June  2017, 37(6): 3123-3160. doi: 10.3934/dcds.2017134

Dynamic rays of bounded-type transcendental self-maps of the punctured plane

1. 

Departament de Matemátiques i Informática, Universitat de Barcelona, Gran Via de les Corts Catalanes 585,08007 Barcelona, Spain

2. 

School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom

* Corresponding author

Received  May 2016 Revised  January 2017 Published  February 2017

Fund Project: The first author was partially supported by the Polish NCN grant decision DEC-2012/06/M/ST1/00168 and by the Spanish grants MTM2011-26995-C02-02 and MTM2014-52209-C2-2-P. The second author was supported by The Open University, by a Formula Santander Scholarship and by the Spanish grant MTM2011-26995-C02-02

We study the escaping set of functions in the class $\mathcal{B}^*$, that is, transcendental self-maps of $\mathbb{C}^*$ for which the set of singular values is contained in a compact annulus of $\mathbb{C}^*$ that separates zero from infinity. For functions in the class $\mathcal{B}^*$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of $\mathbb{C}^*$ (and hence, in the class $\mathcal{B}^*$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every sequence $e∈\{0,∞\}^{\mathbb{N}_0}$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to the set $\{0,∞\}$ according to $e$ under iteration by $f$.

Citation: Núria Fagella, David Martí-Pete. Dynamic rays of bounded-type transcendental self-maps of the punctured plane. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3123-3160. doi: 10.3934/dcds.2017134
References:
[1]

J. M. Aarts and L. G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc., 338 (1993), 897-918. doi: 10.1090/S0002-9947-1993-1182980-3. Google Scholar

[2]

K. Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z., 257 (2007), 33-59. doi: 10.1007/s00209-007-0114-7. Google Scholar

[3]

K. BarańskiX. Jarque and L. Rempe, Brushing the hairs of transcendental entire functions, Topology Appl., 159 (2012), 2102-2114. doi: 10.1016/j.topol.2012.02.004. Google Scholar

[4]

A. M. Benini and N. Fagella, A separation theorem for entire transcendental maps, Proc. Lond. Math. Soc. (3), 110 (2015), 291-324. doi: 10.1112/plms/pdu047. Google Scholar

[5]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. Google Scholar

[6]

W. Bergweiler, On the Julia set of analytic self-maps of the punctured plane, Analysis, 15 (1995), 251-256. doi: 10.1524/anly.1995.15.3.251. Google Scholar

[7]

W. Bergweiler and A. E. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11 (1995), 355-373. doi: 10.4171/RMI/176. Google Scholar

[8]

W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc., 126 (1999), 565-574. doi: 10.1017/S0305004198003387. Google Scholar

[9]

W. BergweilerP. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. (3), 97 (2008), 368-400. doi: 10.1112/plms/pdn007. Google Scholar

[10]

C. J. Bishop, Constructing entire functions by quasiconformal folding, Acta Math., 214 (2015), 1-60. doi: 10.1007/s11511-015-0122-0. Google Scholar

[11]

J. Clunie and T. Kövari, On integral functions having prescribed asymptotic growth. Ⅱ, Canad. J. Math., 20 (1968), 7-20. doi: 10.4153/CJM-1968-002-1. Google Scholar

[12]

A. Deniz, A landing theorem for periodic dynamic rays for transcendental entire maps with bounded post-singular set, J. Difference Equ. Appl., 20 (2014), 1627-1640. doi: 10.1080/10236198.2014.968564. Google Scholar

[13]

R. L. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory Dynam. Systems, 4 (1984), 35-52. doi: 10.1017/S014338570000225X. Google Scholar

[14]

R. L. Devaney and F. Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems, 6 (1986), 489-503. doi: 10.1017/S0143385700003655. Google Scholar

[15]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes. Partie / Publications Mathématiques d'Orsay, 84/85 Université de Paris-Sud, Département de Mathématiques, Orsay, 1984/1985. Google Scholar

[16]

A. E. Eremenko, On the iteration of entire functions Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., PWN, Warsaw, 23 (1989), 339-345. Google Scholar

