##
**Rescaling limits of complex rational maps.**
*(English)*
Zbl 1347.37089

This article is about rescaling limits of families of rational maps. Let \((f_t)_t\) be a family of rational maps of a given degree \(d \geq 2\). A rescaling limit \(g\) for the family \((f_t)_t\) is the limit (outside a finite set) of a family of conjugate maps of the form \( M_t^{-1} \circ f_t^q \circ M_t \), where \(M_t\) is a Möbius transform depending on \(t\) and \(q \geq 1\). The author considers two cases: either \(t\) is a complex parameter in a neighborhood of \(0\), in which case the dependency \(t\mapsto f_t\) is assumed holomorphic and we consider the limit \(t \rightarrow 0\), or \(t=n\) is a natural number and we consider \(n \rightarrow \infty\).

The family \((M_t)_t\) is called a rescaling for \((f_t)_t\) if there is a rescaling limit or simply a moving frame when considered independently of the mappings \((f_t)_t\).

The moving frames \((M_t)_t\) and \((L_t)_t\) are said to be equivalent if there is a Möbius transformation such that \(M_t^{-1}\circ L_t \rightarrow M\) as \(t \rightarrow 0\). Then one can show that holomorphic families \((f_t)_t\) act on the set of equivalence classes of moving frames (Section 3.4). More precisely, for every moving frame \((M_t)_t\) there exists a unique class of moving frames \((L_t)_t\) such that \(L_t^{-1} \circ f_t \circ M_t\) converges to some nonconstant rational map outside a finite set. Two rescalings of \((f_t)_t\) are called dynamically independent if their corresponding classes belong to different orbits in the action of \((f_t)_t\).

The first main result of the article is Theorem 1, which is as follows. Given a holomorphic family \((f_t)_t\) of rational maps of degree \(d\), there are at most \(2 d - 1\) dynamically independent rescalings such that the corresponding rescaling limit is not postcritically finite. Moreover, in the case \(d=2\) there is a total of at most \(2\) rescaling limits. Furthermore, the statement gives some description of the possible rescaling limits when the rescaling period is at least \(2\).

The second result, Theorem 2, concerns sequential rescaling, i.e., \(t = n\) is an integer and \(n \rightarrow \infty\), and gives conclusions similar to Theorem 1.

The article contains several examples of rescaling limits in Section 2, such as cubic polynomials, quadratic rational maps, flexible Lattès maps and family of Cantor cross circles Julia sets. These examples have arisen in previous works of various authors interested in holomorphic families.

The proofs of the main results are based on Berkovich projective line techniques. Recall that the field of formal Puiseux series is the algebraic closure of the field of formal Laurent series. Then any holomorphic family \((f_t)_t\) of rational maps can be seen as a rational map \(\mathbf{f}\) defined on the Berkovich projective line over the completion of the field of formal Puiseux series.

The dynamics of this rational map governs the existence of dynamically independent rescaling limits. In particular, one of the main tools is J. Rivera-Letelier’s result [in: Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday. Paris: Société Mathématique de France. 147–230 (2003; Zbl 1140.37336)] (Proposition 3.3). Assuming that \(({M_t^j})_j\) is a family of dynamically independent moving frames, let \(x_j\) be the image of the Gauss point of the Berkovich projective line by the moving frame \(\mathbf{M_t^j}\). It follows from Proposition 3.3 that the action of the tangent map of \(\mathbf{f}\) at \(x_j\) is conjugate to the rescaling limit \(g\) (Proposition 3.4). This implies that the tangent map is not postcritically finite. Moreover dynamically independent moving frames give rise to pairwise distinct periodic orbits of \(\mathbf{f}\). Since the tangent map is not postcritically finite it follows from Corollary 4.4 that the number of these orbits is at most \(2 d - 2\).

The family \((M_t)_t\) is called a rescaling for \((f_t)_t\) if there is a rescaling limit or simply a moving frame when considered independently of the mappings \((f_t)_t\).

