Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dfc/2009/xtra/timorin/timorin_research_statement.pdf Дата изменения: Sun Oct 25 20:24:37 2009 Дата индексирования: Tue Nov 24 17:37:03 2009 Кодировка: Поисковые слова: http astrokuban.info astrokuban

Vladlen Timorin: Research project I am doing Geometry in a broad sense: my interests range from dynamical systems to algebraic and differential geometry. I have publications in several mathematical fields, and continue to pursue several different lines of research. However, what follows is my most recent and active research pro ject focused on topological dynamics of rational functions. Rational functions and regluing. A rational function of one complex variable (i.e. a ratio of two polynomials) is among the simplest and most basic ob jects in algebra. However, an extremely rich and complicated structure is revealed when one starts to iterate a rational function, i.e. consider it from the point of view of dynamical systems. The Riemann sphere CP 1 , on which a rational function acts, gets divided into two fully invariant sets: an open set, called the Fatou set, on which the dynamics is stable (e.g. in the sense of Lyapunov) and simple, and a closed set, called the Julia set, on which the dynamics is unstable and chaotic. Julia sets tend to have fractal shapes and very intricate geometric properties. The following commutative diagram appears in a great variety of contexts: X -- Y - f X -- Y -


g

()

Suppose e.g. that f : CP 1 CP 1 is a rational function, g : S 2 S 2 is a continuous map, and a homeomorphism. Then g is topological ly conjugate to f . Even if f is given by an explicit formula (say, f (z ) = z 2 - 1.5), it may be very hard to understand its dynamical properties, i.e. what the f -orbits of points are doing. On the other hand, we may have a good model g for f , which is not given by explicit formula but has in a sense explicit dynamics. The meaning of "explicit dynamics" is hard to formalize but, for a good model g , it should be clear what the "model Fatou set" and the "model Julia set" are. Moreover, the topology of the Julia set should be made explicit (e.g. we may know that the Julia set is a Cantor set, or a Sierpinski carpet), and it should also be clear which parts of the Julia set map to which parts (e.g. we may have a Markov partition of the Julia set). Topological models for quadratic polynomials with locally connected Julia sets and all periodic points repelling were constructed by Douady and Hubbard [DH] and, using a different language, by Thurston. Thurston's construction represents Julia sets as quotients of the unit circle by explicitly defined equivalence relations. On the other hand, there are quadratic polynomials of the form f (z ) = z 2 + c, for which no explicit topological models are known. There are also general ways to build new topological models out of known topological models. A well-known mating construction by Douady and Hubbard defines a model of a rational function by gluing the models of two polynomials together, in a certain explicit way. Another construction is capture (it has been defined in a thesis of B. Wittner [W], and extensively studied and generalized by M. Rees [R92]): it allows to define a model for a rational function with a specific periodic or pre-periodic behavior of one critical point. There are few more such constructions, however, much more general principles of making topological models are needed.
1


