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August  2015, 35(8): 3293-3313. doi: 10.3934/dcds.2015.35.3293

On the Hausdorff dimension of the Sierpiński Julia sets

Krzysztof Barański 1, and  Michał Wardal 1,

1. 

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Received  March 2014 Revised  December 2014 Published  February 2015

We estimate the Hausdorff dimension of hyperbolic Julia sets of maps from the well-known family $F_{\lambda,n}(z) = z^n + \lambda/z^n$, $n \ge 2$, $\lambda \in \mathbb{C} \setminus \{0\}$. In particular, we show that $\dim_H J(F_{\lambda,n}) = \mathcal O (1/\ln |\lambda|)$ for large $|\lambda|$, and $\dim_H J(F_{\lambda,n}) = 1 + \mathcal O (1/\ln n)$ for large $n$ in the three cases: when $J(F_{\lambda,n})$ is a Cantor set, a Cantor set of quasicircles and a Sierpiński curve.
Keywords: Hausdorff dimension, Sierpiński Julia sets, Bowen formula..
Mathematics Subject Classification: Primary: 37F10, 37F35; Secondary: 28A78, 28A8.
Citation: Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
References:
[1]

P. Blanchard, R. L. Devaney, D. M. Look, P. Seal and Y. Shapiro, Sierpinski-curve Julia sets and singular perturbations of complex polynomials, Ergodic Theory Dynam. Systems, 25 (2005), 1047-1055. doi: 10.1017/S0143385704000380.  Google Scholar

[2]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25, URL http://www.numdam.org/item?id=PMIHES_1979__50__11_0.  Google Scholar

[3]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

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R. L. Devaney, Cantor and Sierpinski, Julia and Fatou: Complex topology meets complex dynamics, Notices Amer. Math. Soc., 51 (2004), 9-15.  Google Scholar

[5]

R. L. Devaney, Cantor necklaces and structurally unstable Sierpinski curve Julia sets for rational maps, Qual. Theory Dyn. Syst., 5 (2004), 337-359. doi: 10.1007/BF02972685.  Google Scholar

[6]

R. L. Devaney, Structure of the McMullen domain in the parameter planes for rational maps, Fund. Math., 185 (2005), 267-285. doi: 10.4064/fm185-3-5.  Google Scholar

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R. L. Devaney, Singular perturbations of complex polynomials, Bull. Amer. Math. Soc. (N.S.), 50 (2013), 391-429. doi: 10.1090/S0273-0979-2013-01410-1.  Google Scholar

[8]

R. L. Devaney, A Cantor-Mandelbrot-Sierpiński tree in the parameter plane for rational maps, Trans. Amer. Math. Soc., 366 (2014), 1095-1117. doi: 10.1090/S0002-9947-2013-05948-X.  Google Scholar

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R. L. Devaney and A. Garijo, Julia sets converging to the unit disk, Proc. Amer. Math. Soc., 136 (2008), 981-988 (electronic). doi: 10.1090/S0002-9939-07-09084-3.  Google Scholar

[10]

R. L. Devaney, M. Holzer, D. M. Look, M. Moreno Rocha and D. Uminsky, Singular perturbations of $z^n$, in Transcendental dynamics and complex analysis, vol. 348 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, (2008), 111-137. doi: 10.1017/CBO9780511735233.007.  Google Scholar

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R. L. Devaney, K. Josić and Y. Shapiro, Singular perturbations of quadratic maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 161-169. doi: 10.1142/S0218127404009259.  Google Scholar

[12]

R. L. Devaney and D. M. Look, A criterion for Sierpinski curve Julia sets, Spring Topology and Dynamical Systems Conference, Topology Proc., 30 (2006), 163-179.  Google Scholar

[13]

R. L. Devaney, D. M. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.  Google Scholar

[14]

R. L. Devaney and S. M. Marotta, The McMullen domain: rings around the boundary, Trans. Amer. Math. Soc., 359 (2007), 3251-3273 (electronic). doi: 10.1090/S0002-9947-07-04137-2.  Google Scholar

[15]

R. L. Devaney and K. M. Pilgrim, Dynamic classification of escape time Sierpiński curve Julia sets, Fund. Math., 202 (2009), 181-198, doi: 10.4064/fm202-2-5.  Google Scholar

[16]

R. L. Devaney, M. M. Rocha and S. Siegmund, Rational maps with generalized Sierpinski gasket Julia sets, Topology Appl., 154 (2007), 11-27. doi: 10.1016/j.topol.2006.03.024.  Google Scholar

[17]

K. Falconer, Fractal Geometry, 2nd edition, John Wiley & Sons Inc., Hoboken, NJ, 2003, Mathematical foundations and applications. doi: 10.1002/0470013850.  Google Scholar

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P. Haïssinsky and K. M. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.  Google Scholar

[19]

O. Jenkinson and M. Pollicott, Calculating Hausdorff dimensions of Julia sets and Kleinian limit sets, Amer. J. Math., 124 (2002), 495-545, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.3jenkinson.pdf. doi: 10.1353/ajm.2002.0015.  Google Scholar

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P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995, Fractals and rectifiability. doi: 10.1017/CBO9780511623813.  Google Scholar

[21]

C. McMullen, Automorphisms of rational maps, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), vol. 10 of Math. Sci. Res. Inst. Publ., Springer, New York, (1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[22]

C. T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math., 120 (1998), 691-721, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.4mcmullen.pdf. doi: 10.1353/ajm.1998.0031.  Google Scholar

[23]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[24]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, vol. 371 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139193184.  Google Scholar

[25]

W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[26]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[27]

D. Ruelle, Thermodynamic Formalism, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004, The mathematical structures of equilibrium statistical mechanics. doi: 10.1017/CBO9780511617546.  Google Scholar

