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Properties of multicorrelation sequences and large returns under some ergodicity assumptions
Invisible tricorns in real slices of rational maps
1. | Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA |
2. | School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India |
One of the conspicuous features of real slices of bicritical rational maps is the existence of Tricorn-type hyperbolic components. Such a hyperbolic component is called invisible if the non-bifurcating sub-arcs on its boundary do not intersect the closure of any other hyperbolic component. Numerical evidence suggests an abundance of invisible Tricorn-type components in real slices of bicritical rational maps. In this paper, we study two different families of real bicritical maps and characterize invisible Tricorn-type components in terms of suitable topological properties in the dynamical planes of the representative maps. We use this criterion to prove the existence of infinitely many invisible Tricorn-type components in the corresponding parameter spaces. Although we write the proofs for two specific families, our methods apply to generic families of real bicritical maps.
References:
[1] |
G. Ble, A. Douady and C. Henriksen, Round annuli., Contemporary Mathematics: In the Tradition of Ahlfors and Bers, III, 355 (2004), 71–76.
doi: 10.1090/conm/355/06445. |
[2] |
K. Bogdanov, K. Mamayusupov, S. Mukherjee and D. Schleicher,
Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori, Nonlinearity, 30 (2017), 3241-3254.
doi: 10.1088/1361-6544/aa79cf. |
[3] |
A. Bonifant, X. Buff and J. Milnor, On antipode preserving cubic maps, work in progress, 2015. http://www.math.stonybrook.edu/ jack/bbm.pdf Google Scholar |
[4] |
A. Bonifant, X. Buff and J. Milnor,
Antipode preserving cubic maps: The Fjord theorem, Proc. London Math. Soc., 116 (2018), 670-728.
doi: 10.1112/plms.12075. |
[5] |
B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, Vol. 141, Cambridge University Press, Cambridge, 2014.
![]() |
[6] |
X. Buff, G. Cui and L. Tan, Teichmüller spaces and holomorphic dynamics, in Handbook of Teichmüller theory Vol. 4, Société mathématique européenne, 2014,717–756.
doi: 10.4171/117-1/17. |
[7] |
X. Buff and A. L. Epstein,
A parabolic Pommerenke-Levin-Yoccoz inequality, Fund. Math., 172 (2002), 249-289.
doi: 10.4064/fm172-3-3. |
[8] |
J. Canela, N. Fagella and A. Garijo,
On a family of rational perturbations of the doubling map, Journal of Difference Equations and Applications, 21 (2015), 715-741.
doi: 10.1080/10236198.2015.1050387. |
[9] |
A. Douady,
Does a Julia set depend continuously on the polynomial?, Complex dynamical systems, Proc. Sympos. Appl. Math., 49 (1994), 91-138.
doi: 10.1090/psapm/049/1315535. |
[10] |
A. Douady and J. Hubbard,
A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297.
doi: 10.1007/BF02392534. |
[11] |
J. Hubbard, D. Schleicher and S. Sutherland,
How to find all roots of complex polynomials by Newton's method, Inventiones Mathematicae, 146 (2001), 1-33.
doi: 10.1007/s002220100149. |
[12] |
J. H. Hubbard and D. Schleicher, Multicorns are not path connected, in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday, Princeton University Press, 2014, 73–102. |
[13] |
H. Inou and S. Mukherjee, Discontinuity of straightening in antiholomorphic dynamics, preprint, 2016, arXiv: 1605.08061. Google Scholar |
[14] |
H. Inou and S. Mukherjee,
Non-landing parameter rays of the multicorns, Inventiones Mathematicae, 204 (2016), 869-893.
doi: 10.1007/s00222-015-0627-3. |
[15] |
T. Lei, Local properties of the Mandelbrot set at parabolic points, in The Mandelbrot Set, Theme and Variations, Lecture Note Series, Vol. 274, London Mathematical Society, Cambridge, 2000, 130–160. |
[16] |
R. Lodge, Y. Mikulich and D. Schleicher, A classification of postcritically finite Newton maps, preprint, arXiv: 1510.02771. Google Scholar |
[17] |
R. Lodge, Y. Mikulich and D. Schleicher, Combinatorial properties of Newton maps, preprint, arXiv: 1510.02761. Google Scholar |
[18] |
J. Milnor,
On rational maps with two critical points, Experiment. Math., 9 (2000), 333-411.
