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Transactions of the American Mathematical Society

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Algebraic $\mathbb{Z} ^d$-actions of entropy rank one


Authors: Manfred Einsiedler and Douglas Lind
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 1799-1831
MSC (2000): Primary 37A35, 37B40, 54H20; Secondary 37A45, 37D20, 13F20
DOI: https://doi.org/10.1090/S0002-9947-04-03554-8
Published electronically: January 6, 2004
MathSciNet review: 2031042
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Abstract: We investigate algebraic $\mathbb Z^d$-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$-action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.


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Additional Information

Manfred Einsiedler
Affiliation: Department of Mathematics, Pennsylvania State University, State College, Pennsylvania 16802
Address at time of publication: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: einsiedl@math.psu.edu, einsiedl@math.washington.edu

Douglas Lind
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: lind@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03554-8
Keywords: Entropy, skew product, algebraic action, variational principle
Received by editor(s): July 26, 2002
Published electronically: January 6, 2004
Additional Notes: The first author gratefully acknowledges the hospitality of the University of Washington and the Penn State University, and was supported by the FWF research project P14379-MAT and the Erwin Schrödinger Stipendium J2090
The second author thanks the generous hospitality of the Yale Mathematics Department
Article copyright: © Copyright 2004 American Mathematical Society

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