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Puzzles of quasi-finite type, zeta functions and symbolic dynamics for multi-dimensional maps. (English) Zbl 1207.37009

The paper under review is a very interesting and deep mathematical work. The author studies entropy-expanding transformations which define a class of smooth dynamical systems which generalizes positive entropy and expanding maps.
In this paper a symbolic representation of those dynamics in terms of puzzles in the Yoccoz’s sense is constructed. Those puzzles are controlled by a “constraint entropy” bounded by the hypersurface entropy of the aforementioned transformations.
This approach allows the generalization of classical properties of subshifts of finite type: finite multiplicity of maximal entropy measures, almost topological classification, meromorphic extension of Artin-Mazur zetafunctions counting periodic points.

MSC:

37B10 Symbolic dynamics
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37B40 Topological entropy
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References:

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