[17]

A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020. doi: 10.5802/aif.1318. Google Scholar

[18]

N. Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl., 229 (1999), 1-31. doi: 10.1006/jmaa.1998.6134. Google Scholar

[19]

P. Fatou, Sur l'itération des fonctions transcendantes entiéres, Acta Math., 47 (1926), 337-370. doi: 10.1007/BF02559517. Google Scholar

[20]

O. Forster, Lectures on Riemann Surfaces Translated from the German by Bruce Gilligan. Graduate Texts in Mathematics, 81 Springer-Verlag, New York-Berlin, 1981. Google Scholar

[21]

W. K. Hayman, Meromorphic Functions Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. Google Scholar

[22]

M. Heins, Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2), 49 (1948), 200-213. doi: 10.2307/1969122. Google Scholar

[23]

F. Iversen, Recherches sur les fonctions inverses des fonctions méromorphes Ph. D. thesis, Helsingin Yliopisto, 1914.Google Scholar

[24]

L. Keen, Dynamics of holomorphic self-maps of C*, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 9-30, Math. Sci. Res. Inst. Publ. 10, Springer, New York, 1988. doi: 10.1007/978-1-4613-9602-4_2. Google Scholar

[25]

L. Keen, Topology and growth of a special class of holomorphic self-maps of $\textbf{C}^* $, Ergodic Theory Dynam. Systems, 9 (1989), 321-328. doi: 10.1017/S0143385700004995. Google Scholar

[26]

J. Kotus, Iterated holomorphic maps on the punctured plane, Dynamical systems (Sopron, 1985), 10-28, Lecture Notes in Econom. and Math. Systems 287, Springer, Berlin, 1987. doi: 10.1007/978-3-662-00748-8_2. Google Scholar

[27]

J. K. Langley, On the multiple points of certain meromorphic functions, Proc. Amer. Math. Soc., 123 (1995), 1787-1795. doi: 10.1090/S0002-9939-1995-1242092-4. Google Scholar

[28]

P. M. Makienko, Iterations of analytic functions in C* Dokl. Akad. Nauk SSSR, 297 (1987), 35-37; translation in Soviet Math. Dokl. , 36 (1988), 418-420. Google Scholar

[29]

D. Martí-Pete, Structural Theorems for Holomorphic Self-maps of the Punctured Plane Ph. D. thesis, The Open University, 2016.Google Scholar

[30]

D. Martí-Pete, The escaping set of transcendental self-maps of the punctured plane, to appear in Ergodic Theory Dynam. Systems arXiv: 1412.1032.Google Scholar

[31]

D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane, in preparation.Google Scholar

[32]

H. Mihaljević-Brandt and L. Rempe-Gillen, Absence of wandering domains for some real entire functions with bounded singular sets, Math. Ann., 357 (2013), 1577-1604. doi: 10.1007/s00208-013-0936-z. Google Scholar

[33]

J. W. Milnor, Dynamics in One Complex Variable 3rd edition, Annals of Mathematics Studies, 160 Princeton University Press, Princeton, NJ, 2006. doi: 10.1007/978-3-663-08092-3. Google Scholar

[34]

S. B. Nadler, Jr. , Continuum Theory. An introduction Monographs and Textbooks in Pure and Applied Mathematics, 158 Marcel Dekker Inc. , New York, 1992. Google Scholar

[35]

G. Pólya, On an integral function of an integral function, J. London Math. Soc., S1-1 (1925), 12-15. doi: 10.1112/jlms/s1-1.1.12. Google Scholar

[36]

H. Rådström, On the iteration of analytic functions, Math. Scand., 1 (1953), 85-92. doi: 10.7146/math.scand.a-10367. Google Scholar

[37]

L. Rempe, A landing theorem for periodic rays of exponential maps, Proc. Amer. Math. Soc., 134 (2006), 2639-2648 (electronic). doi: 10.1090/S0002-9939-06-08287-6. Google Scholar

[38]

