sér. I Math. 325 (1997) 883-888. | MR 1485910 | Zbl 0896.58043

[32] Bowen R., Ruelle D., The ergodic theory of Axiom A flows, Invent. Math. 29 (1975) 181-202. | MR 380889 | Zbl 0311.58010

[33] Bunimovich L.A., Sinai Ya.G., Stochasticity of the attractor in the Lorenz model, in: Nonlinear Waves, Proc. Winter School, Nauka, Moscow, 1980, pp. 212-226.

[34] Cartwright M., Littlewood J., On non-linear differential equations of the second order, J. London Math. Soc. 20 (1945) 127-153. | MR 16789 | Zbl 0061.18903

[35] Cartwright M., Littlewood J., On non-linear differential equations of the second order, Ann. of Math. 48 (1947) 472-494. | MR 21190 | Zbl 0029.12602

[36] Colli E., Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 539-579. | Numdam | MR 1643393 | Zbl 0932.37015

[37] Coullet P., Tresser C., Itérations d'endomorphims et groupe de renormalization, C. R. Acad. Sci. Paris, Sér. I 287 (1978) 577-580. | MR 512110 | Zbl 0402.54046

[38] Dias Carneiro M.J., Palis J., Bifurcations and global stability of families of gradients, Publ. Math. IHES 70 (1989) 70-163. | Numdam | MR 1067381 | Zbl 0706.58042

[39] De Melo W., Structural stability of diffeomorphisms on two-manifolds, Invent. Math. 21 (1973) 233-246. | MR 339277 | Zbl 0291.58011

[40] Van Der Pol B., On relaxation oscillations, Philos. Mag. Ser. 7 2 (1926) 978-992. | JFM 52.0450.05

[41] Díaz L.J., Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory Dynam. Systems 15 (1995) 291-315. | MR 1332405 | Zbl 0831.58035

[42] Díaz L.J., Pujals E., Ures R., Partial hyperbolicity and robust transitivity, Acta Math. 183 (1999) 1-43. | MR 1719547 | Zbl 0987.37020

[43] Díaz L.J., Rocha J., Large measure of hyperbolic dynamics when unfolding heterodimensional cycles, Nonlinearity 10 (1997) 857-884. | MR 1457749 | Zbl 0908.58042

[44] Dolgopyat D., On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004) 359-449. | MR 2031432 | Zbl 1059.37021

[45] Feigenbaum M., Qualitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978) 25-52. | MR 501179 | Zbl 0509.58037

[46] Franks J., Williams R., Anomalous Anosov Flows. Global Theory of Dynamical Systems, Lecture Notes in Math., vol. 819, Springer, 1980. | MR 591182 | Zbl 0463.58021

[47] Gonchenko S.V., Shil'Nikov L.P., Turaev D.V., Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos 6 (1996) 15-31. | MR 1376892 | Zbl 1055.37578

[48] A. Gorodetski, V. Kaloshin, How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency, in press. | MR 2304335 | Zbl 1110.37034

[49] Graczyk J., Swiatek G., Generic hyperbolicity in the logistic family, Ann. of Math. 146 (1997) 1-52. | MR 1469316 | Zbl 0936.37015

[50] Guckenheimer J., Williams R.F., Structural stability of Lorenz attractors, Publ. Math. IHES 50 (1979) 59-72. | Numdam | MR 556582 | Zbl 0436.58018

[51] Hayashi S., Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows, Ann. of Math. 145 (1997) 81-137. | MR 1432037 | Zbl 0871.58067

[52] Hénon M., A two dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976) 69-77. | MR 422932 | Zbl 0576.58018

[53] Ilyashenko Yu., Li W., Nonlocal Bifurcations, Math. Surveys and Monographs, vol. 66, Amer. Math. Soc., 1999. | MR 1650842 | Zbl 1120.37308 | Zbl 01232921

