On the combinatorics of Fibonacci-like non-renormalizable maps. (English) Zbl 1472.37053
Summary: 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 \(\ell >1\).
MSC:
| 37E15 | Combinatorial dynamics (types of periodic orbits) |
| 37E05 | Dynamical systems involving maps of the interval |
| 37E20 | Universality and renormalization of dynamical systems |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
References:
| [1] | Bruin, H., Combinatorics of the kneading map, Int. J. Bifurc. Chaos., 5, 5, 1339-1349 (1995) · Zbl 0886.58023 · doi:10.1142/S0218127495001010 |
| [2] | Bruin, H., Topological conditions for the existence of absorbing Cantor sets, Trans. Am. Math. Soc., 350, 2229-2263 (1998) · Zbl 0901.58029 · doi:10.1090/S0002-9947-98-02109-6 |
| [3] | Block, L.; Coppel, W., Dynamics in One-Dimension. Lect. Notes in Math., 1513. (1992), Berlin: Springer, Berlin · Zbl 0746.58007 |
| [4] | Blokh, A.; Misiurewicz, M., Wild attractors of polymodal negative Schwarzian maps, Commun. Math. Phys., 192, 2, 397-416 (1998) · Zbl 0943.37019 · doi:10.1007/s002200050506 |
| [5] | Bruin, H.; Keller, G.; Nowicki, T.; van Strien, S., Wild Cantor attractors exist, Ann. Math., 143, 97-130 (1996) · Zbl 0848.58016 · doi:10.2307/2118654 |
| [6] | Bruin, H.; Shen, W.; van Strien, S., Invariant measure exists without a growth condition, Commun. Math. Phys., 241, 287-306 (2003) · Zbl 1098.37034 · doi:10.1007/s00220-003-0928-z |
| [7] | de Melo, W.; van Strien, S., One-Dimensional Dynamics (1993), Berlin: Springer, Berlin · Zbl 0791.58003 |
| [8] | Gao, R.; Shen, W., Decay of correlations for Fibonacci unimodal interval maps, Acta Math. Sin. Eng. Ser.(1), 34, 114-138 (2018) · Zbl 1384.37049 · doi:10.1007/s10114-017-6438-2 |
| [9] | Kozlovski, O., Getting rid of the negative Schwarzian derivative condition, Ann. Math., 152, 743-762 (2000) · Zbl 0988.37044 · doi:10.2307/2661353 |
| [10] | Lyubich, M.; Milnor, J., The Fibonacci unimodal map, J. Am. Math. Soc., 6, 2, 425-457 (1993) · Zbl 0778.58040 · doi:10.1090/S0894-0347-1993-1182670-0 |
| [11] | Li, S.; Shen, W., The topological complexity of cantor attractors for unimodal interval maps, Trans. Am. Math. Soc., 268, 1, 659-688 (2015) · Zbl 1352.37116 · doi:10.1090/S0002-9947-2015-06372-7 |
| [12] | Lyubich, M., Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. Math., 140, 347-404 (1994) · Zbl 0821.58014 · doi:10.2307/2118604 |
| [13] | Li, S.; Wang, Q., A new class of generalized Fibonacci unimodal maps, Nonlinearity, 27, 1633-1643 (2014) · Zbl 1348.37066 · doi:10.1088/0951-7715/27/7/1633 |
| [14] | Martens, M., Distortion results and invariant Cantor sets of unimodal maps, Ergod. Theory. Dyn. Syst., 14, 331-349 (1994) · Zbl 0809.58026 · doi:10.1017/S0143385700007902 |
| [15] | Nowicki, T.; Sands, D., Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132, 3, 633-680 (1998) · Zbl 0908.58016 · doi:10.1007/s002220050236 |
| [16] | Peterson, K., Ergodic Theory (1983), Cambridge: Cambridge University Press, Cambridge · Zbl 0507.28010 |
| [17] | Shen, W., Decay of geometry for unimodal maps: an elementary proof, Ann. Math., 163, 383-404 (2006) · Zbl 1097.37032 · doi:10.4007/annals.2006.163.383 |
| [18] | van Strien, S.; Vargas, E., Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Am. Math. Soc., 17, 4, 749-782 (2004) · Zbl 1073.37043 · doi:10.1090/S0894-0347-04-00463-1 |
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