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Nonconjugate pointed generalized solenoids with shift equivalent $ \pi_1$-actions

Author: Richard Swanson
Journal: Proc. Amer. Math. Soc. 140 (2012), 3581-3586
MSC (2010): Primary 37B10, 37B45, 37B50; Secondary 37B99, 05C20
Published electronically: May 4, 2012
MathSciNet review: 2929026
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Abstract: Going back to the pioneering work of R. F. Williams, it has been widely believed that if elementary presentations $ (K_i, f_i), i=1, 2$, satisfy the Williams axioms and fix the branch points, then there is a pointed conjugacy

$\displaystyle \overline r:(\varprojlim (K_1,f_1),\overline {y}_1)\to (\varprojlim (K_2,f_2),\overline {y}_2)$

between the natural shift maps $ \overline {f}_1$ and $ \overline {f}_2$ if and only if the fundamental group homomorphisms $ \pi _1(f_1, y_1)$ and $ \pi _1(f_2, y_2)$ are shift equivalent. The ``only if'' direction is true. We prove that the ``if'' direction goal of this belief is false by exhibiting a family of counterexamples not previously analyzed. Of course, the associated hyperbolic attractors constructed by R. F. Williams cannot be conjugate in these exceptional cases.

References [Enhancements On Off] (What's this?)

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Additional Information

Richard Swanson
Affiliation: Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717-2400

Received by editor(s): November 20, 2010
Received by editor(s) in revised form: April 11, 2001, and April 17, 2011
Published electronically: May 4, 2012
Communicated by: Bryna Kra
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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