Jakobson theorem

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\\)

\\[\tag{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.

## [edit] References

<a name='BC1'></a>[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

Internal references

* John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.

* Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.

* Nikolai Chernov (2008) Decay of correlations. Scholarpedia, 3(4):4862.

* James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

* David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

## [edit] See Also

Ergodic Theory, Invariant Measure, Logistic Map, SRB Measure