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August  2020, 40(8): 4767-4775. doi: 10.3934/dcds.2020200

Mean dimension of shifts of finite type and of generalized inverse limits

Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

Received  June 2019 Revised  December 2019 Published  May 2020

We study mean dimension of shifts of finite type defined on compact metric spaces and give its lower bound when the shift possesses a certain "periodic block" of arbitrarily large length. The result is applied to shift maps on generalized inverse limits with upper semi-continuous closed set-valued functions. In particular we obtain a refinement of some results due to Banič [1] and Erceg and Kennedy [5] on the dimension of the inverse limit spaces and topological entropy of their shifts.

Citation: Kazuhiro Kawamura. Mean dimension of shifts of finite type and of generalized inverse limits. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4767-4775. doi: 10.3934/dcds.2020200
References:
[1]

I. Banič, On dimension of inverse limits with upper semicontinuous set-valued bonding functions, Top. Appl., 154 (2007), 2771-2778.  doi: 10.1016/j.topol.2007.06.002.

[2]

M. Chacon-Tirado and V. Martínez-de-la-Vega, Closed subsets of the square whose inverse limits result in Hilbert cube, Colloq. Math., 152 (2018), 29-44.  doi: 10.4064/cm6592-3-2017.

[3]

M. Coornaert, Topological Dimension and Dynamical Systems, Univesitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-19794-4.

[4]

A. N. Dranishnikov, Cohomological dimension theory of compact metric spaces, Topology Atlas Invited Contributions, 6 (2001), 61 pp, http://at.yorku.ca/t/a/i/c/43.htm.

[5]

G. Erceg and J. Kennedy, Topological entropy on closed sets in $[0, 1]^{2}$, Top. Appl., 246 (2018), 106-136.  doi: 10.1016/j.topol.2018.06.015.

[6]

M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Mathematical Physics, Analysis and Geometry, 2 (1999), 323-415.  doi: 10.1023/A:1009841100168.

[7]

W. T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, Springer Briefs in Math., Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-4487-9.

[8]

K. Kawamura and J. Kennedy, Shift maps and their variants on inverse limits with set-valued functions, Top. Appl., 239 (2018), 92-114.  doi: 10.1016/j.topol.2018.02.015.

[9]

J. P. Kelly and T. Tennant, Topological entropy of set-valued functions, Houston J. Math., 43 (2017), 263-282.  doi: 10.1007/s40995-017-0443-2.

[10]

J. Kennedy and V. Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Th. Dyn. Sys., 38 (2018), 1499-1524.  doi: 10.1017/etds.2016.73.

[11]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[12]

W. S. Mahavier, Inverse limits with subsets of $[0, 1]\times[0, 1]$, Top. Appl., 141 (2004), 225-231.  doi: 10.1016/j.topol.2003.12.008.

[13]

Y. Shitanda, A fixed point theorem and equivariant points for set-valued mappings, Publ. RIMS Kyoto Univ., 45 (2009), 811-844.  doi: 10.2977/prims/1249478966.

[14]

M. Tsukamoto, Mean dimension of full shifts, Israel J. Math., 230 (2019), 183-193.  doi: 10.1007/s11856-018-1813-y.

[15]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

show all references

References:
[1]

I. Banič, On dimension of inverse limits with upper semicontinuous set-valued bonding functions, Top. Appl., 154 (2007), 2771-2778.  doi: 10.1016/j.topol.2007.06.002.

[2]

M. Chacon-Tirado and V. Martínez-de-la-Vega, Closed subsets of the square whose inverse limits result in Hilbert cube, Colloq. Math., 152 (2018), 29-44.  doi: 10.4064/cm6592-3-2017.

[3]

M. Coornaert, Topological Dimension and Dynamical Systems, Univesitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-19794-4.

[4]

A. N. Dranishnikov, Cohomological dimension theory of compact metric spaces, Topology Atlas Invited Contributions, 6 (2001), 61 pp, http://at.yorku.ca/t/a/i/c/43.htm.

[5]

G. Erceg and J. Kennedy, Topological entropy on closed sets in $[0, 1]^{2}$, Top. Appl., 246 (2018), 106-136.  doi: 10.1016/j.topol.2018.06.015.

[6]

M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Mathematical Physics, Analysis and Geometry, 2 (1999), 323-415.  doi: 10.1023/A:1009841100168.

[7]

W. T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, Springer Briefs in Math., Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-4487-9.

[8]

K. Kawamura and J. Kennedy, Shift maps and their variants on inverse limits with set-valued functions, Top. Appl., 239 (2018), 92-114.  doi: 10.1016/j.topol.2018.02.015.

[9]

J. P. Kelly and T. Tennant, Topological entropy of set-valued functions, Houston J. Math., 43 (2017), 263-282.  doi: 10.1007/s40995-017-0443-2.

[10]

J. Kennedy and V. Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Th. Dyn. Sys., 38 (2018), 1499-1524.  doi: 10.1017/etds.2016.73.

[11]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[12]

W. S. Mahavier, Inverse limits with subsets of $[0, 1]\times[0, 1]$, Top. Appl., 141 (2004), 225-231.  doi: 10.1016/j.topol.2003.12.008.

[13]

Y. Shitanda, A fixed point theorem and equivariant points for set-valued mappings, Publ. RIMS Kyoto Univ., 45 (2009), 811-844.  doi: 10.2977/prims/1249478966.

[14]

M. Tsukamoto, Mean dimension of full shifts, Israel J. Math., 230 (2019), 183-193.  doi: 10.1007/s11856-018-1813-y.

[15]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

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