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Bijective coding of automorphisms of the torus and binary quadratic forms. (English. Russian original) Zbl 0934.37006

Russ. Math. Surv. 53, No. 5, 1106-1107 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 231-232 (1998).
The paper studies a certain class of maps from a symbolic compactum to a two-dimensional torus connecting a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and preserving the maximum entropy measure. Known methods of geometric coding usually make use of the construction of so called Markov partitions, while the coding under consideration is constructed by means of the arithmetic and group theoretic structure. The latter makes it possible to connect the arithmetic of the corresponding algebraic extension with the dynamics of the automorphism.

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B10 Symbolic dynamics
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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