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 and Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246
References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[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.

[3]

I. BaničM. 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.

[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.

[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.

[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.

[7]

S. Greenwood and J. Kennedy, Connected generalized inverse limits over intervals, Fund. Math., 236 (2017), 1-43.  doi: 10.4064/fm241-4-2016.

[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.

[9]

K. P. Hart, J. Nagata and J. E. Vaughan (eds.), Encyclopedia of General Topology, Elsevier Science Publishers, B. V., Amsterdam, 2004.

[10]

W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941.

[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.

[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.

[13]

W. T. Ingram and W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math., 32 (2006), 119-130. 

[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.

[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.

[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.

[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.

[18]

R. P. McGehee and T. Wiandt, Conley decomposition for closed relations, J. Difference Equ. Appl., 12 (2006), 1-47.  doi: 10.1080/00207210500171620.

[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.

[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. 

[21]

A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.

[22]

T. Wiandt, Liapunov functions for closed relations, J. Difference Equ. Appl., 14 (2008), 705-722.  doi: 10.1080/10236190701809315.

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[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.

[3]

I. BaničM. 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.

[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.

[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.

[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.

[7]

S. Greenwood and J. Kennedy, Connected generalized inverse limits over intervals, Fund. Math., 236 (2017), 1-43.  doi: 10.4064/fm241-4-2016.

[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.

[9]

K. P. Hart, J. Nagata and J. E. Vaughan (eds.), Encyclopedia of General Topology, Elsevier Science Publishers, B. V., Amsterdam, 2004.

[10]

W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941.

[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.

[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.

[13]

W. T. Ingram and W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math., 32 (2006), 119-130. 

[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.

[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.

[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.

[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.

[18]

R. P. McGehee and T. Wiandt, Conley decomposition for closed relations, J. Difference Equ. Appl., 12 (2006), 1-47.  doi: 10.1080/00207210500171620.

[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.

[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. 

[21]

A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.

[22]

T. Wiandt, Liapunov functions for closed relations, J. Difference Equ. Appl., 14 (2008), 705-722.  doi: 10.1080/10236190701809315.

Figure 1.  A torus as a GIL on intervals
Figure 2.  A circle as a binary Mahavier product of simply-connected spaces
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