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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.

##### MSC:
 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 70K50 Bifurcations and instability for nonlinear problems in mechanics
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##### References:
 [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
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