Parameter exclusions in Hénon-like systems

M. Viana   ; S. Luzzatto

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 6, pp. 1053-1092

Résumé

This paper is devoted to the Hénon attractor and related systems. The Hénon attractor is observed when the diffeomorphism $H_{a,b}$ of the plane given by the rule $(x,y)\mapsto(1- ax^2+ by,x)$ is iterated. If the number $b$ is small, then this map is close to the one-dimensional map $x\mapsto1-ax^2$. The latter has a fold, which is also reflected by the properties of $H_{a,b}$ with $b\ne0$, namely, this map has some hyperbolicity, though the property is not uniform. Benedicks and Carleson found an approach to the study of such maps that gives fairly complete information about the attractor for “most” values of $a$. Subsequently Mora and Viana, together with many other authors, proved that similar attractors occur in much more general situations, and the theory of these systems has also been well developed. The present paper is an introduction to this area of investigation. While omitting detailed proofs, the authors have tried to single out the crucial points, referring the reader to the original sources for a thorough presentation.

Bibliographie

[1] D. V. Anosov, Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny, Tr. MIAN, 90, 1967 | MR

[2] D. V. Anosov, V. V. Solodov, “Giperbolicheskie mnozhestva”, Dinamicheskie sistemy – IX, Itogi nauki i tekhniki. Sovr. problemy matem. Fund. napr., 66, VINITI, M., 1991, 12–99 | MR | Zbl

[3] V. Baladi, Positive Transfer Operators and Decay of Correlations, Adv. Ser. Nonlinear Dynam., 16, World Sci. Publ., River Edge, NJ, 2000 | MR | Zbl

[4] L. Barreira, Ya. Resin, “Lectures on Lyapunov exponents and smooth ergodic theory”, Proc. Sympos. Pure Math., 69 (2001), 3–106, Appendix A by M. Brin and Appendix B by D. Dolgopyat, H. Hu and Ya. Pesin | MR | Zbl

[5] M. Benedicks, L. Carleson, “On iterations of $1-ax^2$ on $(-1,1)$”, Ann. of Math. (2), 122:1 (1985), 1–25 | DOI | MR | Zbl

[6] M. Benedicks, L. Carleson, “The dynamics of the Hénon map”, Ann. of Math. (2), 133:1 (1991), 73–169 | DOI | MR | Zbl

[7] M. Benedicks, L. Carleson, Parameter selection in the Hénon family, Preprint, Royal Institute of Technology (KTH), Stockholm, 2002 | MR

[8] M. Benedicks, M. Viana, “Solution of the basin problem for Hénon-like attractors”, Invent. Math., 143:2 (2001), 375–434 | DOI | MR | Zbl

[9] M. Benedicks, L. S. Young, “Sinaĭ–Bowen–Ruelle measures for certain Hénon maps”, Invent. Math., 112:3 (1993), 541–576 | DOI | MR | Zbl

[10] M. Benedicks, L. S. Young, “Markov extensions and decay of correlations for certain Hénon maps”, Astérisque, 261 (2000), 13–56 | MR | Zbl

[11] P. Collet, J.-P. Eckmann, “Positive Lyapunov exponents and absolute continuity for maps of the interval”, Ergodic Theory Dynam. Systems, 3:1 (1983), 13–46 | DOI | MR | Zbl

[12] M. J. Costa, “Saddle-node horseshoes giving rise to global Hénon-like attractors”, An. Acad. Brasil. Ciênc., 70:3 (1998), 393–400 | MR | Zbl

[13] L. J. Díaz, J. Rocha, M. Viana, “Strange attractors in saddle-nodescycles: prevalence and globality”, Invent. Math., 125 (1996), 37–74 | DOI | MR | Zbl

[14] M. Hénon, “A two-dimensional mapping with a strange attractor”, Comm. Math. Phys., 50:1 (1976), 69–77 | DOI | MR

[15] M. Holland, A new proof of the stable manifold theorem for hyperbolic fixed points on surfaces, math/0301235

[16] M. Holland, S. Luzzatto, “Dynamics of two-dimensional maps with criticalities and singularities” (to appear)

[17] A. J. Homburg, T. Young, “Intermittency in families of unimodal maps”, Ergodic Theory Dynam. Systems, 22:1 (2002), 203–225 | DOI | MR | Zbl

[18] M. V. Jakobson, “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps”, Comm. Math. Phys., 81:1 (1981), 39–88 | DOI | MR | Zbl

[19] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopaedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[20] E. D. Lorenz, “Deterministic nonperiodic flow”, J. Atmospheric Sci., 20 (1963), 130–141 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[21] S. Luzzatto, “Combinatorial structure of the parameter space for Lorenz-like and Hénon-like maps”, Workshop on Dynamical Systems, International Centre for Theoretical Physics, Trieste, 1998; http://www.ictp.trieste.it/www<nobr>$_-$</nobr>users/math/prog1069.html

