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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

Chris Good 1, , Robin Knight 2, and  Brian Raines 3,

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 $\w(c)$.
Keywords: unimodal, indecomposable., continuum, inverse limits, invariant set, attractor.
Mathematics Subject Classification: 37B45, 37E05, 54F15, 54H2.
Citation: Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059
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