Abstract
We study attracting graphs of step skew products from the topological and ergodic points of view where the usual contracting-like assumptions of the fiber dynamics are replaced by weaker merely topological conditions. In this context, we prove the existence of an attracting invariant graph and study its topological properties. We prove the existence of globally attracting measures and we show that (in some specific cases) the rate of convergence to these measures is exponential.
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Notes
This terminology is borrowed from number theory and its use in our context was coined in [2].
The Hutchinson attractor is the globally attracting fixed point of the Barnsley-Hutchinson operator \(\mathcal {B}_{F}\) when the maps \(f_i\) are uniform contractions, see [9].
The maximal Lyapunov exponent is the rate of growth of the Lipschitz constant of the compositions \(f_\vartheta ^n\).
This means that for every \((x_1,\dots ,x_m)\in [0,1]^m\), every \(s,i\in \{1, \dots , m\}\), and every \(j\in \{1,2\}\), the map \(x\mapsto \pi ^s (f_j(x_1,\dots , x_{i-1},x,x_{i+1},\dots ,x_m))\) is increasing.
Obviously, here the hyperbolicity concerns only the one dimensional fiber direction.
In [4] the set \(S_F\) corresponds to the “subset of sequences in \(\Sigma _2\) with trivial spines”. Indeed, this is a simple remark that can be obtained in more general settings: it is not difficult to see that if \(S_F\ne \emptyset \) then \(S_F\) is a residual subset of \(\Sigma _k\).
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This paper is part of the PhD thesis of EM (PUC-Rio) supported by CAPES. EM was supported by PNPD CAPES. LJD is partially supported by CNPq and CNE-Faperj. The authors thank K. Gelfert for her useful comments on this paper. The authors thank an anonymous referee for the careful reading of the paper and useful observations.
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Díaz, L.J., Matias, E. Attracting graphs of skew products with non-contracting fiber maps. Math. Z. 291, 1543–1568 (2019). https://doi.org/10.1007/s00209-018-2113-2
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DOI: https://doi.org/10.1007/s00209-018-2113-2