Entropy dimension of shifts of finite type on free groups

Jung-Chao Ban; Chih-Hung Chang; Jung-Chao Ban; Chih-Hung Chang

doi:10.3934/math.2020329

AIMS Mathematics

2020, Volume 5, Issue 5: 5121-5139. doi: 10.3934/math.2020329

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Research article

Entropy dimension of shifts of finite type on free groups

Jung-Chao Ban ^1,2,
Chih-Hung Chang ^{3
,
,}

1.
Department of Mathematical Sciences, National Chengchi University, Taipei 11605, Taiwan, ROC
2.
Math. Division, National Center for Theoretical Science, National Taiwan University, Taipei 10617, Taiwan. ROC
3.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 81148, Taiwan, ROC

Received: 08 March 2020 Accepted: 03 June 2020 Published: 12 June 2020
MSC : 37A35, 37B10

This paper considers the topological degree of G-shifts of finite type for the case where G is a finitely generated free group. Topological degree is the logarithm of entropy dimension; that is, topological degree is a characterization for zero entropy systems. Following the conjugacy-invariance of topological degree, we show that it is equivalent to solving a system of nonlinear recurrence equations. More explicitly, the topological degree of G-shift of finite type is achieved as the maximal spectral radius of a collection of matrices corresponding to the shift itself.
- topological degree,
- entropy dimension,
- free group,
- finitely generated group,
- Cayley graph,
- conjugacy-invariant
Citation: Jung-Chao Ban, Chih-Hung Chang. Entropy dimension of shifts of finite type on free groups[J]. AIMS Mathematics, 2020, 5(5): 5121-5139. doi: 10.3934/math.2020329

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Abstract

This paper considers the topological degree of G-shifts of finite type for the case where G is a finitely generated free group. Topological degree is the logarithm of entropy dimension; that is, topological degree is a characterization for zero entropy systems. Following the conjugacy-invariance of topological degree, we show that it is equivalent to solving a system of nonlinear recurrence equations. More explicitly, the topological degree of G-shift of finite type is achieved as the maximal spectral radius of a collection of matrices corresponding to the shift itself.

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