Abstract
In this paper, we extend Fibonacci unimodal map to a wider class. We describe the combinatorial property of these maps by first return map and principal nest. We give the sufficient and necessary condition for the existence of this class of maps. Moreover, for maps with ’bounded combinatorics’, we prove that they have no absolutely continuous invariant probability measure when the critical order \(\ell \) is sufficiently large; for maps with reluctantly recurrent critical point, we prove they have absolutely continuous invariant probability measure whenever the critical order \(\ell >1\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bruin, H.: Combinatorics of the kneading map. Int. J. Bifurc. Chaos. 5(5), 1339–1349 (1995)
Bruin, H.: Topological conditions for the existence of absorbing Cantor sets. Trans. Am. Math. Soc. 350, 2229–2263 (1998)
Block, L., Coppel, W.: Dynamics in One-Dimension. Lect. Notes in Math., 1513. Springer, Berlin (1992)
Blokh, A., Misiurewicz, M.: Wild attractors of polymodal negative Schwarzian maps. Commun. Math. Phys. 192(2), 397–416 (1998)
Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143, 97–130 (1996)
Bruin, H., Shen, W., van Strien, S.: Invariant measure exists without a growth condition. Commun. Math. Phys. 241, 287–306 (2003)
de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993)
Gao, R., Shen, W.: Decay of correlations for Fibonacci unimodal interval maps. Acta Math. Sin. Eng. Ser.(1) 34, 114–138 (2018)
Kozlovski, O.: Getting rid of the negative Schwarzian derivative condition. Ann. Math. 152, 743–762 (2000)
Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6(2), 425–457 (1993)
Li, S., Shen, W.: The topological complexity of cantor attractors for unimodal interval maps. Trans. Am. Math. Soc. 268(1), 659–688 (2015)
Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math. 140, 347–404 (1994)
Li, S., Wang, Q.: A new class of generalized Fibonacci unimodal maps. Nonlinearity 27, 1633–1643 (2014)
Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergod. Theory. Dyn. Syst. 14, 331–349 (1994)
Nowicki, T., Sands, D.: Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132(3), 633–680 (1998)
Peterson, K.: Ergodic Theory. Cambridge University Press, Cambridge (1983)
Shen, W.: Decay of geometry for unimodal maps: an elementary proof. Ann. Math. 163, 383–404 (2006)
van Strien, S., Vargas, E.: Real bounds, ergodicity and negative Schwarzian for multimodal maps. J. Am. Math. Soc. 17(4), 749–782 (2004)
Acknowledgements
Simin Li is supported by NSFC grant 11731003.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ji, H., Li, S. On the Combinatorics of Fibonacci-Like Non-renormalizable Maps. Commun. Math. Stat. 8, 473–496 (2020). https://doi.org/10.1007/s40304-020-00210-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-020-00210-x