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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 & Continuous Dynamical Systems - A, 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. 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. [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. 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. [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.
$\mathcal{M}_2^*$, also known as the tricorn and the parabolic arcs on the boundary of the hyperbolic component of period 1 (in blue)
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 $w\in \mathbb{R}$
$\pi_2 \circ F : w \mapsto b(w)$ is injective in a neighborhood of $\widetilde{u}$ for all but possibly finitely many $\widetilde{u} \in \mathbb{R}$
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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