doi: 10.1080/10586458.2000.10504657. |
[19] |
J. Milnor, Dynamics in one complex variable, 3rd edition, Princeton University Press, NJ, 2006.
![]() |
[20] |
J. Milnor, Hyperbolic components in spaces of polynomial maps, with an appendix by A. Poirier, in Conformal Dynamics and Hyperbolic Geometry, Contemporary Mathematics, Vol. 573, American Mathematical Society, Providence, RI, 2012,183–232.
doi: 10.1090/conm/573/11428. |
[21] |
S. Mukherjee, S. Nakane and D. Schleicher,
On Multicorns and Unicorns II: Bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory and Dynamical Systems, 37 (2015), 859-899.
doi: 10.1017/etds.2015.65. |
[22] |
V. A. Naishul,
Topological invariants of analytic and area preserving mappings and their applications to analytic differential equations in $\mathbb{C}^2$ and $\mathbb{C}\mathbb{P}^2$, Transactions of the Moscow Mathematical Society, 42 (1983), 239-250.
|
[23] |
S. Nakane and D. Schleicher,
On Multicorns and Unicorns I : Antiholomorphic dynamics, hyperbolic components and real cubic polynomials, International Journal of Bifurcation and Chaos, 13 (2003), 2825-2844.
doi: 10.1142/S0218127403008259. |
[24] |
K. Pilgrim, Cylinders for Iterated Rational Maps, Ph.D. thesis, University of California, Berkeley, 1994. |
[25] |
K. Pilgrim and L. Tan,
Combining rational maps and controlling obstructions, Ergodic Theory and Dynamical Systems, 18 (1998), 221-245.
doi: 10.1017/S0143385798100329. |
[26] |
F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, in Dynamical Systems and Ergodic Theory (ed. K. Krzyzewski), Polish Scientific Publishers, Warszawa, 1989,229–235. |
[27] |
J. Rückert and D. Schleicher,
On Newton's method for entire functions, J. London Math. Soc., 75 (2007), 659-676.
doi: 10.1112/jlms/jdm046. |
[28] |
M. Shishikura, The connectivity of the Julia set and fixed points, in Complex Dynamics: Families and Friends, A. K. Peters, Ltd., Massachusetts, 2009,257–276.
doi: 10.1201/b10617. |
[29] |
S. Sutherland, Finding Roots of Complex Polynomials with Newton's Method, Ph.D. thesis, Boston University, 1989. |
[30] |
L. Tan and Y. Yin,
Local connectivity of the Julia set for geometrically finite rational maps, Science China Mathematics, 39 (1996), 39-47.
|
show all references
References:
[1] |
G. Ble, A. Douady and C. Henriksen, Round annuli., Contemporary Mathematics: In the Tradition of Ahlfors and Bers, III, 355 (2004), 71–76.
doi: 10.1090/conm/355/06445. |
[2] |
K. Bogdanov, K. Mamayusupov, S. Mukherjee and D. Schleicher,
Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori, Nonlinearity, 30 (2017), 3241-3254.
doi: 10.1088/1361-6544/aa79cf. |
[3] |
A. Bonifant, X. Buff and J. Milnor, On antipode preserving cubic maps, work in progress, 2015. http://www.math.stonybrook.edu/ jack/bbm.pdf Google Scholar |
[4] |
A. Bonifant, X. Buff and J. Milnor,
Antipode preserving cubic maps: The Fjord theorem, Proc. London Math. Soc., 116 (2018), 670-728.
doi: 10.1112/plms.12075. |
[5] |
B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, Vol. 141, Cambridge University Press, Cambridge, 2014.
![]() |
[6] |
X. Buff, G. Cui and L. Tan, Teichmüller spaces and holomorphic dynamics, in Handbook of Teichmüller theory Vol. 4, Société mathématique européenne, 2014,717–756.
doi: 10.4171/117-1/17. |
[7] |
X. Buff and A. L. Epstein,
A parabolic Pommerenke-Levin-Yoccoz inequality, Fund. Math., 172 (2002), 249-289.
doi: 10.4064/fm172-3-3. |
[8] |
J. Canela, N. Fagella and A. Garijo,
On a family of rational perturbations of the doubling map, Journal of Difference Equations and Applications, 21 (2015), 715-741.
