Escaping endpoints explode. (English) Zbl 1381.37051
The authors study escaping endpoints for complex exponential maps \(f_a: z\mapsto e^z+a\), \(a\in \mathbb{C}\), and also for certain subclasses of entire functions. For a large class of parameter values, they show that, under natural hypotheses, the point at infinity is an explosion point for the set of escaping endpoints for \(f_a\). This answers a question by D. Schleicher; earlier results in this direction have previously been obtained by J. C. Mayer [Ergodic Theory Dyn. Syst. 10, No. 1, 177–183 (1989; Zbl 0668.58032)].
The paper also contains many other results, including connectivity statements for finite-order entire functions and no-wandering-triangles type theorems for exponential maps.
The paper also contains many other results, including connectivity statements for finite-order entire functions and no-wandering-triangles type theorems for exponential maps.
Reviewer: Alan A. Sola (Stockholm)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 54F15 | Continua and generalizations |
| 54G15 | Pathological topological spaces |
Keywords:
transcendental dynamics; exponential map; escaping set; explosion point; Lelek fan; Cantor bouquetCitations:
Zbl 0668.58032References:
| [1] | Aarts, J.M., Oversteegen, L.G.: The geometry of Julia sets. Trans. Am. Math. Soc. 338(2), 897-918 (1993) · Zbl 0809.54034 · doi:10.1090/S0002-9947-1993-1182980-3 |
| [2] | Barański, K.: Trees and hairs for some hyperbolic entire maps of finite order. Math. Z. 257(1), 33-59 (2007) · Zbl 1136.37024 · doi:10.1007/s00209-007-0114-7 |
| [3] | Barański, K., Jarque, X., Rempe, L.: Brushing the hairs of transcendental entire functions. Topol. Appl. 159(8), 2102-2114 (2012) · Zbl 1267.37041 · doi:10.1016/j.topol.2012.02.004 |
| [4] | Benini, A.M.: Triviality of fibers for Misiurewicz parameters in the exponential family. Conform. Geom. Dyn. 15(9), 133-151 (2011) · Zbl 1246.37069 · doi:10.1090/S1088-4173-2011-00227-X |
| [5] | Benini, A.M., Lyubich, M.: Repelling periodic points and landing of rays for post-singularly bounded exponential maps. Annales de l’Institut Fourier 64(4), 1493-1520 (2014) · Zbl 1323.37029 · doi:10.5802/aif.2888 |
| [6] | Blokh, A., Oversteegen, L.: Wandering triangles exist. Comptes Rendus Mathematique 339(5), 365-370 (2004) · Zbl 1052.37036 · doi:10.1016/j.crma.2004.06.024 |
| [7] | Bodelón, C., Devaney, R.L., Hayes, M., Roberts, G., Goldberg, L.R., Hubbard, J.H.: Dynamical convergence of polynomials to the exponential. J. Differ. Equ. Appl. 6(3), 275-307 (2000) · Zbl 0976.37022 · doi:10.1080/10236190008808229 |
| [8] | Bula, W.D., Oversteegen, L.G.: A characterization of smooth Cantor bouquets. Proc. Am. Math. Soc. 108(2), 529-534 (1990) · Zbl 0679.54034 · doi:10.1090/S0002-9939-1990-0991691-9 |
| [9] | Charatonik, W.J.: The Lelek fan is unique. Houst. J. Math. 15(1), 27-34 (1989) · Zbl 0675.54034 |
| [10] | Devaney, R.L.: Cantor bouquets, explosions, and Knaster continua: dynamics of complex exponentials. Publ. Mat. 43(1), 27-54 (1999) · Zbl 0939.37028 · doi:10.5565/PUBLMAT_43199_02 |
| [11] | Devaney, R.L., Jarque, X.: Indecomposable continua in exponential dynamics. Conform. Geom. Dyn. 6, 1-12 (2002) · Zbl 1062.37035 · doi:10.1090/S1088-4173-02-00080-2 |
| [12] | Devaney, R.L., Krych, M.: Dynamics of \[{\rm exp}(z)\] exp(z). Ergod. Theory Dyn. Syst. 4(1), 35-52 (1984) · Zbl 0567.58025 · doi:10.1017/S014338570000225X |
| [13] | Devaney, R.L., Tangerman, F.: Dynamics of entire functions near the essential singularity. Ergod. Theory Dyn. Syst. 6(4), 489-503 (1986) · Zbl 0612.58020 · doi:10.1017/S0143385700003655 |
| [14] | Eremenko, A.E., Lyubich, M.Y.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier Grenoble 42(4), 989-1020 (1992) · Zbl 0735.58031 · doi:10.5802/aif.1318 |
| [15] | Evdoridou, V.: Fatou’s web. Preprint (2015) arXiv:1510.07449 · Zbl 1351.37183 |
| [16] | Förster, M., Rempe, L., Schleicher, D.: Classification of escaping exponential maps. Proc. Am. Math. Soc. 136(2), 651-663 (2008) · Zbl 1134.37019 · doi:10.1090/S0002-9939-07-09073-9 |
| [17] | Kiwi, J.: Non-accessible critical points of Cremer polynomials. Ergod. Theory Dyn. Syst. 20(5), 1391-1403 (2000) · Zbl 0966.37015 · doi:10.1017/S0143385700000754 |
| [18] | Knaster, B., Kuratowski, C.: Sur les ensembles connexes. Fundamenta Mathematicae 2(1), 206-255 (1921) · JFM 48.0209.02 |
| [19] | Lelek, A.: On plane dendroids and their end points in the classical sense. Fund. Math. 49, 301-319 (1960/1961) · Zbl 0099.17701 |