[54] Jakobson M., On smooth mappings of the circle into itself, Math. USSR-Sb. 14 (1971) 161-185. | MR 290406 | Zbl 0241.58006

[55] Jakobson M., Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981) 39-88. | MR 630331 | Zbl 0497.58017

[56] V.Yu. Kaloshin, B.R. Hunt, Stretched exponential estimate on growth of the number of periodic points for prevalent diffeomorphisms, in press. | MR 2276768 | Zbl 1132.37011

[57] Kifer Yu., Ergodic Theory of Random Perturbations, Birkhäuser, 1986. | MR 884892

[58] Kifer Yu., Random Perturbations of Dynamical Systems, Birkhäuser, 1988. | MR 1015933 | Zbl 0659.58003

[59] Kozlovski O., Getting rid of the negative Schwarzian derivative condition, Ann. of Math. 152 (2000) 743-762. | MR 1815700 | Zbl 0988.37044

[60] Kozlovski O., Axiom A maps are dense in the space of unimodal maps in the C k topology, Ann. of Math. 157 (2003) 1-43. | MR 1954263 | Zbl 1215.37022 | Zbl 01933154

[61] O. Kozlovski, W. Shen, S. van Strien, Density of hyperbolicity in dimension one, Preprint, Warwick, 2004. | MR 2342693 | Zbl 1138.37013

[62] O. Kozlovski, S. van Strien, W. Shen, Rigidity for real polynomials, Preprint, Warwick, 2003. | MR 2335796 | Zbl 1129.37020

[63] Labarca R., Pacifico M.J., Stability of singular horseshoes, Topology 25 (1986) 337-352. | MR 842429 | Zbl 0611.58033

[64] Levi M., Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc. 32 (244) (1981). | MR 617687 | Zbl 0448.34032

[65] Levinson N., A second order differential equations with singular solutions, Ann. of Math. 50 (1949) 127-153. | MR 30079 | Zbl 0041.42311 | Zbl 0045.36501

[66] Liao S.-T., Hyperbolicity properties of the non-wandering sets of certain 3-dimensional systems, Acta Math. Sci. 3 (1983) 361-368. | MR 812542 | Zbl 0527.58031

[67] Liao S.T., On the stability conjecture, Chinese Ann. of Math. 1 (1980) 9-30. | MR 591229 | Zbl 0449.58013

[68] Littlewood J., On non-linear differential equations of the second order, III, Acta Math. 97 (1957) 267-308. | MR 88615 | Zbl 0081.08401

[69] Littlewood J., On non-linear differential equations of the second order, IV, Acta Math. 98 (1957) 1-110. | MR 90732 | Zbl 0081.08401

[70] Lorenz E.N., Deterministic nonperiodic flow, J. Atmosph. Sci. 20 (1963) 130-141.

[71] Lyubich M., Almost every real quadratic map is either regular or stochastic, Ann. of Math. 156 (2002) 1-78. | MR 1935840 | Zbl 01850538

[72] Mañé R., An ergodic closing lemma, Ann. of Math. 116 (1982) 503-540. | MR 678479 | Zbl 0511.58029

[73] Mañé R., A proof of the C 1 stability conjecture, Publ. Math. IHES 66 (1988) 161-210. | Numdam | MR 932138 | Zbl 0678.58022

[74] Martens M., Nowicki T., Invariant measures for Lebesgue typical quadratic maps. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque 261 (2001) 239-252. | MR 1755443 | Zbl 0939.37020

[75] May R.M., Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459-467.