The moving frames \((M_t)_t\) and \((L_t)_t\) are said to be equivalent if there is a Möbius transformation such that \(M_t^{-1}\circ L_t \rightarrow M\) as \(t \rightarrow 0\). Then one can show that holomorphic families \((f_t)_t\) act on the set of equivalence classes of moving frames (Section 3.4). More precisely, for every moving frame \((M_t)_t\) there exists a unique class of moving frames \((L_t)_t\) such that \(L_t^{-1} \circ f_t \circ M_t\) converges to some nonconstant rational map outside a finite set. Two rescalings of \((f_t)_t\) are called dynamically independent if their corresponding classes belong to different orbits in the action of \((f_t)_t\).

The first main result of the article is Theorem 1, which is as follows. Given a holomorphic family \((f_t)_t\) of rational maps of degree \(d\), there are at most \(2 d - 1\) dynamically independent rescalings such that the corresponding rescaling limit is not postcritically finite. Moreover, in the case \(d=2\) there is a total of at most \(2\) rescaling limits. Furthermore, the statement gives some description of the possible rescaling limits when the rescaling period is at least \(2\).

The second result, Theorem 2, concerns sequential rescaling, i.e., \(t = n\) is an integer and \(n \rightarrow \infty\), and gives conclusions similar to Theorem 1.

The article contains several examples of rescaling limits in Section 2, such as cubic polynomials, quadratic rational maps, flexible Lattès maps and family of Cantor cross circles Julia sets. These examples have arisen in previous works of various authors interested in holomorphic families.

The proofs of the main results are based on Berkovich projective line techniques. Recall that the field of formal Puiseux series is the algebraic closure of the field of formal Laurent series. Then any holomorphic family \((f_t)_t\) of rational maps can be seen as a rational map \(\mathbf{f}\) defined on the Berkovich projective line over the completion of the field of formal Puiseux series.

The dynamics of this rational map governs the existence of dynamically independent rescaling limits. In particular, one of the main tools is J. Rivera-Letelier’s result [in: Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday. Paris: Société Mathématique de France. 147–230 (2003; Zbl 1140.37336)] (Proposition 3.3). Assuming that \(({M_t^j})_j\) is a family of dynamically independent moving frames, let \(x_j\) be the image of the Gauss point of the Berkovich projective line by the moving frame \(\mathbf{M_t^j}\). It follows from Proposition 3.3 that the action of the tangent map of \(\mathbf{f}\) at \(x_j\) is conjugate to the rescaling limit \(g\) (Proposition 3.4). This implies that the tangent map is not postcritically finite. Moreover dynamically independent moving frames give rise to pairwise distinct periodic orbits of \(\mathbf{f}\). Since the tangent map is not postcritically finite it follows from Corollary 4.4 that the number of these orbits is at most \(2 d - 2\).

Reviewer: Alexandre De Zotti (London)

### MSC:

37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |

12J25 | Non-Archimedean valued fields |

32S99 | Complex singularities |

26E30 | Non-Archimedean analysis |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |

37P20 | Dynamical systems over non-Archimedean local ground fields |

37P50 | Dynamical systems on Berkovich spaces |

### Keywords:

rescaling limits; rational maps; complex dynamics; parameter spaces; Berkovich projective line; moving frames### Citations:

Zbl 1140.37336### References:

[1] | M. Arfeux, Dynamique holomorphe et arbres de sphères , Ph.D. dissertation, Université Paul Sabatier - Toulouse III, Toulouse, 2013. |

[2] | M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line , Math. Surveys Monogr. 159 , Amer. Math. Soc., Providence, 2010. · Zbl 1196.14002 |

[3] | V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields , Math. Surveys Monogr. 33 , Amer. Math. Soc., Providence, 1990. · Zbl 0715.14013 |

[4] | A. Bonifant, J. Kiwi, and J. Milnor, Cubic polynomial maps with periodic critical orbit, II: Escape regions , Conform. Geom. Dyn. 14 (2010), 68-112. · Zbl 1188.37047 · doi:10.1090/S1088-4173-10-00204-3 |

