Countable inverse limits of postcritical $w$-limit sets ofunimodal maps

Chris Good; Robin Knight; Brian Raines

doi:10.3934/dcds.2010.27.1059

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2010, Volume 27, Issue 3: 1059-1078. Doi: 10.3934/dcds.2010.27.1059

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Countable inverse limits of postcritical $w$-limit sets of unimodal maps

1.
School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT
2.
Mathematical Institute, University of Oxford, Oxford, OX1 3LB
3.
Department of Mathematics, Baylor University, Waco, TX 76798–7328

Received: January 31, 2009

Revised: January 31, 2010

Published: August 2010

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Abstract

Let $f$ be a unimodal map of the interval with critical point $c$. If the orbit of $c$ is not dense then most points in lim_←{[0, 1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim_← {w(c), f|_w(c)}. In this paper we consider the relationship between the limit complexity of $w(c)$ and the limit complexity of I. We show that if $w(c)$ is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible $\w(c)$.

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Mathematics Subject Classification: 37B45, 37E05, 54F15, 54H20.

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Countable inverse limits of postcritical $w$-limit sets of unimodal maps

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