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Current trends and open problems in arithmetic dynamics. (English) Zbl 1468.37001

This is an excellent survey paper on the recent developments of the relatively new field of arithmetic dynamics. It is written in a quite friendly style so that a general reader can understand the motivations and the importance of the conjectures and open problems in the field. No advanced knowledge in algebraic geometry is required to read the survey.
The authors begin with a two-page introduction explaining the main two sources of inspiration for arithmetic dynamics: one from the classical problems in arithmetic geometry and the other from the \(p\)-adic analogue of complex dynamical systems. A dictionary about the correspondences between objects in arithmetic geometry and objects in dynamical systems theory is given.
Then, two sections are devoted to the terminology, notations and background on dynamical systems, number theory, and algebraic geometry.
After that, the authors take 19 sections to describe the different important conjectures and open problems in arithmetic dynamics. The latest progresses on the current conjectures and open problems are provided as well. These 19 sections are independent of one another and the reader could jump to the sections of interest. As examples, one can find the dynamical uniform boundedness conjecture in Section 4, the dynamical Mordell-Lang conjecture in Section 7, and the dynamical Lehmer conjecture in Section 15.
The very informative bibliography contains 249 references.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37Pxx Arithmetic and non-Archimedean dynamical systems
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References:

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