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On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
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 |
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 $f$.
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. |
[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. |
[3] |
K. Barański, X. 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. |
[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. |
[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. |
[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. |
[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. |
[8] |
W. Bergweiler and A. Hinkkanen,
On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc., 126 (1999), 565-574.
doi: 10.1017/S0305004198003387. |
[9] |
W. Bergweiler, P. 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. |
[10] |
C. J. Bishop,
Constructing entire functions by quasiconformal folding, Acta Math., 214 (2015), 1-60.
doi: 10.1007/s11511-015-0122-0. |
[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. |
[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. |
[13] |
R. L. Devaney and M. Krych,
Dynamics of exp(z), Ergodic Theory Dynam. Systems, 4 (1984), 35-52.
doi: 10.1017/S014338570000225X. |
[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. |
[15] |
A. Douady and J. H. Hubbard,
Étude Dynamique Des Polynômes Complexes. Partie Ⅰ/Ⅱ Publications Mathématiques d'Orsay, 84/85 Université de Paris-Sud, Département de Mathématiques, Orsay, 1984/1985. |
[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.
|
[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. |
[18] |
N. Fagella,
Dynamics of the complex standard family, J. Math. Anal. Appl., 229 (1999), 1-31.
doi: 10.1006/jmaa.1998.6134. |
[19] |
P. Fatou,
Sur l'itération des fonctions transcendantes entiéres, Acta Math., 47 (1926), 337-370.
doi: 10.1007/BF02559517. |
[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. |
[21] |
W. K. Hayman,
Meromorphic Functions Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. |
[22] |
M. Heins,
Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2), 49 (1948), 200-213.
doi: 10.2307/1969122. |
[23] |
F. Iversen,
Recherches sur les fonctions inverses des fonctions méromorphes Ph. D. thesis, Helsingin Yliopisto, 1914. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
D. Martí-Pete,
Structural Theorems for Holomorphic Self-maps of the Punctured Plane Ph. D. thesis, The Open University, 2016. |
[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. |
[31] |
D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane, in preparation. |
[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. |
[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. |
[34] |
S. B. Nadler, Jr. ,
Continuum Theory. An introduction Monographs and Textbooks in Pure and Applied Mathematics, 158 Marcel Dekker Inc. , New York, 1992. |
[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. |
[36] |
H. Rådström,
On the iteration of analytic functions, Math. Scand., 1 (1953), 85-92.
doi: 10.7146/math.scand.a-10367. |
[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. |
[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. |
[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. |
[40] |
L. Rempe, P. J. Rippon and G. M. Stallard,
Are Devaney hairs fast escaping?, J. Difference Equ. Appl., 16 (2010), 739-762.
doi: 10.1080/10236190903282824. |
[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. |
[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. |
[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. |
[44] |
G. Rottenfusser, J. Rückert, L. 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. |
[45] |
D. Schleicher and J. Zimmer,
Escaping points of exponential maps, J. London Math. Soc. (2), 67 (2003), 380-400.
doi: 10.1112/S0024610702003897. |
[46] |
D. Schleicher and J. Zimmer,
Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.
|
[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. |
[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. |
[49] |
G. Valiron,
Lectures on the General Theory of Integral Functions Chelsea Publishing Company, New York, 1949. |
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. |
[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. |
[3] |
K. Barański, X. 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. |
[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. |
[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. |
[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. |
[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. |
[8] |
W. Bergweiler and A. Hinkkanen,
On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc., 126 (1999), 565-574.
doi: 10.1017/S0305004198003387. |
[9] |
W. Bergweiler, P. 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. |
[10] |
C. J. Bishop,
Constructing entire functions by quasiconformal folding, Acta Math., 214 (2015), 1-60.
doi: 10.1007/s11511-015-0122-0. |
[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. |
[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. |
[13] |
R. L. Devaney and M. Krych,
Dynamics of exp(z), Ergodic Theory Dynam. Systems, 4 (1984), 35-52.
doi: 10.1017/S014338570000225X. |
[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. |
[15] |
A. Douady and J. H. Hubbard,
Étude Dynamique Des Polynômes Complexes. Partie Ⅰ/Ⅱ Publications Mathématiques d'Orsay, 84/85 Université de Paris-Sud, Département de Mathématiques, Orsay, 1984/1985. |
[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.
|
[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. |
[18] |
N. Fagella,
Dynamics of the complex standard family, J. Math. Anal. Appl., 229 (1999), 1-31.
doi: 10.1006/jmaa.1998.6134. |
[19] |
P. Fatou,
Sur l'itération des fonctions transcendantes entiéres, Acta Math., 47 (1926), 337-370.
doi: 10.1007/BF02559517. |
[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. |
[21] |
W. K. Hayman,
Meromorphic Functions Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. |
[22] |
M. Heins,
Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2), 49 (1948), 200-213.
doi: 10.2307/1969122. |
[23] |
F. Iversen,
Recherches sur les fonctions inverses des fonctions méromorphes Ph. D. thesis, Helsingin Yliopisto, 1914. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
D. Martí-Pete,
Structural Theorems for Holomorphic Self-maps of the Punctured Plane Ph. D. thesis, The Open University, 2016. |
[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. |
[31] |
D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane, in preparation. |
[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. |
[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. |
[34] |
S. B. Nadler, Jr. ,
Continuum Theory. An introduction Monographs and Textbooks in Pure and Applied Mathematics, 158 Marcel Dekker Inc. , New York, 1992. |
[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. |
[36] |
H. Rådström,
On the iteration of analytic functions, Math. Scand., 1 (1953), 85-92.
doi: 10.7146/math.scand.a-10367. |
[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. |
[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. |
[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. |
[40] |
L. Rempe, P. J. Rippon and G. M. Stallard,
Are Devaney hairs fast escaping?, J. Difference Equ. Appl., 16 (2010), 739-762.
doi: 10.1080/10236190903282824. |
[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. |
[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. |
[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. |
[44] |
G. Rottenfusser, J. Rückert, L. 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. |
[45] |
D. Schleicher and J. Zimmer,
Escaping points of exponential maps, J. London Math. Soc. (2), 67 (2003), 380-400.
doi: 10.1112/S0024610702003897. |
[46] |
D. Schleicher and J. Zimmer,
Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., 28 (2003), 327-354.
|
[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. |
[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. |
[49] |
G. Valiron,
Lectures on the General Theory of Integral Functions Chelsea Publishing Company, New York, 1949. |





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