L. Rempe, On a question of Eremenko concerning escaping components of entire functions, Bull. Lond. Math. Soc., 39 (2007), 661-666. doi: 10.1112/blms/bdm053. Google Scholar

[39]

L. Rempe, Siegel disks and periodic rays of entire functions, J. Reine Angew. Math., 624 (2008), 81-102. doi: 10.1515/CRELLE.2008.081. Google Scholar

[40]

L. RempeP. J. Rippon and G. M. Stallard, Are Devaney hairs fast escaping?, J. Difference Equ. Appl., 16 (2010), 739-762. doi: 10.1080/10236190903282824. Google Scholar

[41]

L. Rempe-Gillen and D. J. Sixsmith, Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B} $ or not class $\mathcal{B}$?, to appear in Math. Z., (2016), 1-18, arXiv: 1502.00492.Google Scholar

[42]

P. J. Rippon and G. M. Stallard, Dimensions of Julia sets of meromorphic functions, J. London Math. Soc. (2), 71 (2005), 669-683. doi: 10.1112/S0024610705006393. Google Scholar

[43]

P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc., 133 (2005), 1119-1126 (electronic). doi: 10.1090/S0002-9939-04-07805-0. Google Scholar

[44]

G. RottenfusserJ. RückertL. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125. doi: 10.4007/annals.2011.173.1.3. Google Scholar

[45]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2), 67 (2003), 380-400. doi: 10.1112/S0024610702003897. Google Scholar

[46]

D. Schleicher and J. Zimmer, Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354. Google Scholar

[47]

D. J. Sixsmith, A new characterisation of the Eremenko-Lyubich class, J. Anal. Math., 123 (2014), 95-105. doi: 10.1007/s11854-014-0014-9. Google Scholar

[48]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418. doi: 10.2307/1971308. Google Scholar

[49]

G. Valiron, Lectures on the General Theory of Integral Functions Chelsea Publishing Company, New York, 1949.Google Scholar

show all references

References:
[1]

J. M. Aarts and L. G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc., 338 (1993), 897-918. doi: 10.1090/S0002-9947-1993-1182980-3. Google Scholar

[2]

K. Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z., 257 (2007), 33-59. doi: 10.1007/s00209-007-0114-7. Google Scholar

[3]

K. BarańskiX. Jarque and L. Rempe, Brushing the hairs of transcendental entire functions, Topology Appl., 159 (2012), 2102-2114. doi: 10.1016/j.topol.2012.02.004. Google Scholar

[4]

A. M. Benini and N. Fagella, A separation theorem for entire transcendental maps, Proc. Lond. Math. Soc. (3), 110 (2015), 291-324. doi: 10.1112/plms/pdu047. Google Scholar

[5]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.), 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. Google Scholar

[6]

W. Bergweiler, On the Julia set of analytic self-maps of the punctured plane, Analysis, 15 (1995), 251-256. doi: 10.1524/anly.1995.15.3.251. Google Scholar

[7]

W. Bergweiler and A. E. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11 (1995), 355-373. doi: 10.4171/RMI/176. Google Scholar

[8]

W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc., 126 (1999), 565-574. doi: 10.1017/S0305004198003387. Google Scholar

[9]

W. BergweilerP. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. Lond. Math. Soc. (3), 97 (2008), 368-400. doi: 10.1112/plms/pdn007. Google Scholar

[10]

C. J. Bishop, Constructing entire functions by quasiconformal folding, Acta Math., 214 (2015), 1-60. doi: 10.1007/s11511-015-0122-0. Google Scholar

[11]

J. Clunie and T. Kövari, On integral functions having prescribed asymptotic growth. Ⅱ, Canad. J. Math., 20 (1968), 7-20. doi: 10.4153/CJM-1968-002-1. Google Scholar

[12]

A. Deniz, A landing theorem for periodic dynamic rays for transcendental entire maps with bounded post-singular set, J. Difference Equ. Appl., 20 (2014), 1627-1640. doi: 10.1080/10236198.2014.968564. Google Scholar

[13]

