December  2019, 12(8): 2253-2277. doi: 10.3934/dcdss.2019144

Orbit portraits in non-autonomous iteration

5 Lipppitt Road, Room 200, Kingston, RI 02881, USA

* Corresponding author: Mark Comerford

Received  July 2016 Revised  December 2017 Published  January 2019

We extend the definition of an orbit portrait to the context of non-autonomous iteration, both for the combinatorial version involving collections of angles, and for the dynamic version involving external rays where combinatorial portraits can be realized by the dynamics associated with sequences of polynomials with suitably uniformly bounded degrees and coefficients. We show that, in the case of sequences of polynomials of constant degree, the portraits which arise are eventually periodic which is somewhat similar to the classical theory of polynomial iteration. However, if the degrees of the polynomials in the sequence are allowed to vary, one can obtain portraits with complementary arcs of irrational length which are fundamentally different from the classical ones.

Citation: Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144
References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.

[2]

A. BlokhJ. MaloughJ. MayerL. Oversteeegen and D. Parris, Rotational subsets of the circle under $ z^d$, Topology Appl., 153 (2006), 1540-1570.  doi: 10.1016/j.topol.2005.04.010.

[3]

R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.

[4]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $ z^2+c_n$, Pacific J. Math., 198 (2001), 347-372.  doi: 10.2140/pjm.2001.198.347.

[5]

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

[6]

L. CarlesonP. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Bras. Mat., 25 (1994), 1-30.  doi: 10.1007/BF01232933.

[7]

M. Comerford and T. Woodard, Preservation of external rays in non-autonomous iteration, J. Difference Equ. Appl., 19 (2013), 585-604.  doi: 10.1080/10236198.2012.662966.

[8]

M. Comerford, Infinitely many grand orbits, Michigan Math. J., 51 (2003), 47-57.  doi: 10.1307/mmj/1049832892.

[9]

M. Comerford, A survey of results in random iteration, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Part 1, Proc. Sympos. Pure Math., 72 (2004), 435-476. 

[10]

M. Comerford, Conjugacy and counterexample in random iteration, Pacific J. Math., 211 (2003), 69-80.  doi: 10.2140/pjm.2003.211.69.

[11]

M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynam. Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.

[12]

D. Cosper, J. Houghton, J. Mayer, L. Mernik and J. Olson, Central strips of sibling leaves in laminations of the unit disk, Topology Proc., 48 (2016), 69-100, arXiv: 1408.0223.

[13]

A. Douady and J. Hubbard, Etude dynamique des polynômes complexes, (French) [Dynamical Study of Complex Polynomials], Publications Math. d'Orsay, Orsay, France, 1984, 75pp.

[14]

E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.

[15]

M Lyubich, J. Milnor and Y. Minsky, Laminations and Foliations in Dynamics, Geometry and Topology, Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/conm/269.

[16]

J. Milnor, Dynamics in One Complex Variable, Vieweg 1999, 2000; Princeton University Press, Princeton New Jersey, 2006.

[17]

J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: an expository account, Gèométrie complexe et systèmes dynamiques, Astérisques, 261 (2000), 277-333.

[18]

R. Näkki and J. Väiäsäla, John disks, Expo. Math., 9 (1991), 3-43. 

[19]

O. Sester, Hyperbolicité des polynômes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.

[20]

O. Sester, Combinatorial configurations of fibered polynomials, (French) [Hyperbolicity of fibered polynomials], Ergodic Theory Dynam. Systems, 21 (2001), 915–955. doi: 10.1017/S0143385701001456.

[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.

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.

[23]

H. Sumi, Erratum to 'Semi-hyperbolic fibered rational maps and rational semigroups', Ergodic Theory Dynam. Systems, 28 (2008), 1043-1045. 

[24]

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.

show all references

References:
[1]

A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991.

[2]

A. BlokhJ. MaloughJ. MayerL. Oversteeegen and D. Parris, Rotational subsets of the circle under $ z^d$, Topology Appl., 153 (2006), 1540-1570.  doi: 10.1016/j.topol.2005.04.010.

[3]

R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.