The main ob jective of this pro ject is to develop systematic ways of building topological models for rational functions. The procedure I suggest can be described by the same diagram (); however, it must be interpreted differently. It would be too restrictive to regard as a map. Instead, I propose to regard it as a topological correspondence, i.e. a closed subset of X в Y , which is not necessarily a map, but is explicit enough to understand topological dynamics on Y in terms of that on X . It will be instructive to think of topological correspondences as multivalued continuous maps. I plan to use the following class of topological correspondences. Let Z be a topological sphere, D Z a closed topological disk, and X : Z X , Y : Z Y quotient maps that are injective outside of D. The correspondence E = X в Y (Z ) X в Y is called a regluing correspondence (we also say that E is a regluing of X (D) into Y (D)). A simple example of regluing is the following. Set X = Y = CP 1 , and consider the multivalued function z 2 - 1. For every simple curve : [-1, 1] C such that (-t) = -(t) and (1) = 1, there is a branch of z 2 - 1 defined over the complement to [-1, 1]. The closure of the graph of this branch defines a regluing of [-1, 1] into a simple curve connecting i with -i. Consider diagram (), in which is understood as a regluing correspondence. Given a rational function f : CP 1 CP 1 , the diagram defines g ; however, this will also be a topological correspondence rather than a function: it will have multiple values at some points. To have a precise setting, assume that f is a function of z 2 (e.g. any rational function is MЁ obius conjugate to a function of this form), and : [-1, 1] C is a simple curve such that (-t) = -(t). Let be a regluing of given by the multivalued function z 2 - (1)2 . Then f -1 extends to a rational function. However, the correspondence g = f -1 is not a well-defined map, because it has multiple values over f -pullbacks of [-1, 1]. We can consider a regluing of these two curves etc. As a result, we obtain a sequence of regluing correspondences. A limit of this sequence can be understood in the following sense. Consider a sequence (Xn , En ) consisting of topological spaces Xn and topological correspon^ dences En Xn в Xn+1 . Form new spaces Xn consisting of sequences (xn , xn+1 , . . . ) such that (xm , xm+1 ) Em for all m n (this goes like in the inverse limit construction). We define ^ ^ the topology on Xn as that induced from the embedding of Xn into mn Xm . Then we have ^ ^ well-defined maps n : Xn Xn+1 (forgetting the first term). The direct limit X of this sequence of maps is called the limit of (Xn , En ). We now have the following tasks: (1) Under some general assumptions, prove that the limit of regluing correspondences (or a certain quotient of it) is homeomorphic to the 2-sphere. (2) Suppose X1 = CP 1 , and f1 : X1 X1 a rational function. Form a sequence (Xn , En ) of regluing correspondences as above. There are topological correspondences fn : Xn Xn that commute with En . They define a continuous map f : X X . Give criteria for f being topologically conjugate to a rational function. (3) For some interesting classes of rational functions on CP 1 , give topological models in terms of the maps f : X X . (4) Find some conditions, under which the limit X carries a canonical conformal structure, and f is holomorphic with respect to this structure.
2


Progress already made. I have made initial progress in parts (1), (3) and (4). The results described below deal with the following particular case of the regluing construction. Let E1 be the regluing correspondence that "cuts along a simple curve C = [-1, 1] and glues it back in a different way", as explained above. Then E1 defines a sequence En of regluing correspondences (by a version of Thurston's algorithm applied to f1 ). Suppose that all iterated pullbacks of C under f1 are disjoint and do not contain critical points. In this case, we say that (X , f ) is obtained from (X1 , f1 ) by regluing of disjoint simple curves (namely, all pullbacks of C under f1 ). Since the curves are disjoint, all regluings can be made simultaneously: there is a (multivalued) map from X1 to X that is single-valued and continuous on the complement to pullbacks of C , and that "reglues" all these pullbacks. Part (1) (the space X is homeomorphic to the 2-sphere) is proved [2] in this case. It follows from a purely topological fact: Theorem [2]. Consider a countable set Z of disjoint compact connected local ly connected nonseparating sets in S 2 . Suppose that Z forms a nul l-sequence. For every A Z , fix a continuous map A : S 2 S 2 such that A restricts to an orientation-preserving homeomorphism between the complement to a closed disk and S 2 - A, and A () = A. The equalizer XZ of al l maps A is homeomorphic to S 2 . Intuitively, the space XZ is obtained from S 2 by blowing up all elements of A Z according to the maps A . This result uses Moore's axiomatic characterization of topological 2-spheres. More general results are possible (an indication and some preparation is given in [4]). For part (3), I considered slices P erk (0) in the space of MЁ obius conjugacy classes of quadratic rational functions (following the guidelines of M. Rees [R92] and J. Milnor [M93]). The slice P erk (0) is defined by the property that one critical point is periodic of a given period k (the zero in the notation stands for the multiplier of a periodic cycle). The first slice P er1 (0) identifies with the space of quadratic polynomials z z 2 + c, the most studied parameter family. Hyperbolicity is the simplest dynamical behavior: a hyperbolic rational function combines a strong contraction on the Fatou set with a strong expansion on the Julia set. The set of hyperbolic maps is open (in every reasonable parameter space), the components of this set are called hyperbolic components. Hyperbolic components in P erk (0) are classified into four types [R90, M93] A, B, C and D, according to the types of mutual behavior of the two critical points. E.g. a hyperbolic function of type B in P erk (0) is defined by the property that the non-periodic critical point lies in the immediate basin of the critical periodic cycle. A hyperbolic rational function of type C in P erk (0) has the property that the non-periodic critical point is attracted by the critical periodic cycle but is not in the immediate basin. I have proved the following Theorem [2]. Al l rational functions on the boundaries of type C hyperbolic components in P erk (0) but not on the boundaries of type B hyperbolic components are obtained from hyperbolic quadratic rational functions (for which explicit topological models are known) by regluing of disjoint simple curves. For most type C components, this gives topological models for all boundary maps -- the situation is better than that in the family z 2 + c, where all hyperbolic components have many complicated maps on the boundary, whose models are not currently known (there are no type C or B components in P er1 (0)). In P er2 (0), there is only one type B component, and all maps
3


