# American Institute of Mathematical Sciences

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August  2010, 27(3): 1059-1078. doi: 10.3934/dcds.2010.27.1059

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

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

Received  February 2009 Revised  February 2010 Published  March 2010

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 $\w(c)$.
Citation: Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059
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