[4]

R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $ z^2+c_n$, Pacific J. Math., 198 (2001), 347-372.  doi: 10.2140/pjm.2001.198.347.

[5]

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

[6]

L. CarlesonP. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Bras. Mat., 25 (1994), 1-30.  doi: 10.1007/BF01232933.

[7]

M. Comerford and T. Woodard, Preservation of external rays in non-autonomous iteration, J. Difference Equ. Appl., 19 (2013), 585-604.  doi: 10.1080/10236198.2012.662966.

[8]

M. Comerford, Infinitely many grand orbits, Michigan Math. J., 51 (2003), 47-57.  doi: 10.1307/mmj/1049832892.

[9]

M. Comerford, A survey of results in random iteration, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Part 1, Proc. Sympos. Pure Math., 72 (2004), 435-476. 

[10]

M. Comerford, Conjugacy and counterexample in random iteration, Pacific J. Math., 211 (2003), 69-80.  doi: 10.2140/pjm.2003.211.69.

[11]

M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynam. Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.

[12]

D. Cosper, J. Houghton, J. Mayer, L. Mernik and J. Olson, Central strips of sibling leaves in laminations of the unit disk, Topology Proc., 48 (2016), 69-100, arXiv: 1408.0223.

[13]

A. Douady and J. Hubbard, Etude dynamique des polynômes complexes, (French) [Dynamical Study of Complex Polynomials], Publications Math. d'Orsay, Orsay, France, 1984, 75pp.

[14]

E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.

[15]

M Lyubich, J. Milnor and Y. Minsky, Laminations and Foliations in Dynamics, Geometry and Topology, Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2001. doi: 10.1090/conm/269.

[16]

J. Milnor, Dynamics in One Complex Variable, Vieweg 1999, 2000; Princeton University Press, Princeton New Jersey, 2006.

[17]

J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: an expository account, Gèométrie complexe et systèmes dynamiques, Astérisques, 261 (2000), 277-333.

[18]

R. Näkki and J. Väiäsäla, John disks, Expo. Math., 9 (1991), 3-43. 

[19]

O. Sester, Hyperbolicité des polynômes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.

[20]

O. Sester, Combinatorial configurations of fibered polynomials, (French) [Hyperbolicity of fibered polynomials], Ergodic Theory Dynam. Systems, 21 (2001), 915–955. doi: 10.1017/S0143385701001456.

[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.

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532.

[23]

H. Sumi, Erratum to 'Semi-hyperbolic fibered rational maps and rational semigroups', Ergodic Theory Dynam. Systems, 28 (2008), 1043-1045. 

[24]

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.

Figure 1.  A critical arc covering a critical value arc
Figure 2.  A degree $ 6 $ example with critical arc $ (\tfrac{11}{36}, \tfrac{7}{36}) $ and critical value arc $ (\tfrac{5}{6}, \tfrac{1}{6}) $
Figure 4.  The dynamic portrait at times $ 0 $ and $ 1 $ for the example in Proposition 2
Figure 3.  The Juila set for $ P(z) = z^3 + \tfrac{3}{2}z $
Figure 5.  The critical arc at time $ m_0 - 1 $ in the cubic case (in pink)
Figure 6.  The critical arc at time $ m_0 - 1 $ in the quadratic case (in pink)
Figure 7.  The Julia Set of the Hyperbolic Semigroup $ \left\langle {P_0 \circ P_0, P_1 \circ P_0} \right\rangle $
[1]

Sergey Zelik. Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 781-810. doi: 10.3934/dcdsb.2015.20.781

[2]

T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265

[3]

Shan Ma, Chengkui Zhong. The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 53-70. doi: 10.3934/dcds.2007.18.53

[4]

Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations and Control Theory, 2022, 11 (1) : 41-65. doi: 10.3934/eect.2020102

[5]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure and Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415

[6]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[7]

Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010

[8]

JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022053

[9]

Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79

[10]

Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351

[11]

Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281

[12]

Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231

[13]

Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002

[14]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[15]

Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067

[16]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

[17]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701

[18]

Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231

[19]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[20]

Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (186)
  • HTML views (698)
  • Cited by (0)

Other articles
by authors

[Back to Top]