M. Lyubich. Almost every real quadratic map is either regular or stochastic. Ann. of Math., 156: 1-78, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... exists, such a measure a is unique and ergodic, and it gives the time average of Lebesgue almost every x 2 I a , Blokh Lyubich [13] Do these cases exhaust all the possibilities for a full Lebesgue measure set of parameters Remarkably, the answer is affirmative, as shown by Lyubich: Theorem 1 ([59]) For Lebesgue almost every a 2 [ Gamma1=4; 2] the quadratic map f a has either a periodic attractor or an absolutely continuous invariant measure. In particular, Palis finitude conjecture in Section 2 holds in this context: Lebesgue almost every quadratic map admits a unique SRB measure ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint Stony Brook, 1997.

.... on V ; moreover, for n 0, we have jR j B(U) Note that A : A( K;V ) is closed. Furthermore, replacing R by an iterate, if necessary, we can assume that A is invariant by the action of R. Selecting K and properly, by the topological convergence (Proposition 3.2) and Lemma 2. 2 in [Lyu02], for every f in the hybrid class of f , there exists N = N(f) so that R f 2 A. Proposition 3.4 (In nitesimal contraction: cf. Lyu99] There exist 1 and N 0 so that jDR , for every f 2 A( K;V ) Proof. Consider the set S : f(f; v) f 2 A; v 2 E f g. It is sucient to ....

....distance between g and g on U is large. We claim that, provided U is small enough, it is possible to choose a quasiconformal conjugacy between any two maps in U so that the quasiconformality is uniformly bounded outside their lled in Julia sets, using the argument in the proof of Lemma 2. 3 in [Lyu02]: in a small neighborhood V B nor (U) of f , it is possible to nd a domain U 1 so that g : g U 1 U 1 is a quadratic like restriction of g (but note that the Julia sets of these quadratic like restrictions are not, in general, connected) This de ne the holomorphic moving fundamental ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Ann. of Math., 156: 1-78, 2002.

.... on V ; moreover, for n 0, we have jR j B(U) Note that A : A( K;V ) is closed. Furthermore, replacing R by an iteration, if necessary, we can assume that A is invariant by the action of R. Selecting K and properly, by the topological convergence (Proposition 2.5) and Lemma 2. 2 in [3], for every f in the hybrid class of f , there exists N = N(f) so that R f 2 A. Proposition 2.7 (In nitesimal contraction) There exist 1 and N 0 so that jDR , for every f 2 A( K;V ) Proof. Consider the set S : f(f; v) f 2 A; v 2 E f g. It is sucient to verify the ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. preprint ims97-8, IMS-SUNY at Stony Brook, 1997.

....insight into this is due to Jakobson, proved around 1980: THEOREM 3.1 [J] There exists a positive Lebesgue measure set of parameters a with the property that f a has an acim with a positive Lyapunov exponent. The latest result on this topic is due to Lyubich, completed in 1998: THEOREM 3.1. 2 [Ly]. For the family f a (x) 1 ax 2 , a 2 [0; 2] there are sets A; B in parameter space with A [ B = 0; 2] modulo a set of Lebesgue measure 0 such that (i) A is open and dense in [0; 2] and for all a 2 A, f a has an attractive periodic orbit which attracts Lebesgue a.e. x 2 [ 1; 1] ii) B ....

Lyubich, M., Almost every real quadratic map is either regular or stochastic, preprint.

....wins, and on an open and dense set, contraction wins. This intermingling of parameters with antipodal behaviors underscores the complexity of the dynamical picture. It has been announced very recently 6 that these two types of dynamical behaviors account for a full measure set of parameters [27]. To control the dynamics, it is natural to go to the source of nonhyperbolicity, i.e. to impose conditions on the critical orbit. The following ideas go back to [13] we follow the formulation in [3] 4] where conditions (i) and (ii) are used to produce chaotic behavior for a positive measure ....

Lyubich, M., Almost every real quadratic map is either regular or stochastic, preprint.

....of all SRB measures cover a full Lebesgue measure subset of the whole manifold. This question has an armative answer in the context of uniformly hyperbolic systems, after [Sin72] BR75] Bow75] Rue76] A detailed picture is also available for maps of the circle or the interval, see [Jak81] [Lyu]. However, in higher dimensions the problem is mostly open, outside the uniformly hyperbolic setting, despite substantial progress in the study of certain classes of maps and ows with some properties of non uniform hyperbolicity, including the Lorenzlike attractors, see e.g. Pes92] Sat92] ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint, 1997.

....of attraction include almost every orbit Moreover, these attractors should have nice ergodic properties, including existence of physical measures and stochastic stability (stability under small random noise) I shall return to this later. The niteness conjecture has been established by Lyubich [34] for real quadratic maps x 7 a x 2 , in a very strong form: for almost every parameter value of a there is a unique attractor, which is either periodic or chaotic (non uniformly hyperbolic) There is an ongoing extension for rather general families of unimodal maps of the interval, by ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint, 1997.

....all SRB measures cover a full Lebesgue measure subset of the whole manifold. This question has an affirmative answer in the context of uniformly hyperbolic systems, after [Sin72] BR75] Bow75] Rue76] A detailed picture is also available for maps of the circle or the interval, see [Jak81] [Lyu]. However, in higher dimensions the problem is mostly open, outside the uniformly hyperbolic setting, despite substantial progress in the study of certain classes of maps and flows with some properties of non uniform hyperbolicity, including the Lorenzlike attractors, see e.g. Pes92] Sat92] ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint, 1997.

