# American Institute of Mathematical Sciences

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

Received  January 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

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.

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
##### 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.

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.
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 $\mathcal{U}$
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 $\mathcal{U}$
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)
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
The $N_a^{\circ 2n}$-orbit of $c_a$ exits through the gate and lands in $W_a$, which is contained in the basin $\mathcal{B}_1$
The main cardioid of the "baby Mandelbrot set" (in black) is a Mandelbrot component
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.
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
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
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 $\partial U_1^-$
A blow-up of the dynamical plane of a near-parabolic parameter showing the eggbeater dynamics "near" an invisible parabolic point
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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