Figure 2.
For every parameter in the shaded region, the two free critical points of $ N_a $ are complex conjugate. The upper red component is denoted by $ \mathcal{U} $
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April
2021, 41(4): 1755-1797.
doi: 10.3934/dcds.2020340
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.
Mathematics Subject Classification:
37F10, 37F20, 37F46, 30D05.
Citation:
Russell Lodge, Sabyasachi Mukherjee. Invisible tricorns in real slices of rational maps. Discrete and Continuous Dynamical Systems,
2021, 41
(4)
: 1755-1797.
doi: 10.3934/dcds.2020340
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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. |
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A. Bonifant, X. Buff and J. Milnor, On antipode preserving cubic maps, work in progress, 2015. http://www.math.stonybrook.edu/ jack/bbm.pdf |
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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. |
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B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, Vol. 141, Cambridge University Press, Cambridge, 2014.
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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. |
| [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. |
| [17] |
R. Lodge, Y. Mikulich and D. Schleicher, Combinatorial properties of Newton maps, preprint, arXiv: 1510.02761. |
| [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 |
| [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. |
| [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. |
| [17] |
R. Lodge, Y. Mikulich and D. Schleicher, Combinatorial properties of Newton maps, preprint, arXiv: 1510.02761. |
| [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.
|
Figure 1.
A part of the parameter plane of the family $ \mathcal{N}_4^* $ that contains a Tricorn component (enclosed by white curves) with two visible and one invisible boundary arcs. The white region at the bottom represents the complement of $ \mathcal{U} $
Figure 10.
Left: A Tricorn component (enclosed by white parabolic arcs) with an invisible parabolic arc. Right: A blow-up of the parameter plane near the invisible parabolic arc (on the left) shows an invisible Tricorn component (enclosed by white parabolic arcs)
Figure 11.
Left: A tongue component of period $ 2 $ with an invisible co-root parabolic arc $ \mathcal{C}_2 $ . The parabolic arcs $ \mathcal{C}_1 $ and $ \mathcal{C}_3 $ contain bare regions. Right: A bounded Tricorn component with all three co-root parabolic arcs invisible
Figure 9.
The $ N_a^{\circ 2n} $ -orbit of $ c_a $ exits through the gate and lands in $ W_a $ , which is contained in the basin $ \mathcal{B}_1 $
Figure 3.
The main cardioid of the "baby Mandelbrot set" (in black) is a Mandelbrot component
Figure 5.
Subdivision rule that defines the sequence of critically finite branched covers $ f_{n}: \mathbb{S}^2\to \mathbb{S}^2 $ where $ n>1 $ . Black edges $ \alpha, \beta, \gamma, \delta $ terminate at $ \infty $ in both the domain (top) and the range (bottom). Light blue edges in the domain represent preimage edges of $ f_{n} $ that are not edges in the range. Both domain and range graphs are symmetric about the horizontal and vertical coordinate axes.
Figure 4.
The dynamical plane of the map in $ \mathcal{N}_4^* $ corresponding to $ n = 3 $ from Proposition 5.3. The basins of $ 1 $ and $ -1 $ are colored red and green. The non-fixed critical points are in a period 6 cycle indicated by the white arrows
Figure 6.
Left: The characteristic Fatou component of a parameter belonging to a Tricorn component. The three dynamical co-roots, two of which are visible and one invisible, are marked. Right: The characteristic Fatou component of a parameter belonging to a Tricorn component. The three dynamical co-roots, each of which is invisible, are marked
Figure 7.
Left: A blow-up of the dynamical plane (of a parameter on a parabolic arc of a Tricorn component) around a visible characteristic parabolic point. Here, the characteristic parabolic point is on the boundary of $ \mathcal{B}^{\mathrm{imm}}_{-1} $ . Right: A blow-up of the dynamical plane (of a parameter on a parabolic arc of a Tricorn component) around an invisible characteristic parabolic point. Here, the characteristic point is in the accumulation set of the pre-periodic Fatou components of $ \mathcal{B}_1 $ (and of $ \mathcal{B}_{-1} $ ), but not on the boundary of any single component thereof. Consequently, there is a 'Julia path' in the repelling cylinder connecting $ \partial U_1^+ $ and $ \partial U_1^- $
Figure 8.
A blow-up of the dynamical plane of a near-parabolic parameter showing the eggbeater dynamics "near" an invisible parabolic point
Figure 12.
Left: The dynamical plane of a parameter on the co-root arc $ \mathcal{C}_2 $ of a tongue of period $ 2 $ . The only dynamical co-root on the boundary of $ U_1 $ is $ p_2(U_1) $ (which is also the characteristic parabolic point for parameters on $ \mathcal{C}_2 $ ), and it is invisible. The dynamical roots $ p_1(U_1) $ and $ p_3(U_1) $ on $ \partial U_1 $ are visible. Right: The dynamical plane of a parameter on a co-root arc $ \mathcal{C}_2 $ of a bounded Tricorn component. For this parameter, each dynamical co-root $ p_k(U_1) $ is invisible. In particular, the characteristic parabolic point $ p_2(U_1) $ is invisible (marked in black)
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