[Author]
 A. M. Vershik

[Title]
 Dynamic Theory of Growth in Groups: Entropy, Boundaries, Examples

[AMS Subj-class]
 20F69 Asymptotic properties of groups
 28D05 Dynamical systems
 28D20 Entropy

[Abstract]
 We study the numerical constants of the countable groups, - logariphmic
 volume ($v$), entropy ($h$), escape ($l$) - and establish the fundamental
 inequality $h \leq l\cdot v$. The new interpretation of those constants
 using boundary polymorphisms is given. We consider the new classes of the
 groups from the point of view of the boundary and fundamental inequality.
 In particulary we find the boundary of the free solvable groups of degree
 two and define so called locally free groups for which estimations of the
 constants are given. This gives some estimations for the braid groups.
 We discuss a various notion of entropy of decreasing sequences of measurable
 partitions and so called secondary entropy of the K-automorphisms in the
 context.

[Keywords]
 entropy, fundamental inequality, group boundaries, polymorphisms,
 local groups, free solvable groups, secondary entropy

[Comments]
 LaTeX, 93 pp., Russian

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