on its boundary (except for parabolics) can be obtained from 1/z 2 by regluing of disjoint simple curves [1]. Moreover, Theorem [1]. Al l maps on the boundary of the only type B component in P er2 (0) are simultaneously matings and anti-matings. Part (4) is done [3] in the case, where a simple regluing of a quadratic polynomial leads to a quadratic polynomial with totally disconnected Julia set. Since we know which quadratic polynomial we obtain as f , the issue is to prove that the map is holomorphic in a certain generalized sense. Let Z be a countable union of disjoint simple curves. Assume that Z has zero Lebesgue measure. We say that a map : C - Z C is holomorphic modulo Z if there is a function : Z C such that =
C-Z Z



for every smooth (1,0)-form on C with compact support. Intuitively, this definition says that the distributional differential must be a sum of countably many -like (0,1)-currents supported in Z . Theorem [3]. Consider a quadratic polynomial f : z z 2 + c with connected Julia set such that the critical value c is accessible from the basin of infinity. There exists a countable union Z of disjoint simple curves of zero area, and a quadratic polynomial g with total ly disconnected Julia set such that f = g on C - Z , where : C - Z C is a holomorphic map modulo Z. Future plans. First, the requirement that all pullbacks of C under f1 be disjoint should be relaxed. Assume only that 1) the forward orbits of critical points and of the endpoints of C are disjoint from C , and 2) no point of C returns to C infinitely many times (under the dynamics of f1 ). Conjecture (Part (1)). A suitable topological quotient Y of X is homeomorphic to the sphere, and f descends to a branched covering g : Y Y . To prove this conjecture, I plan to use a relative version of Moore's theorem introduced in [4]. Conjecture (Part (2)). Suppose additional ly that the map g is critical ly finite, i.e. the forward orbits of al l critical points are finite. Then g has no Thurston obstructions (therefore, it is Thurston equivalent to a rational function h). The function g is even topological ly conjugate to h. Checking that a critically finite branched covering has no Thurston obstructions is a technical task, which is sometimes very complicated. However, in this case, I do not expect principal difficulties. This result can be used as follows: consider a rational function f1 that is not critically finite (and whose combinatorics may be complicated). Do a regluing surgery to obtain a critically finite branched covering g topologically conjugate to a critically finite rational function h. In general, h is much simpler than f1 . Assume that the topological dynamics of h is known. Then we can often describe the topological dynamics of f1 by "undoing" the surgery. It is clear that all intermediate spaces Xn have canonical conformal structures. Task (Part (4)). Define a "degenerate limit conformal structure" on X in terms of conformal structures on Xn . Prove that f is holomorphic with respect to this "degenerate" conformal
4


