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May  2017, 37(5): 2565-2588. doi: 10.3934/dcds.2017110

Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves

1. 

Jacobs University Bremen, Campus Ring 1, Bremen 28759, Germany

2. 

Institute for Mathematical Sciences, Stony Brook University, Stony Brook, 11794, NY, USA

Received  May 2016 Revised  January 2017 Published  February 2017

Fund Project: The author was supported by Deutsche Forschungsgemeinschaft DFG.

The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves $ \mathrm{Per}_n(1)$ of a suitably complexified family of polynomials. As another application of this deformation step, we show that the dynamically natural parametrization of the parabolic arcs has a non-vanishing derivative at all but (possibly) finitely many points.

We also look at the algebraic sets $ \mathrm{Per}_n(1)$ in various families of polynomials, the nature of their singularities, and the 'dynamical' behavior of these singular parameters.

Citation: Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110
References:
[1]

S. Basu, R. Pollack and M. -F. Coste-Roy, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 2003. doi: 10.1007/978-3-662-05355-3.

[2]

A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems Series: Graduate Texts in Mathematics, Vol. 132, Springer-Verlag, 1991.

[3]

W. Bergweiler and A. Eremenko, Green's function and anti-holomorphic dynamics on a torus, Proc. Amer. Math. Soc., 144 (2016), 2911-2922.  doi: 10.1090/proc/13044.

[4]

A. Bonifant, X. Buff and J. Milnor, Antipode preserving cubic maps: The fjord theorem, preprint, arXiv: 1512.01850.

[5]

A. Bonifant, X. Buff and J. Milnor, Antipode preserving cubic maps Ⅱ: tongues and the ring locus, work in progress.

[6]

J. CanelaN. 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.

[7]

M. Denker and M. Urbanski, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118.  doi: 10.1112/jlms/s2-43.1.107.

[8]

N. Dobbs, Nice sets and invariant densities in complex dynamics, Math. Proc. Cambridge Philos. Soc., 150 (2011), 157-165.  doi: 10.1017/S0305004110000265.

[9]

D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd Edition, John Wiley and Sons, Inc. , 2003.

[10]

J. H. Hubbard and D. Schleicher, Multicorns are not path connected, in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday (eds. A. Bonifant, M. Lyubich and S. Sutherland), Princeton University Press, (2014), 73-102 doi: 10.1515/9781400851317-007.

[11]

H. Inou and J. Kiwi, Combinatorics and topology of straightening maps, Ⅰ: Compactness and bijectivity, Advances in Mathematics, 231 (2012), 2666-2733.  doi: 10.1016/j.aim.2012.07.014.

[12]

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.

[13]

H. Inou and S. Mukherjee, Discontinuity of straightening in antiholomorphic dynamics, arXiv: 1605.08061.

[14] F. Kirwan, Complex Algebraic Curves, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511623929.
[15] D. R. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543050.
[16]

C. T. McMullen, Hausdorff dimension and conformal dynamics Ⅱ: Geometrically finite rational maps, Commentarii Mathematici Helvetici, 75 (2000), 535-593.  doi: 10.1007/s000140050140.

[17] J. Milnor, Dynamics in one Complex Variable, 3rd Edition, Princeton University Press, New Jersey, 2006. 
[18]

J. Milnor, Remarks on iterated cubic maps, Experiment. Math., 1 (1992), 5-24. 

[19]

J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies. Princeton University Press, New Jersey, 1968.

[20]

S. Mukherjee, S. Nakane and D. Schleicher, On multicorns and unicorns Ⅱ: Bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory and Dynamical Systems, to appear, 2015, http://dx.doi.org/10.1017/etds.2015.65 doi: 10.1017/etds.2015.65.

[21]

S. Mukherjee, Orbit portraits of unicritical antiholomorphic polynomials, Conformal Geometry and Dynamics of the AMS, 19 (2015), 35-50.  doi: 10.1090/S1088-4173-2015-00276-3.

[22]

S. Nakane, Connectedness of the tricorn, Ergodic Theory and Dynamical Systems, 13 (1993), 349-356.  doi: 10.1017/S0143385700007409.

[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]

J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no-growth condition, Ergodic Theory and Dynamical Systems, 27 (2007), 595-636.  doi: 10.1017/S0143385706000629.

[25]

D. Ruelle, Repellers for real analytic maps, Turbulence, Strange Attractors and Chaos, (1995), 351-359.  doi: 10.1142/9789812833709_0023.

[26]

B. Skorulski and M. Urbanski, Finer fractal geometry for analytic families of conformal dynamical systems, Dynamical Systems, 29 (2014), 369-398.  doi: 10.1080/14689367.2014.903385.

[27]

M. Urbanski, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc., 40 (2003), 281-321.  doi: 10.1090/S0273-0979-03-00985-6.

[28]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1979), 937-971.  doi: 10.2307/2373682.

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Volume 79, Springer, 1982.

[30]

C. T. C. Wall, Singular Points of Plane Curves, London Mathematical Society Student Texts (vol. 63), Cambridge University Press, 2004. doi: 10.1017/CBO9780511617560.

[31]

M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, SMF/AMS Texts and Monographs, Volume 2,2000.

show all references

References:
[1]

S. Basu, R. Pollack and M. -F. Coste-Roy, Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 2003. doi: 10.1007/978-3-662-05355-3.

[2]

A. F. Beardon, Iteration of Rational Functions, Complex Analytic Dynamical Systems Series: Graduate Texts in Mathematics, Vol. 132, Springer-Verlag, 1991.

