Jakobson theorem

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Michael Jakobson (2008), Scholarpedia, 3(4):2060.

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Curator: Dr. Michael Jakobson, Department of Mathematics, University of MD, College Park

Let $q_{\lambda}: x \rightarrow \lambda x(1-x)$ , $x \in [0,1]$ , $0 \le \lambda \le 4$ be the one-parameter family of quadratic maps. Let $f_{\lambda} : [0,1] \rightarrow [0,1]$ , $f_{\lambda}(0)=f_{\lambda}(1)=0$ , $\lambda \in [\lambda_0,\lambda_1]$ be a family $C^2$ -close to $q_{\lambda}$ , and suppose $f_{\lambda_1}$ is a map topologically equivalent to the Chebyshev polynomial $x \rightarrow 4 x(1-x)$ (Logistic Map). The following theorem was proved in [J1].

Theorem. There is a set $\Lambda$ of positive Lebesgue measure such that for $\lambda \in \Lambda$ the map $f_{\lambda}$ has an invariant measure $\mu_{\lambda}$ absolutely continuous with respect to the Lebesgue measure (acim). Moreover for $a < \lambda_1$

(1)

$\lim_{a \rightarrow \lambda_1}\frac{\mid \lambda \in [a,\lambda_1]\cap \Lambda \mid} {\mid \lambda_1-a \mid } = 1$

In [J2] this Theorem was generalized to families of piecewise smooth maps and $\mid \Lambda \mid$ was estimated through finitely many parameters of the family $f_{\lambda}$ . That makes possible computer assisted proofs of the existence of positive measure sets $\Lambda$ and estimates of their measures, see also [LT].

The proof of the Theorem is based on an inductive construction of an increasing sequence of partitions $\xi_n(\lambda)$ in the phase space. For each $\lambda \in \Lambda$ there is a limit partition $\xi_{\lambda} = \lim_{n \rightarrow \infty}\xi_n(\lambda)$ of an interval $I \subset [0,1]$ . Elements of $\xi_{\lambda}$ are countably many intervals $\Delta$ which are domains of a piecewise smooth power map $F_{\lambda} : \Delta \rightarrow I$ such that $F_{\lambda} \mid \Delta = f_{\lambda}^k, \ k = k(\Delta)$ . Inductive construction implies that for $\lambda \in \Lambda$ the maps $\ F_{\lambda}$ are expanding and have uniformly bounded distortions. According to the Folklore Theorem, see [J3], [JS], $F_{\lambda}$ has an acim $\nu_{\lambda}$ with continuous density bounded away from $0$ . Then $\mu_{\lambda}$ is obtained from $\nu_{\lambda}$ by a tower construction.

At step $n$ of induction partitions $\xi_n(\lambda)$ are defined for $\lambda \in \Lambda_n$ . By using parameter exclusion one constructs a decreasing sequence of sets $\Lambda_n$ in the parameter space such that $\Lambda = \bigcap_n \Lambda_n$ .

For $\lambda \in \Lambda$ the systems $(f_{\lambda},\mu_{\lambda})$ have strong mixing properties. The rate of decay of correlations is faster than polynomial. However there are $\lambda \in \Lambda$ such that $f_{\lambda}$ do not satisfy Collet-Eckmann condition (CE) and have the rate of decay of correlations slower than exponential, see [J2]. Several alternative proofs of the Theorem were obtained in subsequent works, see references in [J2], [JS]. Properties of $f_{\lambda}$ can vary depending on the construction. In particular for $\lambda \in \Lambda$ obtained by Benedicks-Carleson construction [BC1] $F_{\lambda}$ do not satisfy Markov property, and $f_{\lambda}$ satisfy CE condition. For $\lambda \in \Lambda$ obtained by Yoccoz construction, see [S], [Y], both Markov property and CE condition are satisfied. Property (1) implies that most $\lambda$ close to $\lambda_1$ belong to the intersection of $\Lambda$ obtained by different constructions.

See [J3], [JS] for an overview of related topics in one-dimensional dynamics.

In [BC2], [MV] similar sets $\Lambda$ were constructed for Henon-like maps, which were small perturbations of one-dimensional maps. Respective $f_{\lambda}$ have attractors carrying Sinai-Ruelle-Bowen measures, see [BY] .

See [LV] for a survey of results on Henon-like maps.

An important technical ingredient in the above results are distortion estimates for compositions of hyperbolic and parabolic maps, and maps with unbounded derivatives, see [JN],[PY] for related results.

Unsolved problems in that direction include construction of similar sets $\Lambda$ for families of 2-dim conservative maps, in particular Standard Family, for multidimensional quadratic-like families and for multidimensional Henon-like families.

References

[BC1]	M. Benedicks and L. Carleson. On iterations of $1-ax^2$ on $(-1,1)$ . Annals of Math., 122: 1--25, 1985.
[BC2]	M. Benedicks and L. Carleson. The dynamics of the Henon map. Annals of Math., 133: 73--169, 1991.
[BY]	M. Benedicks and L.-S. Young. Sinai-Bowen-Ruelle measures for certain Henon maps. Invent. Math., 112: 541--576, 1993.
[J1]	M.V. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Communications Math. Phys., 81:39--88, 1981.
[J2]	M. Jakobson. Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions. Proceedings in Symposia in Pure Math., 69: 825--881, 2001.
[J3]	M.V. Jakobson. Ergodic theory of one-dimensional mappings. Dynamical Systems, Ergodic Theory and Applications, Encyclopaedia of Math. Sciences, Springer,Volume 100, Part II, Chapter 9 : 234--263, 2000.
[JN]	M.V. Jakobson and S.E. Newhouse. Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle. Asterisque, 261 : 103--159, 2000.
[JS]	M. Jakobson and G. Swiatek. One-dimensional maps. Handbook of Dynamical Systems, Elsevier Science B.V., Volume 1A, Chapter 8: 599--664, 2002.
[LV]	S. Luzzatto and M. Viana. Parameter exclusion in Henon-like systems. Russian Math. Surveys 58, no. 6: 1053--1092, 2003.
[LT]	S. Luzzatto and H. Takahasi. Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps. Nonlinearity 19 , no. 7 : 1657--1695, 2006.
[MV]	L. Mora and M. Viana. Abundance of strange attractors. Acta Math. 171: 1--71.
[PY]	J. Palis and J.-C. Yoccoz. Implicit formalism for affine-like maps and parabolic compositions. Global Analysis of Dynamical Systems : Festschrift dedicated to Floris Takens, 67--88, 2001.
[S]	S. Senti. Dimension of weakly expanding points for quadratic maps. Bull. Soc. Math. France 131 (3): 399--420, 2003.
[Y]	J.-C. Yoccoz. Jakobson's theorem. Manuscript of the course at College de France, 1997

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