[76] Mcmullen C., Complex Dynamics and Renormalization, Ann. of Math. Stud., vol. 142, Princeton University Press, 1994. | MR 1312365 | Zbl 0822.30002

[77] Metzger R., Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 247-276. | Numdam | MR 1753089 | Zbl 0983.37022

[78] Metzger R., Stochastic stability for contracting Lorenz maps and flows, Comm. Math. Phys. 212 (2000) 277-296. | MR 1772247 | Zbl 1052.37018

[79] Mora L., Viana M., Abundance of strange attractors, Acta Math. 171 (1993) 1-71. | MR 1237897 | Zbl 0815.58016

[80] Morales C., Pacifico M.J., A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems 23 (2003) 1575-1600. | MR 2018613 | Zbl 1040.37014

[81] Morales C., Pacifico M.J., Pujals E., Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999) 3393-3401. | MR 1610761 | Zbl 0924.58068

[82] Morales C., Pacifico M.J., Pujals E., Robust transitive singular sets for 3-flows are partially hyperbolic attractors and repellers, Ann. of Math. 160 (2004) 1-58. | MR 2123928 | Zbl 1071.37022

[83] Moreira C.G., Palis J., Viana M., Homoclinic tangencies and fractal invariants in arbitrary dimension, C. R. Acad. Sci. Paris, Sér. I Math. 333 (2001) 475-480. | MR 1859240 | Zbl 01686435

[84] Moreira C.G., Yoccoz J.-C., Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math. 154 (2001) 45-96. | MR 1847588 | Zbl 01657116

[85] Newhouse S., Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9-18. | MR 339291 | Zbl 0275.58016

[86] Newhouse S., The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. IHES 50 (1979) 101-151. | Numdam | MR 556584 | Zbl 0445.58022

[87] Newhouse S., Palis J., Cycles and bifurcation theory, Astérisque 31 (1976) 44-140. | MR 516408 | Zbl 0322.58009

[88] Newhouse S., Palis J., Takens F., Bifurcations and stability of families of diffeomorphisms, Publ. Math. IHES 57 (1983) 5-71. | Numdam | MR 699057 | Zbl 0518.58031

[89] Palis J., On Morse-Smale dynamical systems, Topology 8 (1969) 385-405. | MR 246316 | Zbl 0189.23902

[90] Palis J., A global view of Dynamics and a conjecture on the denseness of finitude of attractors. Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque 261 (1995) 335-347. | MR 1755446 | Zbl 1044.37014

[91] Palis J., Smale S., Structural stability theorems, in: Global Analysis, Berkeley 1968, Proc. Sympos. Pure Math., vol. XIV, Amer. Math. Soc., 1970, pp. 223-232. | MR 267603 | Zbl 0214.50702

[92] Palis J., Takens F., Stability of parametrized families of gradient vector fields, Ann. of Math. 118 (1983) 383-421. | MR 727698 | Zbl 0533.58018

[93] Palis J., Takens F., Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms, Invent. Math. 82 (1985) 397-422. | MR 811543 | Zbl 0579.58005

[94] Palis J., Takens F., Hyperbolic and the creation of homoclinic orbits, Ann. of Math. 1987 (1987) 337-374. | MR 881272 | Zbl 0641.58029

[95] Palis J., Takens F., Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993. | MR 1237641 | Zbl 0790.58014

[96] Palis J., Viana M., High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. 140 (1994) 207-250. | MR 1289496 | Zbl 0817.58004

[97] Palis J., Yoccoz J.-C., Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math. 172 (1994) 91-136. | MR 1263999 | Zbl 0801.58035

[98] Palis J., Yoccoz J.-C., Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs, C. R. Acad. Sci. Paris, Sér. I Math. 333 (2001) 867-871. | MR 1873226 | Zbl 1015.37024

[99] Peixoto M., Structural stability on two-dimensional manifolds, Topology 1 (1962) 101-120. | MR 142859 | Zbl 0107.07103

[100] Pesin Ya., Sinai Ya., Gibbs measures for partially hyperbolic attractors, Ergodic Theory Dynam. Systems 2 (1982) 417-438. | MR 721733 | Zbl 0519.58035

[101] E. Pujals, Density of hyperbolicity and homoclinic bifurcation for 3d-diffeomorphisms in attracting regions, IMPA's preprint server, http://www.preprint.impa.br. | MR 2221742

[102] Pujals E., Sambarino M., Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000) 961-1023. | MR 1779562 | Zbl 0959.37040