[3]

W. Bergweiler and A. Eremenko, Green's function and anti-holomorphic dynamics on a torus, Proc. Amer. Math. Soc., 144 (2016), 2911-2922.  doi: 10.1090/proc/13044.

[4]

A. Bonifant, X. Buff and J. Milnor, Antipode preserving cubic maps: The fjord theorem, preprint, arXiv: 1512.01850.

[5]

A. Bonifant, X. Buff and J. Milnor, Antipode preserving cubic maps Ⅱ: tongues and the ring locus, work in progress.

[6]

J. CanelaN. 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.

[7]

M. Denker and M. Urbanski, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118.  doi: 10.1112/jlms/s2-43.1.107.

[8]

N. Dobbs, Nice sets and invariant densities in complex dynamics, Math. Proc. Cambridge Philos. Soc., 150 (2011), 157-165.  doi: 10.1017/S0305004110000265.

[9]

D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd Edition, John Wiley and Sons, Inc. , 2003.

[10]

J. H. Hubbard and D. Schleicher, Multicorns are not path connected, in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday (eds. A. Bonifant, M. Lyubich and S. Sutherland), Princeton University Press, (2014), 73-102 doi: 10.1515/9781400851317-007.

[11]

H. Inou and J. Kiwi, Combinatorics and topology of straightening maps, Ⅰ: Compactness and bijectivity, Advances in Mathematics, 231 (2012), 2666-2733.  doi: 10.1016/j.aim.2012.07.014.

[12]

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.

[13]

H. Inou and S. Mukherjee, Discontinuity of straightening in antiholomorphic dynamics, arXiv: 1605.08061.

[14] F. Kirwan, Complex Algebraic Curves, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511623929.
[15] D. R. Mauldin and M. Urbanski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511543050.
[16]

C. T. McMullen, Hausdorff dimension and conformal dynamics Ⅱ: Geometrically finite rational maps, Commentarii Mathematici Helvetici, 75 (2000), 535-593.  doi: 10.1007/s000140050140.

[17] J. Milnor, Dynamics in one Complex Variable, 3rd Edition, Princeton University Press, New Jersey, 2006. 
[18]

J. Milnor, Remarks on iterated cubic maps, Experiment. Math., 1 (1992), 5-24. 

[19]

J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies. Princeton University Press, New Jersey, 1968.

[20]

S. Mukherjee, S. Nakane and D. Schleicher, On multicorns and unicorns Ⅱ: Bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory and Dynamical Systems, to appear, 2015, http://dx.doi.org/10.1017/etds.2015.65 doi: 10.1017/etds.2015.65.

[21]

S. Mukherjee, Orbit portraits of unicritical antiholomorphic polynomials, Conformal Geometry and Dynamics of the AMS, 19 (2015), 35-50.  doi: 10.1090/S1088-4173-2015-00276-3.

[22]

S. Nakane, Connectedness of the tricorn, Ergodic Theory and Dynamical Systems, 13 (1993), 349-356.  doi: 10.1017/S0143385700007409.

[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]

J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no-growth condition, Ergodic Theory and Dynamical Systems, 27 (2007), 595-636.  doi: 10.1017/S0143385706000629.

[25]

D. Ruelle, Repellers for real analytic maps, Turbulence, Strange Attractors and Chaos, (1995), 351-359.  doi: 10.1142/9789812833709_0023.

[26]

B. Skorulski and M. Urbanski, Finer fractal geometry for analytic families of conformal dynamical systems, Dynamical Systems, 29 (2014), 369-398.  doi: 10.1080/14689367.2014.903385.

[27]

M. Urbanski, Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc., 40 (2003), 281-321.  doi: 10.1090/S0273-0979-03-00985-6.

[28]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1979), 937-971.  doi: 10.2307/2373682.

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Volume 79, Springer, 1982.

[30]

C. T. C. Wall, Singular Points of Plane Curves, London Mathematical Society Student Texts (vol. 63), Cambridge University Press, 2004. doi: 10.1017/CBO9780511617560.

[31]

M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, SMF/AMS Texts and Monographs, Volume 2,2000.

Figure 1.  $\mathcal{M}_2^*$, also known as the tricorn and the parabolic arcs on the boundary of the hyperbolic component of period 1 (in blue)
Figure 2.  Pictorial representation of the image of $\left[0,1\right]$ under the quasiconformal map $L_w$; for $w=1+i/8$ (top) and $w=1$ (bottom). The Fatou coordinates of $c_0$ and $f_{c_0}^{\circ k} (c_0)$ are $1/4$ and $3/4$ respectively. For $w=1+i/8$, $L_w(1/4)=1/8+i$ and $L_w(3/4)=7/8-i$, and for $w=1$, $L_w(1/4)=1/4+i$ and $L_w(3/4)=3/4-i$. Observe that $L_w$ commutes with $z\mapsto \overline{z}+1/2$ only when $w\in \mathbb{R}$
Figure 3.  $\pi_2 \circ F : w \mapsto b(w)$ is injective in a neighborhood of $\widetilde{u}$ for all but possibly finitely many $\widetilde{u} \in \mathbb{R}$
Figure 4.  The outer yellow curve indicates part of $\mathrm{Per}_1(1)\cap \lbrace a=\overline{b}\rbrace$, and the inner blue curve (along with the red point) indicates part of the deformation $\mathrm{Per}_1(r)\cap \lbrace a=\overline{b}\rbrace$ for some $r\in (1-\epsilon,1)$. The cusp point $c_0$ on the yellow curve is a critical point of $h_1$, i.e. a singular point of $\mathrm{Per}_1(1)$, and the red point is a critical point of $h_r$; i.e a singular point of $\mathrm{Per}_1(r)$
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