Tan Lei and Shishikura s example of obstructed polynomial mating without a levy cycle.
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1 Tan Lei and Shishikura s example of obstructed polynomial mating without a levy cycle. Arnaud Chéritat CNRS, Univ. Toulouse Feb A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
2 Origins = + A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
3 Origins A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
4 Topological (instant) mating K(P 1 ) K(P 2 ) / with : relation generated by identifying endpoints of external rays. A dynamics is well defined thereon. When is the quotient a sphere? When is the dynamics conjugated to a rational map? A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
5 Formal mating Since PCF (post-critically finite) rational maps are characterized by Thurston s theorem, it is tempting to try and guess the Th-equivalence class of a potential mating of P 1 and P 2. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
6 Formal mating Since PCF (post-critically finite) rational maps are characterized by Thurston s theorem, it is tempting to try and guess the Th-equivalence class of a potential mating of P 1 and P 2. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
7 Formal mating Since PCF (post-critically finite) rational maps are characterized by Thurston s theorem, it is tempting to try and guess the Th-equivalence class of a potential mating of P 1 and P 2. In good cases, it is unobstructed and Th-equivalent to a rational map and to the topological mating. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
8 Degenerate (assisted) mating However sometimes the formal mating has a Th-obstruction yet the topological mating is conjugated to a rational map. Rees, Shishikura and Tan Lei have devised a way to detect this on the formal mating and to correct the latter by collapsing some post critical points together, yielding a new ramified cover that is unobstructed, and proved that it is Th-equivalent to a rational map conjugated to the topological mating. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
9 Obstructed matings The last case is when the obstruction cannot be removed. Then, the topological mating cannot be equivalent to a rational map (even though the quotient still may be a sphere, or not). A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
10 Slow mating Define a Riemann surface S R by cutting & pasting along equipotential e R, R > 1. Glue according to external angle. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
11 Slow mating Uniformize to Ĉ. Here: stereographicy projected to S 2. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
12 Slow mating There is a natural holomorphic map (rational of degree d after uniformization) F R : S R S R d. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
13 Slow mating S R 1/2 S R 2 F R F R 1/2 F R 2 S R S R 4 A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
14 Slow mating Question: Do the maps F R converge as R 1 to a rational map of the same degree? It is then tempting to define the latter as a mating of P 1 and P 2. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
15 Slow mating In the PCF case, the post-critical set of P 1 and P 2 map to Riemann surfaces S R, so we get Riemann surfaces with marked points. The sequence of marked S R 1/d n for n N is an orbit under Thurston s pull-back map associated to the formal mating. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
16 Comparison Corrected Formal mating Th-equiv class of the Formal mating Topological mating Slow mating PCF polyn J connected and locally connected J connected A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
17 The example It is a mating of two PCF polynomials of degree 3 whose formal mating has a non removable Th-obstruction. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
18 The example A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
19 The example A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
20 The example Matrix of the multicurve {orange,green}: [ 1/2 1/2 1 0 Spectrum: {1, 1/2}. ] A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
21 The example Remark: Shishikura and Tan Lei have proved that the ray equivalence relation is closed and that classes are trees with a bounded number of equator crossing: thus the topological mating gives a sphere. Aslo, the topol mating is Th-equivalent to the formal mating (and thus not to a rational map). A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
22 Pinching curves A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
23 Showtime Show movie. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
24 Flat view A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
25 Three normalizations A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
26 Three normalizations A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
27 Three normalizations A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
28 Three normalizations A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
29 Interpretation: limit dynamical system. There is a limit dynamical system on a tree of spheres: the tree of three spheres obtained when the canonical obstruction gets completely pinched. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
30 Interpretation: limit dynamical system. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
31 Interpretation: limit dynamical system. The third iterate of the limit maps each sphere to itself, by three semi-conjugated degree 6 rational maps. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
32 Interpretation: limit dynamical system. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
33 3 Interpretation: tubes and mess A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
34 Interpretation: tubes and mess. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
35 Interpretation: tubes and mess. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
36 Interpretation: tubes and mess. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
37 Interpretation: tubes and mess. A. Chéritat (CNRS, UPS) Obstructed mating Feb / 15
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