....: 1 p p X i=1 ffi x i is again a natural measure for f a which describes the asymptotics for almost all points ( BL91] For a long time it was conjectured that for almost all a, f a either has a periodic attractor or admits an acip. This fundamental result was recently proved by M. Lyubich ([Lyu97]) It follows that the mapping Psi : a 7 a = the natural measure of f a is well defined for almost all a in (0; 2] Of course, Psi is continuous on the set of hyperbolic attracting maps. In what follows we discuss the structure in parameter space and the parameter dependence of a near ....

M. Lyubich, Almost every real quadratic map is either regular or stochastic, SUNY preprint IMS 97--8, SUNY Stony-Brook, New York, 1997.

.... exists, such a measure a is unique and ergodic, and it gives the time average of Lebesgue almost every x 2 I a , Blokh Lyubich [13] Do these cases exhaust all the possibilities for a full Lebesgue measure set of parameters Remarkably, the answer is affirmative, as shown by Lyubich: Theorem 1 ([59]) For Lebesgue almost every a 2 [ Gamma1=4; 2] the quadratic map f a has either a periodic attractor or an absolutely continuous invariant measure. In particular, Palis finitude conjecture in Section 2 holds in this context: Lebesgue almost every quadratic map admits a unique SRB measure ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint Stony Brook, 1997.

....of all SRB measures cover a full Lebesgue measure subset of the whole manifold. This question has an affirmative answer in the context of uniformly hyperbolic systems, after [Sin72] BR75] Bow75] Rue76] A detailed picture is also available for maps of the circle or the interval, see [Jak81] [Lyu97]. However, in higher dimensions the problem is mostly open, outside the uniformly hyperbolic setting, despite substantial progress in the study of certain classes of maps and flows with some properties of non uniform hyperbolicity, including the Lorenzlike attractors, see e.g. Pes92] Sat92] ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint Stony Brook, 1997.

....for every combinatorial type. Indeed, if f and g are real analytic infinitely renormalizable maps, by the complex bounds in Theorem A of Levin van Strien in [11] there exists an integer N such that R N (f) and R N (g) have quadratic like extensions. Then we can use Lyubich s Theorem 1. 1 in [12] to conclude the exponential convergence. However, as we pointed out before, this is not sufficient to give the C 1 ff rigidity. Finally, at the moment, we cannot prove the exponential convergence of the operator for C 2 mappings with unbounded combinatorics. Acknowledgments Alberto Adrego ....

M. Lyubich, Almost Every real quadratic map is either regular or stochastic. SUNY Stony Brook IMS preprint 8, 1997.

....renormalization operator. Extending to this setting the analytic estimates of [D] we show that the hyperbolicity feature persists in the space of C r fold maps for r big enough with C 1 local stable manifolds (and real analytic local unstable manifolds given by Lyubich) In a remarkable paper [Lyb], Lyubich proved the hyperbolicity of the renormalization operator in the space of all renormalizable maps including those of unbounded type. In particular he was able to extend Theorem 3.5 to maps with any combinatorial type. However, this is not sufficient to establish a rigidity result which, ....

M. Lyubich, Almost every real quadratic map is either regular or stochastic, IMS Stony Brook preprint 97/8 (1998).

.... family [J, BC] In fact, in the case of the quadratic family fQ g much more is known: The set of regular parameter values is open and dense in the quadratic family [L3, GS2] The set of stochastic parameter values has full Lebesgue measure in the complement of the regular parameters [L6]. The former result was extended by Kozlovski [K1] to any non trivial real analytic family ff t g of real analytic unimodal maps: For an open and dense set of parameter values t in such a family, the map f t has a nite number of periodic attractors whose basin has full Lebesgue measure. One of ....

....x 2 X and any 0 there exists an interval J (x ; x )nX dist(x; J) jJ j C dist(x; J) By the Lebesgue Density Theorem, such a set has zero measure. But unlike the measure zero property, the property to have de nite gaps everywhere is preserved by quasisymmetric maps. Theorem 2. 27 ([L6], x4.1) The set I has de nite gaps everywhere and hence has zero Lebesgue measure in the parameter interval [1=2; 2] Putting together the last two theorems, we obtain: Theorem 2.28. Almost any real quadratic map q is either regular or stochastic. 2.17. Invariant line elds and equivariance. ....

M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint IMS at Stony Brook, # 1997/8. To appear in Ann. Math.

....the Conjecture. A characteristic feature of this development is that it is almost completely based upon the methods of holomorphic dynamics, though the nal results can be formulated in purely real terms. A generalization of the Renormalization Theory to all possible combinatorial types given in [L7] leads to the following assertion: NOTATIONS AND TERMINOLOGY 7 Theorem B [L7] The set of in nitely renormalizable real parameters has zero Lebesgue measure Theorems A and B together imply the Regular or Stochastic Theorem. Theory of quadratic like maps originated by Douady Hubbard [DH2] has ....

....completely based upon the methods of holomorphic dynamics, though the nal results can be formulated in purely real terms. A generalization of the Renormalization Theory to all possible combinatorial types given in [L7] leads to the following assertion: NOTATIONS AND TERMINOLOGY 7 Theorem B [L7]. The set of in nitely renormalizable real parameters has zero Lebesgue measure Theorems A and B together imply the Regular or Stochastic Theorem. Theory of quadratic like maps originated by Douady Hubbard [DH2] has played a crucial role in all stages of the above development. The 2nd lecture ....

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M. Lyubich. Almost every real quadratic map is either regular or stochastic. to appear in Annals Math.

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M. Lyubich. Almost every real quadratic map is either regular or stochastic. Ann. of Math., 156: 1-78, 2002.

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M. Lyubich, Almost every real quadratic map is either regular or stochastic, Ann. of Math. (2001)

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M. Lyubich, Almost every real quadratic map is either regular or stochastic, IMS Stony Brook preprint 97/8 (1998).

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M. Lyubich. Almost every real quadratic map is either regular or stochastic. Preprint, 1997.

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