| [20] | Mayer, J.C.: An explosion point for the set of endpoints of the Julia set of \[\lambda \exp (z)\] λexp(z). Ergod. Theory Dyn. Syst. 10(1), 177-183 (1990) · Zbl 0668.58032 · doi:10.1017/S0143385700005460 |
| [21] | Mihaljević-Brandt, H.: Topological dynamics of transcendental entire functions. Ph.D. thesis, University of Liverpool, Liverpool, UK, September 2009 · Zbl 1256.37001 |
| [22] | Mihaljević-Brandt, H.: A landing theorem for dynamic rays of geometrically finite entire functions. J. Lond. Math. Soc. (2) 81(3), 696-714 (2010) · Zbl 1221.37096 |
| [23] | Mihaljević-Brandt, H.: Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Trans. Am. Math. Soc. 364(8), 4053-4083 (2012) · Zbl 1291.37063 · doi:10.1090/S0002-9947-2012-05541-3 |
| [24] | Nadler, Jr., S.B.: Continuum theory. An introduction. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 158. Marcel Dekker Inc., New York (1992) · Zbl 0757.54009 |
| [25] | Newman, M.H.A.: Elements of the Topology of Plane Sets of Points, 2nd edn. Cambridge University Press (1951) · Zbl 0045.44003 |
| [26] | Pérez-Marco, R.: Topology of Julia sets and hedgehogs. Université Paris-Sud (1994). Preprint · Zbl 1116.37033 |
| [27] | Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975). With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV · Zbl 0298.30014 |
| [28] | Rempe, L.: Dynamics of exponential maps. Doctoral thesis, Christian-Albrechts-Universität Kiel (2003). http://macau.uni-kiel.de/receive/dissertation_diss_00000781 · Zbl 1138.37025 |
| [29] | Rempe, L.: A landing theorem for periodic rays of exponential maps. Proc. Am. Math. Soc 134(9), 2639-2648 (2006) · Zbl 1099.37040 · doi:10.1090/S0002-9939-06-08287-6 |
| [30] | Rempe, L.: Topological dynamics of exponential maps on their escaping sets. Ergod. Theory Dyn. Syst. 26(6), 1939-1975 (2006) · Zbl 1138.37025 · doi:10.1017/S0143385706000435 |
| [31] | Rempe, L.: On nonlanding dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 32(2), 353-369 (2007) · Zbl 1116.37033 |
| [32] | Rempe, L.: Rigidity of escaping dynamics for transcendental entire functions. Acta Math. 203(2), 235-267 (2009) · Zbl 1226.37027 · doi:10.1007/s11511-009-0042-y |
| [33] | Rempe, L.: Connected escaping sets of exponential maps. Ann. Acad. Sci. Fenn. Math. 36(1), 71-80 (2011) · Zbl 1291.37070 · doi:10.5186/aasfm.2011.3604 |
| [34] | Rempe, L., Schleicher, D.: Bifurcation loci of exponential maps and quadratic polynomials: local connectivity, triviality of fibers, and density of hyperbolicity. In: Holomorphic Dynamics and Renormalization. Fields Institute Communications, vol. 53, pp. 177-196. American Mathematical Society, Providence (2008) · Zbl 1154.37352 |
| [35] | Rempe, L.; Schleicher, D.; Rippon, P. (ed.); Stallard, G. (ed.), Combinatorics of bifurcations in exponential parameter space, No. 348, 317-370 (2008), Cambridge · Zbl 1178.37043 · doi:10.1017/CBO9780511735233.014 |
| [36] | Rempe, L., Schleicher, D.: Bifurcations in the space of exponential maps. Invent. Math. 175(1), 103-135 (2009) · Zbl 1165.37014 · doi:10.1007/s00222-008-0147-5 |
| [37] | Rippon, P.J., Stallard, G.M.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105(4), 787-820 (2012) · Zbl 1291.30160 |
| [38] | Rottenfußer, G., Rückert, J., Rempe, L., Schleicher, D.: Dynamic rays of bounded-type entire functions. Ann. Math. (2) 173(1), 77-125 (2011) · Zbl 1232.37025 |
| [39] | Schleicher, D.: Attracting dynamics of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 3-34 (2003) · Zbl 1088.30016 |
| [40] | Schleicher, D.: The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Math. J. 136(2), 343-356 (2007) · Zbl 1114.37024 · doi:10.1215/S0012-7094-07-13625-1 |
| [41] | Schleicher, D.: On fibers and local connectivity of compact sets in \[\mathbb{C}C\]. IMS Preprint 1998/13b (2007). arXiv:math/9902154 · Zbl 1291.37063 |
| [42] | Schleicher, D., Zimmer, J.: Escaping points of exponential maps. J. Lond. Math. Soc. (2) 67(2), 380-400 (2003) · Zbl 1042.30012 |
| [43] | Schleicher, D., Zimmer, J.: Periodic points and dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 327-354 (2003) · Zbl 1088.30017 |
| [44] | Steen, L.A., Seebach, J.A.: Counterexamples in Topology. Dover Publications Inc, Mineola (1995) · Zbl 1245.54001 |
| [45] | Wilder, R.L.: A point set which has no true quasicomponents, and which becomes connected upon the addition of a single point. Bull. Am. Math. Soc. 33(4), 423-427 (1927) · JFM 53.0570.03 · doi:10.1090/S0002-9904-1927-04395-6 |
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