zbMATH — the first resource for mathematics

Measure and dimension of solenoidal attractors of one-dimensional dynamical systems. (English) Zbl 0721.58033
A solenoid attractor of a map f in a one-dimensional compact manifold with boundary is considered. Let f: \(M\to M\) be a \(C^{\infty}\)-smooth transformation with non-flat critical points, i.e. for each critical point c there exists n such that \(f^{(n)}(c)\neq 0\). By a solenoid attractor they mean a closed f-invariant subset \(A\subset M\) of a following structure \(A=\cap^{\infty}_{n=1}M^{(n)}\), \(M^{(1)}\supset M^{(2)}\supset...\), where \(M^{(n)}=\cup^{p_ n- 1}_{k=0}I_ k^{(n)}\) and \(I_ k^{(n)}\), \(k=0,...,p_ n-1\) are disjoint intervals such that \(fI_ k^{(n)}\subset I_{k+1}^{(n)}\); \(p_ n\) are the periods, \(I_{p_ n}^{(n)}=I_ 0^{(n)}\), \(p_ n\to \infty.\)
The type of the solenoid attractor A is the maximal possible sequence \(\{p_ n\}^{\infty}_{n=1}\) of pairwise distinct periods \(p_ n\). Let \(\lambda\) denote the Lebesgue measure on M and dim X denote the Hausdorff dimension of a subset \(X\subset M.\)
The main result of the paper is the following assertion. Let A be a solenoidal attractor of type \(\{p_ n\}^{\infty}_{n=0}\) of a map f. Then \(1)\quad \lambda (A)=0;\quad 2)\quad if\sup (p_{n+1}/p_ n)<\infty\), then dim A\(<1.\)
For unimodal maps the theorem was proved earlier.

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
70K50 Bifurcations and instability for nonlinear problems in mechanics
Full Text: DOI
[1] [B 1] Blokh, A.M.: Decomposition of dynamical systems on an interval. Russ. Math. Surv.38; N. 5, 133–134 (1983) [see also Blokh, A.M.: Letter to the editors. Russ. Math. Surv.42, (1987)] · Zbl 0557.28017
[2] [B 2] Blokh, A.M.: On dynamical systems on one dimensional branched manifolds (in Russian), I, II, III; Theory Funct. Functional Anal. Appl.46, 8–18 (1986);47, 67–77 (1987);48, 32–46 (1987)
[3] [BL 1] Blokh, A.M., Lyubich, M.Yu.: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. II. The Smooth case, preprint (1987) · Zbl 0665.58024
[4] [BL 2] Blokh, A.M., Lyubich, M.Yu.: Measure of solenoidal attractors of unimodal maps of an interval (1987). Matem. Zametki (to appear) · Zbl 0774.58026
[5] [BL 3] Blokh, A.M., Lyubich, M.Yu.: Attractors of maps of the interval. Funct. Anal. Appl.21, 70–71 (1987) · Zbl 0653.58022
[6] [G 1] Guckenheimer, J.: Sensitive dependence on initial conditions for one dimensional maps. Commun. Math. Phys.70, 133–160 (1979) · Zbl 0429.58012
[7] [G 2] Guckenheimer, J.: Limit sets ofS-unimodal maps with zero entropy. Commun. Math. Phys.110, 655–659 (1987) · Zbl 0625.58027
[8] [L] Lyubich, M.Yu.: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. I. The case of negative Schwarzian derivative, preprint (1987) · Zbl 0665.58023
[9] [MS] De Melo, W., van Strien, S.J.: A structure theorem in one dimensional dynamics. Preprint (1986)
[10] [MMS] Martens, M., de Melo, W., van Strien, S.J.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Preprint (1988) · Zbl 0761.58007
[11] [MMSS] Martens, M., de Melo, W., van Strien, S.J., Sullivan, D.: Bounded geometry and measure of the attracting Cantor set of quadratic-like maps, preprint (1988)
[12] [M] Milnor, J.: On the concept of attractor. Commun. Math. Phys.99, 177–195 (1985) · Zbl 0595.58028
[13] [MT] Milnor, J., Thurston, W.: On iterated maps of the interval. I, II, preprint (1977)
[14] [S] Van Strien, S.: Hyperbolicity and invariant measures for generalC 2-interval maps satisfying Misiurewicz condition. Preprint Delft University of Technology, pp. 27–46
[15] [Y] Yoccoz, J.-C.: Il n’y a pas de contre-exemples de Denjoy analytiques. C.R. Acad. Sci. Paris,289, 141–144 (1984) · Zbl 0573.58023
[16] [VSK] Vul, E.B., Sinai, Ya.G., Khanin, K.M.: Universality of Feigenbaum and thermodynamical formalism. Russ. Math. Surv.39, 3–37 (1984) · Zbl 0561.58033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.