# Jakobson theorem

Curator and Contributors

1.00 - Michael Jakobson

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.

## 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

Internal references

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