doi: 10.1080/10236198.2015.1050387. |
[9] |
A. Douady,
Does a Julia set depend continuously on the polynomial?, Complex dynamical systems, Proc. Sympos. Appl. Math., 49 (1994), 91-138.
doi: 10.1090/psapm/049/1315535. |
[10] |
A. Douady and J. Hubbard,
A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297.
doi: 10.1007/BF02392534. |
[11] |
J. Hubbard, D. Schleicher and S. Sutherland,
How to find all roots of complex polynomials by Newton's method, Inventiones Mathematicae, 146 (2001), 1-33.
doi: 10.1007/s002220100149. |
[12] |
J. H. Hubbard and D. Schleicher, Multicorns are not path connected, in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday, Princeton University Press, 2014, 73–102. |
[13] |
H. Inou and S. Mukherjee, Discontinuity of straightening in antiholomorphic dynamics, preprint, 2016, arXiv: 1605.08061. Google Scholar |
[14] |
H. Inou and S. Mukherjee,
Non-landing parameter rays of the multicorns, Inventiones Mathematicae, 204 (2016), 869-893.
doi: 10.1007/s00222-015-0627-3. |
[15] |
T. Lei, Local properties of the Mandelbrot set at parabolic points, in The Mandelbrot Set, Theme and Variations, Lecture Note Series, Vol. 274, London Mathematical Society, Cambridge, 2000, 130–160. |
[16] |
R. Lodge, Y. Mikulich and D. Schleicher, A classification of postcritically finite Newton maps, preprint, arXiv: 1510.02771. Google Scholar |
[17] |
R. Lodge, Y. Mikulich and D. Schleicher, Combinatorial properties of Newton maps, preprint, arXiv: 1510.02761. Google Scholar |
[18] |
J. Milnor,
On rational maps with two critical points, Experiment. Math., 9 (2000), 333-411.
doi: 10.1080/10586458.2000.10504657. |
[19] |
J. Milnor, Dynamics in one complex variable, 3rd edition, Princeton University Press, NJ, 2006.
![]() |
[20] |
J. Milnor, Hyperbolic components in spaces of polynomial maps, with an appendix by A. Poirier, in Conformal Dynamics and Hyperbolic Geometry, Contemporary Mathematics, Vol. 573, American Mathematical Society, Providence, RI, 2012,183–232.
doi: 10.1090/conm/573/11428. |
[21] |
S. Mukherjee, S. Nakane and D. Schleicher,
On Multicorns and Unicorns II: Bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory and Dynamical Systems, 37 (2015), 859-899.
doi: 10.1017/etds.2015.65. |
[22] |
V. A. Naishul,
Topological invariants of analytic and area preserving mappings and their applications to analytic differential equations in $\mathbb{C}^2$ and $\mathbb{C}\mathbb{P}^2$, Transactions of the Moscow Mathematical Society, 42 (1983), 239-250.
|
[23] |
S. Nakane and D. Schleicher,
On Multicorns and Unicorns I : Antiholomorphic dynamics, hyperbolic components and real cubic polynomials, International Journal of Bifurcation and Chaos, 13 (2003), 2825-2844.
doi: 10.1142/S0218127403008259. |
[24] |
K. Pilgrim, Cylinders for Iterated Rational Maps, Ph.D. thesis, University of California, Berkeley, 1994. |
[25] |
K. Pilgrim and L. Tan,
Combining rational maps and controlling obstructions, Ergodic Theory and Dynamical Systems, 18 (1998), 221-245.
doi: 10.1017/S0143385798100329. |
[26] |
F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, in Dynamical Systems and Ergodic Theory (ed. K. Krzyzewski), Polish Scientific Publishers, Warszawa, 1989,229–235. |
[27] |
J. Rückert and D. Schleicher,
On Newton's method for entire functions, J. London Math. Soc., 75 (2007), 659-676.
doi: 10.1112/jlms/jdm046. |
[28] |
M. Shishikura, The connectivity of the Julia set and fixed points, in Complex Dynamics: Families and Friends, A. K. Peters, Ltd., Massachusetts, 2009,257–276.
doi: 10.1201/b10617. |
[29] |
S. Sutherland, Finding Roots of Complex Polynomials with Newton's Method, Ph.D. thesis, Boston University, 1989. |
[30] |
L. Tan and Y. Yin,
Local connectivity of the Julia set for geometrically finite rational maps, Science China Mathematics, 39 (1996), 39-47.
|












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