2-manifolds and inverse limits of set-valued functions on intervals

Sina Greenwood; Rolf Suabedissen

doi:10.3934/dcds.2017246

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2017, Volume 37, Issue 11: 5693-5706. Doi: 10.3934/dcds.2017246

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2-manifolds and inverse limits of set-valued functions on intervals

Sina Greenwood^1, , and
Rolf Suabedissen^2,

1.
University of Auckland, Private Bag 92019, Auckland, New Zealand
2.
University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom

* Corresponding author: Sina Greenwood
* Corresponding author: Sina Greenwood

Received: March 31, 2017

Published: October 2017

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Abstract

Suppose for each $n\in\mathbb{N}$, $f_n \colon [0,1] \to 2^{[0,1]}$ is a function whose graph $\Gamma(f_n) = \left\lbrace (x,y) \in [0,1]^2 \colon y \in f_n(x)\right\rbrace$ is closed in $[0,1]^2$ (here $2^{[0,1]}$ is the space of non-empty closed subsets of $[0,1]$). We show that the generalized inverse limit $\varprojlim (f_n) = \left\lbrace (x_n) \in [0,1]^\mathbb{N} \colon \forall n \in \mathbb{N},\ x_n \in f_n(x_{n+1})\right\rbrace$ of such a sequence of functions cannot be an arbitrary continuum, answering a long-standing open problem in the study of generalized inverse limits. In particular we show that if such an inverse limit is a 2-manifold then it is a torus and hence it is impossible to obtain a sphere.

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Mathematics Subject Classification: Primary: 54C08, 54E45; Secondary: 54F15, 54F65.

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Figure 1. A torus as a GIL on intervals

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Figure 2. A circle as a binary Mahavier product of simply-connected spaces

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