doi: 10.3934/dcdss.2019144

Orbit portraits in non-autonomous iteration

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* 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 & Continuous Dynamical Systems - S, 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 $
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