structure. Final ly, the degenerate conformal structure on X induces a non-degenerate conformal structure on Y , and g is holomorphic with respect to this structure. A part of the problem is to define the notion of "degenerate conformal structure" on a topological space (that is not necessarily nice). A possible progress in part (3) includes the description of the boundaries of type B hyperbolic components in P erk (0) in terms of regluing. A rather straightforward transfer can be made from P erk (0) to P erk () (parameter slice of quadratic rational functions having a k -cycle with multiplier ) with || < 1. The case || = 1 is much subtler. However, it is very interesting, and I believe the regluing surgery can be done in these parameter slices in the same way as in P erk (0) (the Siegel case is of course easier than the Cremer case). Regluing methods can also be tested on parameter spaces of rational functions with a preperiodic critical point (say, of given period and preperiod). I believe that all (or, at least, all sufficiently nice) such functions can be obtained from polynomials by regluing (which makes the fixed critical point at infinity into a preperiodic critical point). There is a combinatorial aspect in this. Namely, starting with a critically finite polynomial and a path connecting infinity to a preperiodic point, we can define a Thurston equivalence class of branched coverings (e.g. by applying a path homeomorphism in the sense of M. Rees). Regluing surgery associated with this path (if it leads to a rational function) must belong to the same class. The question is to find whether the given Thurston equivalence class represents a rational function, and, if yes, which rational function (these questions are similar to those addressed in [BN, T, R95, R] etc.). Related joint pro jects. Finally, let me mention several joint pro jects that are related to this pro ject of mine. In a joint pro ject with A. Blokh, we study "combinatorial models" for cubic polynomials. Our goal is to obtain a description of the "combinatorial cubic Mandelbrot set" similar to well-known descriptions of "combinatorial quadratic Mandelbrot set" (which is homeomorphic to the actual Mandelbrot set provided that the latter is locally connected -- local connectivity of the Mandelbrot set is a ma jor open problem). Since cubic Mandelbrot set lives in 4-dimensional space and is hard to visualize, we approach the problem by considering certain generic 2-dimensional slices of it. A possible topological picture of these slices includes a repeated regluing surgery in the parameter picture. In a joint pro ject with M. Rees, we try to find persistent Markov partitions for rational functions in P erk (0) -- the first case to consider is the "airplane region" in P er3 (0). In a joint pro ject with M. Lyubich, we are trying to prove a simple criterion of hyperbolicity for 2D Henon maps. Teaching exp erience and plans. I have been teaching various mathematical courses at all levels in Russia, Canada, USA and Germany. My current employment at the State University Higher School of Economics comes with teaching and supervising students. I also plan to give a special topics course in holomorphic dynamics at the Independent University of Moscow. References
[BN] L. Bartholdi, V. Nekrashevich, "Thurston equivalence of topological polynomials", Acta Math., 197, (2006), p. 1­51. ґ [DH] A. Douady and J. Hubbard, "Etude dynamique des polyn^ omes complexes I & II" Publ. Math. Orsay (1984­85) [M93] J. Milnor, "Geometry and dynamics of quadratic rational maps", Exper. Math. 2 (1993) 37­83.
5


[Mo16] R.L. Moore, "On the foundations of plane analysis situs", Trans. AMS 17 (1916), 131­164 [R90] M. Rees, "Components of degree 2 hyperbolic rational maps" Invent. Math. 100 (1990), 357­382 [R92] M. Rees, "A partial description of the Parameter Space of Rational Maps of Degree Two: Part 1" Acta Math. 168 (1992), 11­87 [R95] M.Rees, "A partial description of the Parameter Space of Rational Maps of degree two: Part 2" Proc. LMS 70 (1995), 644­690 [R] M. Rees, "A Fundamental Domain for V3 ", preprint. [T] Tan Lei, "Matings of quadratic polynomials", Erg. Th. and Dyn. Sys. 12 (1992) 589­620 [1] V. Timorin, "The external boundary of M2 ", Fields Institute Communications Vol. 53: "Holomorphic Dynamics and Renormalization, A Volume in Honour of John Milnor's 75th Birthday" [2] V. Timorin, "Topological regluing of holomorphic functions", to appear in Inventiones Math. [3] V. Timorin, "On partial semi-conjugacies of quadratic polynomials", preprint [4] V. Timorin, "Moore's theorem", preprint [W] B. Wittner, "On the bifurcation loci of rational maps of degree two", PhD Thesis, Cornell University, 1988.

6