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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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