## Expansion properties for finite subdivision rules II

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- by William Floyd, Walter Parry and Kevin M. Pilgrim PDF
- Conform. Geom. Dyn.
**24**(2020), 29-50 Request permission

## Abstract:

We prove that every sufficiently large iterate of a Thurston map which is not doubly covered by a torus endomorphism and which does not have a Levy cycle is isotopic to the subdivision map of a finite subdivision rule. We determine which Thurston maps doubly covered by a torus endomorphism have iterates that are isotopic to subdivision maps of finite subdivision rules. We give conditions under which no iterate of a given Thurston map is isotopic to the subdivision map of a finite subdivision rule.## References

- Laurent Bartholdi and Dzmitry Dudko,
*Algorithmic aspects of branched coverings*, Ann. Fac. Sci. Toulouse Math. (6)**26**(2017), no. 5, 1219–1296 (English, with English and French summaries). MR**3746628**, DOI 10.5802/afst.1566 - Laurent Bartholdi and Dzmitry Dudko,
*Algorithmic aspects of branched coverings IV/V. Expanding maps*, Trans. Amer. Math. Soc.**370**(2018), no. 11, 7679–7714. MR**3852445**, DOI 10.1090/tran/7199 - Mario Bonk and Daniel Meyer,
*Expanding Thurston maps*, Mathematical Surveys and Monographs, vol. 225, American Mathematical Society, Providence, RI, 2017. MR**3727134**, DOI 10.1090/surv/225 - J. W. Cannon, W. J. Floyd, and W. R. Parry,
*Finite subdivision rules*, Conform. Geom. Dyn.**5**(2001), 153–196. MR**1875951**, DOI 10.1090/S1088-4173-01-00055-8 - J. W. Cannon, W. J. Floyd, and W. R. Parry,
*Conformal modulus: the graph paper invariant or the conformal shape of an algorithm*, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 71–102. MR**1714840** - J. W. Cannon, W. J. Floyd, and W. R. Parry,
*Constructing subdivision rules from rational maps*, Conform. Geom. Dyn.**11**(2007), 128–136. MR**2329140**, DOI 10.1090/S1088-4173-07-00167-1 - J. W. Cannon, W. J. Floyd, and W. R. Parry,
*Lattès maps and finite subdivisison rules*, Conform. Geom. Dyn.**14**(2010), 113–140. MR**2629972**, DOI 10.1090/S1088-4173-10-00203-1 - J. W. Cannon, W. J. Floyd, W. R. Parry, and K. M. Pilgrim,
*Nearly Euclidean Thurston maps*, Conform. Geom. Dyn.**16**(2012), 209–255. MR**2958932**, DOI 10.1090/S1088-4173-2012-00248-2 - Guizen Cui, Yan Gao, and Jinsong Zeng,
*Invariant graphs of rational maps*, https://arxiv.org/abs/1907.02870, 2019. - Adrien Douady and John H. Hubbard,
*A proof of Thurston’s topological characterization of rational functions*, Acta Math.**171**(1993), no. 2, 263–297. MR**1251582**, DOI 10.1007/BF02392534 - A. Douady and J. H. Hubbard,
*Étude dynamique des polynômes complexes. Partie I*, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR**762431** - William J. Floyd, Walter R. Parry, and Kevin M. Pilgrim,
*Expansion properties for finite subdivision rules I*, Sci. China Math.**61**(2018), no. 12, 2237–2266. MR**3881956**, DOI 10.1007/s11425-016-9265-y - Yan Gao, Peter Haïssinsky, Daniel Meyer, and Jinsong Zeng,
*Invariant Jordan curves of Sierpiński carpet rational maps*, Ergodic Theory Dynam. Systems**38**(2018), no. 2, 583–600. MR**3774834**, DOI 10.1017/etds.2016.47 - Peter Haïssinsky and Kevin M. Pilgrim,
*An algebraic characterization of expanding Thurston maps*, J. Mod. Dyn.**6**(2012), no. 4, 451–476. MR**3008406**, DOI 10.3934/jmd.2012.6.451 - Curtis T. McMullen,
*Complex dynamics and renormalization*, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR**1312365** - John Milnor,
*Dynamics in one complex variable*, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR**1721240** - John Milnor,
*Hyperbolic components*, Conformal dynamics and hyperbolic geometry, Contemp. Math., vol. 573, Amer. Math. Soc., Providence, RI, 2012, pp. 183–232. With an appendix by A. Poirier. MR**2964079**, DOI 10.1090/conm/573/11428 - Kevin Michael Pilgrim,
*Cylinders for iterated rational maps*, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–University of California, Berkeley. MR**2691488** - Kevin M. Pilgrim,
*Combinations of complex dynamical systems*, Lecture Notes in Mathematics, vol. 1827, Springer-Verlag, Berlin, 2003. MR**2020454**, DOI 10.1007/b14147 - Nikita Selinger and Michael Yampolsky,
*Constructive geometrization of Thurston maps and decidability of Thurston equivalence*, Arnold Math. J.**1**(2015), no. 4, 361–402. MR**3434502**, DOI 10.1007/s40598-015-0024-4

## Additional Information

**William Floyd**- Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
- MR Author ID: 67750
- Email: floyd@math.vt.edu
**Walter Parry**- Affiliation: Department of Mathematics and Statistics, Eastern Michigan University, Ypsilanti, Michigan 48197
- MR Author ID: 136390
- Email: walter.parry@emich.edu
**Kevin M. Pilgrim**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- MR Author ID: 614176
- Email: pilgrim@indiana.edu
- Received by editor(s): August 20, 2019
- Received by editor(s) in revised form: November 19, 2019
- Published electronically: January 14, 2020
- Additional Notes: The third author was supported by Simons grant #245269.
- © Copyright 2020 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**24**(2020), 29-50 - MSC (2010): Primary 37F10, 52C20; Secondary 57M12
- DOI: https://doi.org/10.1090/ecgd/347
- MathSciNet review: 4051834