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\noindent A famous Polish mathematician and one-time editor of Fundamenta Mathematicae is sometimes quoted as saying that he had never accepted an article more than eight pages in length, because a longer proof could not possibly be correct. The reviewer has never written an article which was less than nine pages, so needless to say, he doesn't agree. But as many a provocative overstatement, this one has some truth to it, particularly if eight is replaced with eighty. From writing a few papers longer even than that, I recall the certain numbness of mind which might set in on page seventy-seven, when I no longer could remember notations or definitions from the first ten pages, and I didn't care enough to look them up. If one believes in miracles, such a paper may still be right. But will it ever find a devoted enough reader to be read, not just skimmed or quoted?

The paper by Lyubich has 113 pages and while lengthy by any standard it is hardly an exception in the field of conformal dynamics. In fact, some recent works in this area have never been published as articles, but appeared straightaway as books. Great complexity seems to be the staple of works in one-dimensional dynamics. Some papers from back in the 1980s, to mention just one [M. V. Jakobson, Comm. Math. Phys. 81 (1981), no. 1, 39--88; MR 83j:58070], were long and complicated when written, have not been much simplified since, and yet have been read and have given inspiration for further research. One could speculate about reasons for this state of affairs. First of all, proofs in dynamical systems tend to be complicated. They depend on the "picture" of the dynamics which exists in the author's mind and is typically not easy to explain to the reader. When such an explanation is undertaken it often leads to lengthy combinatorial, topological or geometric constructions. Somewhat paradoxically, the technical side of one-dimensional proofs also tends to be very complicated. There are a number of "principles" that have underlain the development of the 1990s, and in particular the present paper. Examples of these are "treat one-dimensional real and complex systems jointly, looking to explore the real and complex facets of the problem at hand", "use quasiconformal deformations and pullback", "try inducing or renormalization". Without getting into the exact meaning of these principles, they are not theorems or even well-defined techniques, but merely indications of directions in which to search. For reasons not completely understood, they work very well on concrete problems, but require work in order to be made into proofs. In addition to these general factors, the paper is also made lengthy by its ambitious scope. The four theorems stated in the introduction together with Corollary 1.1 cover a large part of the development of the field in the 1990s.

Among these results, the density theorem is probably the most famous. It states that in the logistic family $x\to ax(1-x)$ an attracting periodic orbit exists for values of $a$ in a dense subset of $(0,4)$. The complex bounds theorem is a long-sought-for and powerful technical result. The statements of Theorem III and the rigidity theorem cannot be understood without going more deeply into the paper. Roughly, Theorem III means that high iterates of the quadratic polynomial become increasingly expanding (an example of the "inducing principle" first used by Jakobson [op. cit.]), and the rigidity theorem is an attempt at a generalization of the density theorem to a class of complex polynomials. Such a generalization is not likely to be relevant in the study of complex quadratic polynomials. The approach used in the paper requires complex bounds. These are true for real polynomials, but are generally not satisfied for infinitely renormalizable complex quadratic polynomials. At the top of page 188, the author observes that the Hausdorff dimension of the complex parameters to which his rigidity theorem applies is at least 1, the Hausdorff dimension of the boundary of the Mandelbrot set is $2$, and $1$ is one-half of $2$. This is not convincing. Hausdorff dimension 1 is attained on the real line and being a half of the full dimension is hardly the sign of the set being robust, let alone typical.

The density theorem has another proof by J. Graczyk and the reviewer [see Ann. of Math. (2) 146 (1997), no. 1, 1--52; MR 99b:58079]. This surely affects the way I look at the present paper, thinking first of all similarities with and differences from my own work. The approaches of both proofs are very similar. Our work also uses a version of Theorem III (Theorem C in [J. Graczyk and G. Swiatek, Ann. Sci. Ecole Norm. Sup. (4) 29 (1996), no. 4, 399--482; MR 98d:58152]) and includes a proof of the complex bounds theorem, together with quasiconformal pullback techniques. There are two main differences. The first concerns the technique of proof of the growth of moduli (Theorem III in the present paper). Both approaches use an invariant based on "separating annuli" but these invariants are quite different. The second difference occurs in the way partial conjugacies obtained on subsequent levels of renormalization for an infinitely renormalizable polynomial are pasted together. Our technique uses the fact that polynomials are real. Since the author's method covers some complex polynomials, he has to use a more subtle complex construction.

To read the paper critically will not be an easy task. The size is one obvious problem. Sometimes, a very lengthy paper may not be so hard to go through if its structure is modular, more like a sequence of related papers which can be checked independently. Such is not the case with the paper at hand. The proof depends on a number of constructions, techniques and combinatorial schemes which are the basis of a fairly sophisticated language developed by the author. A simple attempt of going through the statements of the paper named "theorems", and presumably representing the main steps of the proof, reveals that most of them cannot be understood by an expert without extensive reading through earlier parts of the paper. For example, we find Theorem 8.1 on page 241, which is attributed to Martens. The problem is, Martens never states any such result. He does not talk of "tails of long central cascades" or intervals $I\sb m$. As far as I understand the author's intention and Martens' paper, I believe that a proof of the statement needed for the present paper can be recovered from Martens' work. However, a critical reader has no choice but to go through Martens' work in some detail. Besides, the statement of Theorem 8.1 uses colloquial language and is not clear at all (are the bounds referred to uniform in some sense?). The impossibility of skimming through the paper without reading the whole might be particularly annoying for an expert, and imprecise statements will bother a devoted non-expert reader.

Nevertheless, it may be rewarding to go through the paper. The density theorem alone is worth 113 pages. Other results included in the paper (complex bounds and local connectivity) make the work more interesting still. After all, the density problem can be traced over seventy years back to Fatou, is actually relevant even to non-mathematicians, and represents a very rare circumstance in which we can make a statement about what is typical and what is not in a whole class of non-hyperbolic dynamical systems. It is true that most of the results of this paper can also be found in works by other authors [J. Graczyk and J. Swiatek, op. cit., 1996; op. cit., 1997; G. Levin and S. J. van Strien, "Local connectivity of Julia sets of real polynomials", Preprint No. 5, Inst. Math. Sci., SUNY, Stony Brook, NY, 1995; per bibl.], but those are technically involved as well. The present paper represents a great deal of work and some elements of its approach are different and original.

The progress in conformal dynamics of the 1990s has been impressive if measured by problems solved. The price of these results has been the incredible complexity of proofs. For this reason, the achievements are less than completely satisfying. Hopefully, the works of the 1990s will help others to find better proofs as well as contribute to nonlinear dynamics in general.

Cited in: 98d:58152 99g:58106 99e:58143 99b:58079 98e:58070

Item: 7 of 53