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**Dynamics of quadratic polynomials. I, II.**
*(English)*
Zbl 0908.58053

The theory of quadratic polynomials has been an area of great activity during the past decade. The paper summarizes a large part of the progress that has been made and provides proofs of several important results. These main results are the density of periodic windows in the logistic family and the local connectivity of Julia sets for real quadratic polynomials. The proof of the first result can also be found in [J. Graczyk and G. Świątek, Ann. Math., II. Ser. 146, No. 1, 1-52 (1997)]. The second theorem is true more generally for real unimodal polynomials, not necessarily quadratic, see [LvS] as quoted in the paper.

Given the complexity of the subject, it would be very useful to have all proofs in one paper. The current paper’s size, though large (113 pages), seems actually modest for such an undertaking. Unfortunately, the paper is written in a fashion that makes it hard to read for anyone. First, close familiarity with the existing literature is expected. For example, for Theorem 8.1 on page 241 the reader is referred to a paper by Martens. But nothing close to Theorem 8.1 is stated there, so a critical reader is left with the task of constructing the proof by himself, based on Martens’ paper. Furthermore, frequent use of colloquial language where inequalities and quantifiers would normally appear does not help to understand the statements and proofs.

Lemma 8.7 and its proof provide a glaring and somewhat amusing example.

Given the complexity of the subject, it would be very useful to have all proofs in one paper. The current paper’s size, though large (113 pages), seems actually modest for such an undertaking. Unfortunately, the paper is written in a fashion that makes it hard to read for anyone. First, close familiarity with the existing literature is expected. For example, for Theorem 8.1 on page 241 the reader is referred to a paper by Martens. But nothing close to Theorem 8.1 is stated there, so a critical reader is left with the task of constructing the proof by himself, based on Martens’ paper. Furthermore, frequent use of colloquial language where inequalities and quantifiers would normally appear does not help to understand the statements and proofs.

Lemma 8.7 and its proof provide a glaring and somewhat amusing example.

Reviewer: G.Swiatek (University Park)

### MSC:

37F99 | Dynamical systems over complex numbers |

37E99 | Low-dimensional dynamical systems |

30D10 | Representations of entire functions of one complex variable by series and integrals |

### Keywords:

Yoceoz puzzle; rigidity; quadratic polynomials; logistic family; local connectivity; Julia sets
Full Text:
DOI

### References:

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