[22] S. Luzzatto, “Bounded recurrence of critical points and Jakobson's theorem”, London Math. Soc. Lecture Note Ser., 274 (2000), 173–210 | MR | Zbl

[23] S. Luzzatto, W. Tucker, “Non-uniformly expanding dynamics in maps with singularities and criticalities”, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 179–226 | DOI | MR | Zbl

[24] S. Luzzatto, M. Viana, “Positive Lyapunov exponents for Lorenz-like families with criticalities”, Astérisque, 261 (2000), 201–237 | MR | Zbl

[25] S. Luzzatto, M. Viana, Lorenz-like attractors without continuous invariant foliations, Preprint, 2003 | MR

[26] R. Mañé, “Hyperbolicity, sinks and measure in one-dimensional dynamics”, Comm. Math. Phys., 100:4 (1985), 495–524 ; “Erratum”, Comm. Math. Phys., 112:4 (1987), 721–724 | DOI | MR | Zbl | DOI | MR | Zbl

[27] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987 | MR

[28] W. de Melo, S. van Strien, “One-dimensional dynamics: the Schwarzian derivative and beyond”, Bull. Amer. Math. Soc. (N.S.), 18:2 (1988), 159–162 | DOI | MR | Zbl

[29] L. Mora, M. Viana, “Abundance of strange attractors”, Acta Math., 171:1 (1993), 1–71 | DOI | MR | Zbl

[30] T. Nowicki, S. van Strien, “Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet–Eckmann conditions”, Invent. Math., 93:3 (1988), 619–635 | DOI | MR | Zbl

[31] M. J. Pacifico, A. Rovella, M. Viana, “Infinite-modal maps with global chaotic behavior”, Ann. of Math. (2), 148:2 (1998), 441–484 ; “Corrigendum”, Ann. of Math. (2), 149:2 (1999), 705 | DOI | MR | DOI | MR | Zbl

[32] J. Palis, J.-C. Yokkoz, “Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs”, C. R. Acad. Sci. Paris Sér. I Math., 333:9 (2001), 867–871 | MR | Zbl

[33] Ya. B Pesin, “Kharakteristicheskie eksponenty Lyapunova i gladkaya ergodicheskaya teoriya”, UMN, 32:4 (1977), 55–112 | MR | Zbl

[34] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, London Math. Soc. Lecture Note Ser., 180, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl

[35] A. Pumariño, J. A. Rodríguez, “Coexistence and persistence of infinitely many strange attractors”, Ergodic Theory Dynam. Systems, 21:5 (2001), 1511–1523 | DOI | MR | Zbl

[36] M. Rychlik, “Another proof of Jakobson's theorem and related results”, Ergodic Theory Dynam. Systems, 8:1 (1988), 93–109 | DOI | MR | Zbl

[37] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987 | MR

[38] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | DOI | MR

[39] Ph. Thieullen, C. Tresser, L.-S. Young, “Positive Lyapunov exponent for generic one-parameter families of unimodal maps”, J. Anal. Math., 64 (1994), 121–172 | DOI | MR | Zbl

[40] M. Tsujii, “Positive Lyapunov exponents in families of one-dimensional dynamical systems”, Invent. Math., 111:1 (1993), 113–137 | DOI | MR | Zbl

[41] M. Tsujii, “A proof of Benedicks–Carleson–Jacobson theorem”, Tokyo J. Math., 16:2 (1993), 295–310 | MR | Zbl

[42] M. Tsujii, “Small random perturbations of one-dimensional dynamical systems and Margulis–Pesin entropy formula”, Random Comput. Dynam., 1:1 (1992/93), 59–89 | MR

[43] M. Viana, “Strange attractors in higher dimensions”, Bol. Soc. Brasil. Mat. (N.S.), 24:1 (1993), 13–62 | DOI | MR | Zbl

[44] M. Viana, “Stochastic dynamics of deterministic systems”, Lecture Notes XXI Braz. Math. Colloq., IMPA, Rio de Janeiro, 1997

[45] Q. Wang, L. S. Young, “Strange attractors with one direction of instability”, Comm. Math. Phys., 218:1 (2001), 1–97 | DOI | MR | Zbl

[46] Q. Wang, L.-S. Young, “From invariant curves to strange attractors”, Comm. Math. Phys., 225:2 (2002), 275–304 | DOI | MR | Zbl

[47] J.-C. Yoccoz, “Introduction to hyperbolic dynamics”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464 (1995), 265–291 | MR | Zbl

[48] L.-S. Young, “Statistical properties of dynamical systems with some hyperbolicity”, Ann. of Math. (2), 147:3 (1998), 585–650 | DOI | MR | Zbl