# American Institute of Mathematical Sciences

January  2020, 40(1): 33-46. doi: 10.3934/dcds.2020002

## Hereditarily non uniformly perfect non-autonomous Julia sets

 1 Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA 2 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA 3 Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan

Received  October 2018 Revised  June 2019 Published  October 2019

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15].

Citation: Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002
##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.  Google Scholar [2] Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034.  Google Scholar [3] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.  Google Scholar [4] 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 [5] M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.  Google Scholar [6] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.  Google Scholar [7] A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar [8] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.  Google Scholar [9] J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.  Google Scholar [10] A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192.  Google Scholar [11] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.   Google Scholar [12] R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2.  Google Scholar [13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.   Google Scholar [14] C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar [15] L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490.  Google Scholar [16] O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.   Google Scholar [17] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar [18] R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar [19] R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150.  Google Scholar [20] H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar [21] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286.  Google Scholar [22] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.  Google Scholar [23] H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923.  Google Scholar [24] W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.  Google Scholar

show all references

##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.  Google Scholar [2] Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034.  Google Scholar [3] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.  Google Scholar [4] 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 [5] M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.  Google Scholar [6] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.  Google Scholar [7] A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. Google Scholar [8] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.  Google Scholar [9] J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.  Google Scholar [10] A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192.  Google Scholar [11] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.   Google Scholar [12] R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2.  Google Scholar [13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.   Google Scholar [14] C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3.  Google Scholar [15] L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490.  Google Scholar [16] O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.   Google Scholar [17] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2.  Google Scholar [18] R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583.  Google Scholar [19] R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150.  Google Scholar [20] H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302.  Google Scholar [21] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286.  Google Scholar [22] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.  Google Scholar [23] H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923.  Google Scholar [24] W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.  Google Scholar
How the survival sets ${\mathcal S}_k$ are nested. The pictures show preimages of $\overline {\mathrm D}(0,2)$ at stages $M_k$ (in red) and $M_{k-1}$ (in blue) with $m_k = 3$. The dashed blue circle is ${\mathrm C}(0,2)$ while the unit circle is shown in black. Observe how $Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1)$ as in Remark 1(c) is shown in red at Stage $M_{k-1}$
Schematic for the proof of Theorem 1.6 in the case where $\limsup |c_k| = +\infty$. Note how the round annulus ${\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|})$ at stage $M_{k-1} + m_k$ (in this case $M_1+m_2$) is pulled back conformally first by the preimage branches of $Q_{M_{k-1},M_{k-1}+m_k}$ to form half the members of the collection $\mathcal C$ at Stage $M_1$. Then the preimage branches of $Q_{M_{k-1}}$ pull back the annuli in $\mathcal C$ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $\mathcal{S}_k$ at stage $0$
 [1] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 [2] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [3] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [4] Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61 [5] Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449 [6] Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134 [7] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [8] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [9] Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021 [10] Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004 [11] John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004 [12] Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017 [13] Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161 [14] Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

2019 Impact Factor: 1.338

## Metrics

• PDF downloads (96)
• HTML views (111)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]