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The algebraic entropy of the special linear character automorphisms of a free group on two generators


Author: Richard J. Brown
Journal: Trans. Amer. Math. Soc. 359 (2007), 1445-1470
MSC (2000): Primary 32M05
DOI: https://doi.org/10.1090/S0002-9947-06-04117-1
Published electronically: October 17, 2006
MathSciNet review: 2272133
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Abstract: In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of $ \mathbb{R}^3$, and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus $ S$ on the relative $ \operatorname{SU}(2)$-character varieties of $ S$ embedded in $ \mathbb{R}^3$. It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is $ 2$). Here we calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of $ S$.


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Additional Information

Richard J. Brown
Affiliation: Department of Mathematics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2686
Email: brown@math.jhu.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04117-1
Received by editor(s): December 21, 2004
Published electronically: October 17, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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