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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
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 \(\Sigma _k\).
References
Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998). Springer Monographs in Mathematics
Barnsley, M.F., Leśniak, K.: The chaos game on a general iterated function system from a topological point of view. Int. J. Bifurc. Chaos Appl. Sci. Eng. 24(11), 1450139 (2014). 10
Bhattacharya, R.N., Lee, O.: Asymptotics of a class of Markov processes which are not in general irreducible. Ann. Probab. 16(3), 1333–1347 (1988)
Díaz, L.J., Gelfert, K.: Porcupine-like horseshoes: transitivity, Lyapunov spectrum, and phase transitions. Fund. Math. 216(1), 55–100 (2012)
Díaz, L.J., Matias, E.: Stability of the markov operator and synchronization of markovian random products. Nonlinearity 31(5), 1782–1806 (2018)
Dubins, L.E., Freedman, D.A.: Invariant probabilities for certain Markov processes. Ann. Math. Stat. 37, 837–848 (1966)
Fernandez, B., Quas, A.: Statistical properties of invariant graphs in piecewise affine discontinuous forced systems. Nonlinearity 24(9), 2477–2488 (2011)
Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)
Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)
Jäger, T., Keller, G.: Random minimality and continuity of invariant graphs in random dynamical systems. Trans. Am. Math. Soc. 368(9), 6643–6662 (2016)
Keller, G., Otani, A.: Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing. Dyn. Syst. 28(2), 123–139 (2013)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York (1976). (Reprinting of the 1960 original, Undergraduate Texts in Mathematics)
Kleptsyn, V., Volk, D.: Physical measures for nonlinear random walks on interval. Mosc. Math. J. 14(2), 339–365 (2014). 428
Kravchenko, A.S.: Completeness of the space of separable measures in the Kantorovich-Rubinshteĭn metric. Sibirsk. Mat. Zh. 47(1), 85–96 (2006)
Kudryashov, Y.: Des orbites périodiques et des attracteurs des systémes dynamiques. PhD thesis, ENS Lyon (2010)
Kudryashov, Y.G.: Bony attractors. Funktsional. Anal. i Prilozhen. 44(3), 73–76 (2010)
Letac, G.: A contraction principle for certain Markov chains and its applications. In: Random matrices and their applications (Brunswick,Maine), vol. 50 of Contemp. Math. Amer. Math. Soc., Providence, RI, 1986, pp. 263–273 (1984)
Okunev, A., Shilin, I.: On the attractors of step skew products over the Bernoulli shift. arXiv:1703.01763
Revuz, D.: Markov Chains, 2nd edn., vol. 11 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1984)
Stark, J.: Regularity of invariant graphs for forced systems. Ergod. Theory Dyn. Syst. 19(1), 155–199 (1999)
Sturman, R., Stark, J.: Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity 13(1), 113–143 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-018-2113-2