# American Institute of Mathematical Sciences

November  2017, 37(11): 5693-5706. doi: 10.3934/dcds.2017246

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

 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

Received  April 2017 Published  July 2017

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.

Citation: Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
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##### References:
 [1] E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. Google Scholar [2] I. Banič and J. Kennedy, Inverse limits with bonding functions whose graphs are arcs, Topology Appl., 190 (2015), 9-21. doi: 10.1016/j.topol.2015.04.009. Google Scholar [3] I. Banič, M. Črepnjak and V. Nall, Some results about inverse limits with set-valued bonding functions, Topology Appl., 202 (2016), 106-111. doi: 10.1016/j.topol.2016.01.007. Google Scholar [4] R. Engelking, General Topology, vol. 6 of Sigma Series in Pure Mathematics, 2nd edition, Heldermann Verlag, Berlin, 1989, Translated from the Polish by the author. Google Scholar [5] J. Gallier and D. Xu, A Guide to the Classification Theorem for Compact Surfaces, vol. 9 of Geometry and Computing, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-34364-3. Google Scholar [6] S. Greenwood and J. Kennedy, Connectedness and Ingram-Mahavier products, Topology Appl., 166 (2014), 1-9. doi: 10.1016/j.topol.2014.01.016. Google Scholar [7] S. Greenwood and J. Kennedy, Connected generalized inverse limits over intervals, Fund. Math., 236 (2017), 1-43. doi: 10.4064/fm241-4-2016. Google Scholar [8] G. Guzik, Minimal invariant closed sets of set-valued semiflows, J. Math. Anal. Appl., 449 (2017), 382-396. doi: 10.1016/j.jmaa.2016.11.072. Google Scholar [9] K. P. Hart, J. Nagata and J. E. Vaughan (eds.), Encyclopedia of General Topology, Elsevier Science Publishers, B. V., Amsterdam, 2004. Google Scholar [10] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. Google Scholar [11] A. Illanes, A circle is not the generalized inverse limit of a subset of $[0, 1]^2$, Proc. Amer. Math. Soc., 139 (2011), 2987-2993. doi: 10.1090/S0002-9939-2011-10876-1. Google Scholar [12] W. T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, SpringerBriefs in Mathematics, Springer, New York, 2012. doi: 10.1007/978-1-4614-4487-9. Google Scholar [13] W. T. Ingram and W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math., 32 (2006), 119-130. Google Scholar [14] H. Kato, On dimension and shape of inverse limits with set-valued functions, Fund. Math., 236 (2017), 83-99. doi: 10.4064/fm233-4-2016. Google Scholar [15] J. Kennedy and V. Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Theory and Dynamical Systems, (2016), 1-26. doi: 10.1017/etds.2016.73. Google Scholar [16] R. Langevin and F. Przytycki, Entropie de l'image inverse d'une application, Bull. Soc. Math. France, 120 (1992), 237-250. doi: 10.24033/bsmf.2185. Google Scholar [17] W. S. Mahavier, Inverse limits with subsets of $[0, 1]×[0, 1]$, Topology Appl., 141 (2004), 225-231. doi: 10.1016/j.topol.2003.12.008. Google Scholar [18] R. P. McGehee and T. Wiandt, Conley decomposition for closed relations, J. Difference Equ. Appl., 12 (2006), 1-47. doi: 10.1080/00207210500171620. Google Scholar [19] R. McGehee, Attractors for closed relations on compact hausdorff spaces, Indiana Univ. Math. J., 41 (1992), 1165-1209. doi: 10.1512/iumj.1992.41.41058. Google Scholar [20] V. Nall, More continua which are not the inverse limit with a closed subset of a unit square, Houston J. Math., 41 (2015), 1039-1050. Google Scholar [21] A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. Google Scholar [22] T. Wiandt, Liapunov functions for closed relations, J. Difference Equ. Appl., 14 (2008), 705-722. doi: 10.1080/10236190701809315. Google Scholar
A torus as a GIL on intervals
A circle as a binary Mahavier product of simply-connected spaces
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