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.
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.
See Also
Ergodic Theory, Invariant Measure, Logistic Map, SRB Measure