R. L. Devaney and M. Krych, Dynamics of exp(z), Ergodic Theory Dynam. Systems, 4 (1984), 35-52. doi: 10.1017/S014338570000225X. Google Scholar

[14]

R. L. Devaney and F. Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems, 6 (1986), 489-503. doi: 10.1017/S0143385700003655. Google Scholar

[15]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes. Partie / Publications Mathématiques d'Orsay, 84/85 Université de Paris-Sud, Département de Mathématiques, Orsay, 1984/1985. Google Scholar

[16]

A. E. Eremenko, On the iteration of entire functions Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., PWN, Warsaw, 23 (1989), 339-345. Google Scholar

[17]

A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020. doi: 10.5802/aif.1318. Google Scholar

[18]

N. Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl., 229 (1999), 1-31. doi: 10.1006/jmaa.1998.6134. Google Scholar

[19]

P. Fatou, Sur l'itération des fonctions transcendantes entiéres, Acta Math., 47 (1926), 337-370. doi: 10.1007/BF02559517. Google Scholar

[20]

O. Forster, Lectures on Riemann Surfaces Translated from the German by Bruce Gilligan. Graduate Texts in Mathematics, 81 Springer-Verlag, New York-Berlin, 1981. Google Scholar

[21]

W. K. Hayman, Meromorphic Functions Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. Google Scholar

[22]

M. Heins, Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2), 49 (1948), 200-213. doi: 10.2307/1969122. Google Scholar

[23]

F. Iversen, Recherches sur les fonctions inverses des fonctions méromorphes Ph. D. thesis, Helsingin Yliopisto, 1914.Google Scholar

[24]

L. Keen, Dynamics of holomorphic self-maps of C*, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 9-30, Math. Sci. Res. Inst. Publ. 10, Springer, New York, 1988. doi: 10.1007/978-1-4613-9602-4_2. Google Scholar

[25]

L. Keen, Topology and growth of a special class of holomorphic self-maps of $\textbf{C}^* $, Ergodic Theory Dynam. Systems, 9 (1989), 321-328. doi: 10.1017/S0143385700004995. Google Scholar

[26]

J. Kotus, Iterated holomorphic maps on the punctured plane, Dynamical systems (Sopron, 1985), 10-28, Lecture Notes in Econom. and Math. Systems 287, Springer, Berlin, 1987. doi: 10.1007/978-3-662-00748-8_2. Google Scholar

[27]

J. K. Langley, On the multiple points of certain meromorphic functions, Proc. Amer. Math. Soc., 123 (1995), 1787-1795. doi: 10.1090/S0002-9939-1995-1242092-4. Google Scholar

[28]

P. M. Makienko, Iterations of analytic functions in C* Dokl. Akad. Nauk SSSR, 297 (1987), 35-37; translation in Soviet Math. Dokl. , 36 (1988), 418-420. Google Scholar

[29]

D. Martí-Pete, Structural Theorems for Holomorphic Self-maps of the Punctured Plane Ph. D. thesis, The Open University, 2016.Google Scholar

[30]

D. Martí-Pete, The escaping set of transcendental self-maps of the punctured plane, to appear in Ergodic Theory Dynam. Systems arXiv: 1412.1032.Google Scholar

[31]

D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane, in preparation.Google Scholar

[32]

H. Mihaljević-Brandt and L. Rempe-Gillen, Absence of wandering domains for some real entire functions with bounded singular sets, Math. Ann., 357 (2013), 1577-1604. doi: 10.1007/s00208-013-0936-z. Google Scholar

[33]

J. W. Milnor, Dynamics in One Complex Variable 3rd edition, Annals of Mathematics Studies, 160 Princeton University Press, Princeton, NJ, 2006. doi: 10.1007/978-3-663-08092-3. Google Scholar

[34]

S. B. Nadler, Jr. , Continuum Theory. An introduction Monographs and Textbooks in Pure and Applied Mathematics, 158 Marcel Dekker Inc. , New York, 1992. Google Scholar

[35]

G. Pólya, On an integral function of an integral function, J. London Math. Soc., S1-1 (1925), 12-15. doi: 10.1112/jlms/s1-1.1.12. Google Scholar