[103] Robbin J., A structural stability theorem, Ann. of Math. 94 (1971) 447-493. | MR 287580 | Zbl 0224.58005

[104] Robinson C., Structural stability of vector fields, Ann. of Math. 99 (1974) 154-175. | MR 334283 | Zbl 0275.58012

[105] Robinson C., Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity 2 (1989) 495-518. | MR 1020439 | Zbl 0704.58031

[106] Rovella A., The dynamics of perturbations of the contracting Lorenz attractor, Bull. Braz. Math. Soc. 24 (1993) 233-259. | MR 1254985 | Zbl 0797.58051

[107] Ruelle D., A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976) 619-654. | MR 415683 | Zbl 0355.58010

[108] Rychlik M., Lorenz attractors through Shil'nikov-type bifurcation. Part 1, Ergodic Theory Dynam. Systems 10 (1990) 793-821. | MR 1091428 | Zbl 0715.58027

[109] Sannami A., The stability theorems for discrete dynamical systems on two-dimensional manifolds, Nagoya Math. J. 90 (1983) 1-55. | MR 702250 | Zbl 0515.58022

[110] Shil'Nikov L.P., On the generation of periodic motion from a trajectory doubly asymptotic to an equilibrium state of saddle type, Math. USSR-Sb. 6 (1968) 428-438. | Zbl 0188.15303

[111] Simon R., A 3-dimensional Abraham-Smale example, Proc. Amer. Math. Soc. 34 (1972) 629-630. | MR 295391 | Zbl 0259.58006

[112] Sinai Ya., Gibbs measures in ergodic theory, Russian Math. Surveys 27 (1972) 21-69. | MR 399421 | Zbl 0255.28016

[113] Smale S., Diffeomorphisms with many periodic points, in: Differential and Combinatorial Topology, Princeton University Press, 1965. | MR 182020 | Zbl 0142.41103

[114] Smale S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. | MR 228014 | Zbl 0202.55202

[115] Sullivan D., Bounds, quadratic differentials and renormalization conjectures, Ann. Math. Soc. Centennial Publ. 2 (1992) 417-466. | MR 1184622 | Zbl 0936.37016

[116] Tedeschini-Lalli L., Yorke J., How often do simple dynamical systems have infinitely many coexisting sinks?, Comm. Math. Phys. 106 (1986) 635-657. | MR 860314 | Zbl 0602.58036

[117] M. Tsujii, Physical measures for partially hyperbolic surface endomorphisms, Acta Math., in press. | MR 2231338 | Zbl 1105.37022

[118] Tucker W., A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math. 2 (2002) 53-117. | MR 1870856 | Zbl 1047.37012

[119] Ures R., Abundance of hyperbolicity in C 1 topology, Ann. Sci. École Norm. Sup. 28 (1995) 747-760. | Numdam | MR 1355140 | Zbl 0989.37013

[120] C. Vasquez, Statistical stability for diffeomorphisms with dominated splitting, in press. | Zbl 05144565

[121] Viana M., Strange attractors in higher dimensions, Bull. Braz. Math. Soc. 24 (1993) 13-62. | MR 1224299 | Zbl 0784.58044

[122] Viana M., Multidimensional nonhyperbolic attractors, Publ. Math. IHES 85 (1997) 63-96. | Numdam | MR 1471866 | Zbl 1037.37016

[123] Viana M., What's new on Lorenz strange attractors?, Math. Intelligencer 22 (2000) 6-19. | MR 1773551 | Zbl 1052.37026

[124] Wen L., Homoclinic tangencies and dominated splittings, Nonlinearity 15 (2002) 1445-1469. | MR 1925423 | Zbl 1011.37011

[125] Wen L., Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. 35 (2004) 419-452. | MR 2106314 | Zbl 1099.37024

[126] Young L.-S., Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems 6 (1986) 311-319. | MR 857204 | Zbl 0633.58023