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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 6/1998

N. SIDOROV and A. VERSHIK
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BIJECTIVE ARITHMETIC CODINGS
OF HYPERBOLIC AUTOMORPHISMS OF THE 2-TORUS,
AND BINARY QUADRATIC FORMS

This preprint was accepted February, 1998

Contact:
` N. Sidorov and A. Vershik`

ABSTRACT:
We study arithmetic codings of hyperbolic
automorphisms of the 2-torus, i.e. mappings
acting from the symbolic space of sequences with a finite alphabet
endowed with an appropriate structure
of additive group onto the torus which preserves this structure and
turns the two-sided shift into an automorphism of the torus.
The necessary and sufficient
condition of the existence of a bijective arithmetic coding is obtained; it
is formulated in terms of a certain binary quadratic
form constructed by means of a given automorphism. Furthermore,
we describe all bijective arithmetic codings in terms
the Dirichlet group of the corresponding qudratic field. The arithmetic
minimum of that binary quadratic form
in the general case equals the minimal possible number of
preimages for a.e. point of the torus under such an arithmetical
coding.

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