[28]

N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183 (electronic). doi: 10.1090/S1088-4173-06-00149-4.  Google Scholar

[29]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418. doi: 10.2307/1971308.  Google Scholar

[30]

E. Szpilrajn, La dimension et la mesure, Fund. Math., 28 (1937), 81-89. Google Scholar

[31]

X. Wang and F. Yang, The Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 551-562. doi: 10.1007/s12044-014-0203-6.  Google Scholar

show all references

References:
[1]

P. Blanchard, R. L. Devaney, D. M. Look, P. Seal and Y. Shapiro, Sierpinski-curve Julia sets and singular perturbations of complex polynomials, Ergodic Theory Dynam. Systems, 25 (2005), 1047-1055. doi: 10.1017/S0143385704000380.  Google Scholar

[2]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25, URL http://www.numdam.org/item?id=PMIHES_1979__50__11_0.  Google Scholar

[3]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[4]

R. L. Devaney, Cantor and Sierpinski, Julia and Fatou: Complex topology meets complex dynamics, Notices Amer. Math. Soc., 51 (2004), 9-15.  Google Scholar

[5]

R. L. Devaney, Cantor necklaces and structurally unstable Sierpinski curve Julia sets for rational maps, Qual. Theory Dyn. Syst., 5 (2004), 337-359. doi: 10.1007/BF02972685.  Google Scholar

[6]

R. L. Devaney, Structure of the McMullen domain in the parameter planes for rational maps, Fund. Math., 185 (2005), 267-285. doi: 10.4064/fm185-3-5.  Google Scholar

[7]

R. L. Devaney, Singular perturbations of complex polynomials, Bull. Amer. Math. Soc. (N.S.), 50 (2013), 391-429. doi: 10.1090/S0273-0979-2013-01410-1.  Google Scholar

[8]

R. L. Devaney, A Cantor-Mandelbrot-Sierpiński tree in the parameter plane for rational maps, Trans. Amer. Math. Soc., 366 (2014), 1095-1117. doi: 10.1090/S0002-9947-2013-05948-X.  Google Scholar

[9]

R. L. Devaney and A. Garijo, Julia sets converging to the unit disk, Proc. Amer. Math. Soc., 136 (2008), 981-988 (electronic). doi: 10.1090/S0002-9939-07-09084-3.  Google Scholar

[10]

R. L. Devaney, M. Holzer, D. M. Look, M. Moreno Rocha and D. Uminsky, Singular perturbations of $z^n$, in Transcendental dynamics and complex analysis, vol. 348 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, (2008), 111-137. doi: 10.1017/CBO9780511735233.007.  Google Scholar

[11]

R. L. Devaney, K. Josić and Y. Shapiro, Singular perturbations of quadratic maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 161-169. doi: 10.1142/S0218127404009259.  Google Scholar

[12]

R. L. Devaney and D. M. Look, A criterion for Sierpinski curve Julia sets, Spring Topology and Dynamical Systems Conference, Topology Proc., 30 (2006), 163-179.  Google Scholar

[13]

R. L. Devaney, D. M. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.  Google Scholar

[14]

R. L. Devaney and S. M. Marotta, The McMullen domain: rings around the boundary, Trans. Amer. Math. Soc., 359 (2007), 3251-3273 (electronic). doi: 10.1090/S0002-9947-07-04137-2.  Google Scholar

[15]

R. L. Devaney and K. M. Pilgrim, Dynamic classification of escape time Sierpiński curve Julia sets, Fund. Math., 202 (2009), 181-198, doi: 10.4064/fm202-2-5.  Google Scholar

[16]

R. L. Devaney, M. M. Rocha and S. Siegmund, Rational maps with generalized Sierpinski gasket Julia sets, Topology Appl., 154 (2007), 11-27. doi: 10.1016/j.topol.2006.03.024.  Google Scholar

[17]

K. Falconer, Fractal Geometry, 2nd edition, John Wiley & Sons Inc., Hoboken, NJ, 2003, Mathematical foundations and applications. doi: 10.1002/0470013850.  Google Scholar

[18]

P. Haïssinsky and K. M. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.  Google Scholar

[19]

O. Jenkinson and M. Pollicott, Calculating Hausdorff dimensions of Julia sets and Kleinian limit sets, Amer. J. Math., 124 (2002), 495-545, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.3jenkinson.pdf. doi: 10.1353/ajm.2002.0015.  Google Scholar

[20]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995, Fractals and rectifiability. doi: 10.1017/CBO9780511623813.  Google Scholar

[21]

C. McMullen, Automorphisms of rational maps, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), vol. 10 of Math. Sci. Res. Inst. Publ., Springer, New York, (1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[22]

C. T. McMullen, Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math., 120 (1998), 691-721, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.4mcmullen.pdf. doi: 10.1353/ajm.1998.0031.  Google Scholar

[23]

J. Milnor, Dynamics in One Complex Variable, vol. 160 of Annals of Mathematics Studies, 3rd edition, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[24]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, vol. 371 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9781139193184.  Google Scholar

[25]

W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[26]

D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems, 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar

[27]

D. Ruelle, Thermodynamic Formalism, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004, The mathematical structures of equilibrium statistical mechanics. doi: 10.1017/CBO9780511617546.  Google Scholar

[28]

N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183 (electronic). doi: 10.1090/S1088-4173-06-00149-4.  Google Scholar

[29]

D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2), 122 (1985), 401-418. doi: 10.2307/1971308.  Google Scholar

[30]

E. Szpilrajn, La dimension et la mesure, Fund. Math., 28 (1937), 81-89. Google Scholar

[31]

X. Wang and F. Yang, The Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 551-562. doi: 10.1007/s12044-014-0203-6.  Google Scholar

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