[36]

H. Rådström, On the iteration of analytic functions, Math. Scand., 1 (1953), 85-92. doi: 10.7146/math.scand.a-10367. Google Scholar

[37]

L. Rempe, A landing theorem for periodic rays of exponential maps, Proc. Amer. Math. Soc., 134 (2006), 2639-2648 (electronic). doi: 10.1090/S0002-9939-06-08287-6. Google Scholar

[38]

L. Rempe, On a question of Eremenko concerning escaping components of entire functions, Bull. Lond. Math. Soc., 39 (2007), 661-666. doi: 10.1112/blms/bdm053. Google Scholar

[39]

L. Rempe, Siegel disks and periodic rays of entire functions, J. Reine Angew. Math., 624 (2008), 81-102. doi: 10.1515/CRELLE.2008.081. Google Scholar

[40]

L. RempeP. J. Rippon and G. M. Stallard, Are Devaney hairs fast escaping?, J. Difference Equ. Appl., 16 (2010), 739-762. doi: 10.1080/10236190903282824. Google Scholar

[41]

L. Rempe-Gillen and D. J. Sixsmith, Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B} $ or not class $\mathcal{B}$?, to appear in Math. Z., (2016), 1-18, arXiv: 1502.00492.Google Scholar

[42]

P. J. Rippon and G. M. Stallard, Dimensions of Julia sets of meromorphic functions, J. London Math. Soc. (2), 71 (2005), 669-683. doi: 10.1112/S0024610705006393. Google Scholar

[43]

P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc., 133 (2005), 1119-1126 (electronic). doi: 10.1090/S0002-9939-04-07805-0. Google Scholar

[44]

G. RottenfusserJ. RückertL. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, Ann. of Math. (2), 173 (2011), 77-125. doi: 10.4007/annals.2011.173.1.3. Google Scholar

[45]

D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2), 67 (2003), 380-400. doi: 10.1112/S0024610702003897. Google Scholar

[46]

D. Schleicher and J. Zimmer, Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354. Google Scholar

[47]

D. J. Sixsmith, A new characterisation of the Eremenko-Lyubich class, J. Anal. Math., 123 (2014), 95-105. doi: 10.1007/s11854-014-0014-9. Google Scholar

[48]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418. doi: 10.2307/1971308. Google Scholar

[49]

G. Valiron, Lectures on the General Theory of Integral Functions Chelsea Publishing Company, New York, 1949.Google Scholar

Figure 1.  Period 8 cycle of rays landing on a repelling period 4 orbit in the unit circle for the function $f_{\alpha\beta}(z)=ze^{i\alpha}e^{\beta(z-1/z)/2}$ from the Arnol'd standard family, with $\alpha=0.19725$ and $\beta=0.48348$. Such points lie in the set $I_e(f_{\alpha\beta})$ with $e=\overline{\infty 0 \infty\infty 0 \infty 0 0}$ (see (2)).
Figure 2.  Logarithmic coordinates for a function $f\in\mathcal{B}^*$.
Figure 3.  Phase space of the function $f(z)=\exp(0.3(z+1/z))$ which has a disjoint-type logarithmic transform (see Example 3.12). In orange, the basin of attraction of the fixed point $z_0\simeq 2.2373$. Left, $z\in [-16,16]+i[-16,16]$; right, $z\in [-0.3,0.3]+i[-0.3,0.3]$.
Figure 4.  Logarithmic tracts of functions of finite order with $\rho_\infty(f)=3$ and $\rho_0(f)=2$ (left) and infinite order (right). The color of every point $z\in\mathbb{C}^*$ has been chosen according to the modulus (luminosity) and argument (hue) of $f(z)$.
Figure 5.  Fundamental domains of a function $f$ in the class $\mathcal{B}^*$.
Figure 6.  In the left, we have the phase space of the function $f(z)=z\exp(z^2+\exp(-1/z^2))$ from Example 7.3. In the right, the graph of the restriction of this function to the positive real line
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