# American Institute of Mathematical Sciences

May  2017, 37(5): 2285-2300. doi: 10.3934/dcds.2017100

## Uncountably many planar embeddings of unimodal inverse limit spaces

 1 Faculty of Electrical Engineering and Computing, Unska 3,10000 Zagreb, Croatia 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

* Corresponding author

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: AA was supported in part by Croatian Science Foundation under the project IP-2014-09-2285. HB and JČ were supported by the FWF stand-alone project P25975-N25. We gratefully acknowledge the support of the bilateral grant Strange Attractors and Inverse Limit Spaces, Österreichische Austauschdienst (OeAD) -Ministry of Science, Education and Sport of the Republic of Croatia (MZOS), project number HR 03/2014.

For a point $x$ in the inverse limit space $X$ with a single unimodal bonding map we construct, with the use of symbolic dynamics, a planar embedding such that $x$ is accessible. It follows that there are uncountably many non-equivalent planar embeddings of $X$.

Citation: Ana Anušić, Henk Bruin, Jernej Činč. Uncountably many planar embeddings of unimodal inverse limit spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2285-2300. doi: 10.3934/dcds.2017100
##### References:

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##### References:
Example of two basic arcs having a boundary point in common.
Coding the Cantor set with respect to $(a)$ $L=\ldots 111.$ and $(b)$ $L=\ldots 101.$
Case Ⅰ in the proof of Proposition 3.
Case Ⅱ in the proof of Proposition 3.
The planar representation of an arc in $X$ with the corresponding kneading sequence $\nu=100110010\ldots$. The ordering on basic arcs is such that the basic arc coded by $L=1^{\infty}.$ is the largest.
The planar representation of the same arc as in Figure 5 in $X$ with the corresponding kneading sequence $\nu=100110010\ldots$. The ordering on basic arcs is such that the basic arc coded by $L=(101)^{\infty}.$ is the largest.
Set-up in Lemma 4.1.
Sets constructed in the proof of Lemma 4.1.
Point $a = (a_0, \psi_L(L))$ is accessible.
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