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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 |
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.
Keywords:
Generalized inverse limit,
upper semicontinuous function,
continuum,
2-manifold,
closed relation.
Mathematics Subject Classification:
Primary: 54C08, 54E45; Secondary: 54F15, 54F65.
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
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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. |
| [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. |
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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. |
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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. |
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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. 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. |
| [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 2.
A circle as a binary Mahavier product of simply-connected spaces
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