[5] | E. Casas-Alvero, Singularities of Plane Curves , London Math. Soc. Lecture Note Ser. 276 , Cambridge Univ. Press, Cambridge, 2000. · Zbl 0967.14018 |

[6] | J. W. S. Cassels, Local Fields , London Math. Soc. Stud. Texts 3 , Cambridge Univ. Press, Cambridge, 1986. |

[7] | L. De Marco, Iteration at the boundary of the space of rational maps , Duke Math. J. 130 (2005), 169-197. · Zbl 1183.37086 · doi:10.1215/S0012-7094-05-13015-0 |

[8] | L. De Marco, The moduli space of quadratic rational maps , J. Amer. Math. Soc. 20 (2007), 321-355. · Zbl 1158.37020 · doi:10.1090/S0894-0347-06-00527-3 |

[9] | A. L. Epstein, Bounded hyperbolic components of quadratic rational maps , Ergodic Theory Dynam. Systems 20 (2000), 727-748. · Zbl 0963.37041 · doi:10.1017/S0143385700000390 |

[10] | A. L. Epstein, Boundedness and unboundedness in the moduli space of quadratic rational maps , talk at “Ph.D. Euroconference on Complex Analysis and Holomorphic Dynamics,” Platja d’Aro, Spain, 2000. · Zbl 0999.90016 |

[11] | X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, I , Manuscripta Math. 142 (2013), 439-474. · Zbl 1288.14014 · doi:10.1007/s00229-013-0611-4 |

[12] | X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, II , Math Ann. 356 (2013), 819-844. · Zbl 1277.14020 · doi:10.1007/s00208-012-0872-3 |

[13] | H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I , Ann. of Math. (2) 79 (1964), 109-203; II , 205-326. · Zbl 0122.38603 · doi:10.2307/1970486 |

[14] | M. Jonsson, “Dynamics on Berkovich spaces in low dimensions” in Berkovich Spaces and Applications , Lecture Notes in Math. 2119 , Springer, Cham, 2015, 205-366. · Zbl 1401.37103 |

[15] | J. Kiwi, Puiseux series dynamics of quadratic rational maps , Israel J. Math. 201 (2014), 631-700. · Zbl 1350.37082 · doi:10.1007/s11856-014-1047-6 |

[16] | J. Kollár, Lectures on Resolution of Singularities , Ann. of Math. Stud. 166 , Princeton Univ. Press, Princeton, 2007. |

[17] | C. McMullen, “Automorphisms of rational maps” in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986) , Math. Sci. Res. Inst. Publ. 10 , Springer, New York, 1988, 31-60. · doi:10.1007/978-1-4613-9602-4_3 |

[18] | J. Milnor, Dynamics in One Complex Variable: Introductory Lectures , Vieweg, Braunschweig, 1999. · Zbl 0946.30013 |

[19] | J. Milnor, “On Lattès maps” in Dynamics on the Riemann Sphere , Eur. Math. Soc., Zürich, 2006, 9-43. · Zbl 1235.37015 · doi:10.4171/011-1/1 |

[20] | J. Milnor, “Cubic polynomial maps with periodic critical orbit, I” in Complex Dynamics , A K Peters, Wellesley, Mass., 2009, 333-411. · Zbl 1180.37073 · doi:10.1201/b10617-13 |

[21] | P. Ribenboim, The Theory of Classical Valuations , Springer Monogr. Math., Springer, New York, 1999. · Zbl 0957.12005 |

[22] | J. Rivera-Letelier. “Dynamique des fonctions rationnelles sur des corps locaux” in Geometric Methods in Dynamics, II , Astérisque 287 , Soc. Math. France, Montrouge, 2003, 147-230. |

[23] | J. H. Silverman, The Arithmetic of Dynamical Systems , Grad. Texts in Math. 241 , Springer, New York, 2007. |

[24] | J. Stimson, Degree two rational maps with a periodic critical point , Ph.D. dissertation, University of Liverpool, Liverpool, 1993. |

[25] | E. Trucco, Wandering Fatou components and algebraic Julia sets , Bull. Soc. Math. France 142 (2014), 411-464. · Zbl 1392.37117 |

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