# American Institute of Mathematical Sciences

## 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. Google Scholar [2] A. Blokh, J. Malough, J. Mayer, L. 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. Google Scholar [3] R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252. doi: 10.1007/BF03321035. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. Carleson, P. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Bras. Mat., 25 (1994), 1-30. doi: 10.1007/BF01232933. Google Scholar [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. Google Scholar [8] M. Comerford, Infinitely many grand orbits, Michigan Math. J., 51 (2003), 47-57. doi: 10.1307/mmj/1049832892. Google Scholar [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. Google Scholar [10] M. Comerford, Conjugacy and counterexample in random iteration, Pacific J. Math., 211 (2003), 69-80. doi: 10.2140/pjm.2003.211.69. Google Scholar [11] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynam. Systems, 26 (2006), 353-377. doi: 10.1017/S0143385705000441. Google Scholar [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. Google Scholar [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. Google Scholar [14] E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428. Google Scholar [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. Google Scholar [16] J. Milnor, Dynamics in One Complex Variable, Vieweg 1999, 2000; Princeton University Press, Princeton New Jersey, 2006. Google Scholar [17] J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: an expository account, Gèométrie complexe et systèmes dynamiques, Astérisques, 261 (2000), 277-333. Google Scholar [18] R. Näkki and J. Väiäsäla, John disks, Expo. Math., 9 (1991), 3-43. Google Scholar [19] O. Sester, Hyperbolicité des polynômes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428. Google Scholar [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. 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, Erratum to 'Semi-hyperbolic fibered rational maps and rational semigroups', Ergodic Theory Dynam. Systems, 28 (2008), 1043-1045. Google Scholar [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. Google Scholar

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##### References:
 [1] A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. Google Scholar [2] A. Blokh, J. Malough, J. Mayer, L. 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. Google Scholar [3] R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252. doi: 10.1007/BF03321035. Google Scholar [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. Google Scholar [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. Google Scholar [6] L. Carleson, P. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Bras. Mat., 25 (1994), 1-30. doi: 10.1007/BF01232933. Google Scholar [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. Google Scholar [8] M. Comerford, Infinitely many grand orbits, Michigan Math. J., 51 (2003), 47-57. doi: 10.1307/mmj/1049832892. Google Scholar [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. Google Scholar [10] M. Comerford, Conjugacy and counterexample in random iteration, Pacific J. Math., 211 (2003), 69-80. doi: 10.2140/pjm.2003.211.69. Google Scholar [11] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynam. Systems, 26 (2006), 353-377. doi: 10.1017/S0143385705000441. Google Scholar [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. Google Scholar [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. Google Scholar [14] E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428. Google Scholar [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. Google Scholar [16] J. Milnor, Dynamics in One Complex Variable, Vieweg 1999, 2000; Princeton University Press, Princeton New Jersey, 2006. Google Scholar [17] J. Milnor, Periodic orbits, external rays, and the Mandelbrot set: an expository account, Gèométrie complexe et systèmes dynamiques, Astérisques, 261 (2000), 277-333. Google Scholar [18] R. Näkki and J. Väiäsäla, John disks, Expo. Math., 9 (1991), 3-43. Google Scholar [19] O. Sester, Hyperbolicité des polynômes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428. Google Scholar [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. 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, Erratum to 'Semi-hyperbolic fibered rational maps and rational semigroups', Ergodic Theory Dynam. Systems, 28 (2008), 1043-1045. Google Scholar [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. Google Scholar
A critical arc covering a critical value arc
A degree $6$ example with critical arc $(\tfrac{11}{36}, \tfrac{7}{36})$ and critical value arc $(\tfrac{5}{6}, \tfrac{1}{6})$
The dynamic portrait at times $0$ and $1$ for the example in Proposition 2
The Juila set for $P(z) = z^3 + \tfrac{3}{2}z$
The critical arc at time $m_0 - 1$ in the cubic case (in pink)
The critical arc at time $m_0 - 1$ in the quadratic case (in pink)
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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