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Dynamics of one-dimensional maps. Transl. from the Russian by A. G. Sivak, P. Malyshev and D. Malyshev. Rev. and upd. transl. Rev. and upd. transl. (English) Zbl 0881.58020

Mathematics and its Applications (Dordrecht). 407. Dordrecht: Kluwer Academic Publishers. ix, 261 p. (1997).
[This is an updated translation of the original 1989 Russian version, reviewed in Zbl 0759.58001.]
Many people consider Professor A. N. Sharkovsky to be the father of modern discrete one-dimensional dynamical systems. This book, co-authored by some authoritative members of his Kiev group, aims to survey the most significant developments in the field. Topics are (among others): symbolic dynamics, coexistence of periodic orbits (including, of course, famous Sharkovsky Theorem), topological and metrical aspects of dynamics (with an emphasis on unimodal maps), structural stability, bifurcations in one-parameter families of unimodal maps and universality.
While having such a large range of interests, the book cannot be comprehensive, the reader can always expect a deep and original approach to each item. The list of references, not being comprehensive either, provides quite a few papers of historical relevance hardly known by the interested readership.
Just one small “but” for this otherwise recommendable book: many proofs are omitted. This is unimportant in the case of western works (appropriate references are always provided, including several well established standard monographs), but one sorely misses the proofs of some significant results by the authors themselves presently only available in shortened Russian-written papers.

MSC:

37E99 Low-dimensional dynamical systems
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory

Citations:

Zbl 0759.58001
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