September  2020, 40(9): 5117-5129. doi: 10.3934/dcds.2020221

Notes on the values of volume entropy of graphs

1. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA

2. 

Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea

* Corresponding author: Seonhee Lim

Received  September 2018 Revised  December 2019 Published  June 2020

Fund Project: This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1601-03 and the National Research Foundation of Korea(NRF) (NRF-2017R1E1A1A03070779, 2017R1A5A1015626)

Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph or when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm for calculating the persistent volume entropy of graphs.

Citation: Wooyeon Kim, Seonhee Lim. Notes on the values of volume entropy of graphs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5117-5129. doi: 10.3934/dcds.2020221
References:
[1]

F. Balacheff, Volume entropy, weighted girths and stable balls on graphs, Journal of Graph Theory, 55 (2007), 291-305.  doi: 10.1002/jgt.20236.  Google Scholar

[2]

G. BessonG. Courtois and S. Gallot, Volume et entropie minimale des espaces localement symétriques, Inventiones Mathematicae, 103 (1991), 417-445.  doi: 10.1007/BF01239520.  Google Scholar

[3]

A. Broise-Alamichel, J. Parkkonen and F. Paulin, Equidistribution and Counting under Equilibrium States in Negatively Curved Spaces and Graphs of Groups, Progress Mathematics, 329. Birkhäuser, 2019. Google Scholar

[4]

G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X.  Google Scholar

[5]

S. Karam, Growth of balls in the universal cover of surfaces and graphs, Trans. Amer. Math. Soc., 367 (2015), 5355-5373.  doi: 10.1090/S0002-9947-2015-06189-3.  Google Scholar

[6]

H. Lee, E. Kim, H. Kang, Y. Huh, Y. Lee, S. Lim and D. S. Lee, Volume entropy and information flow in a brain graph, Sci. Rep., 9 (2019), 256. doi: 10.1038/s41598-018-36339-7.  Google Scholar

[7]

S. Lim, Minimal volume entropy for graphs, Trans. Amer. Math. Soc., 360 (2008), 5089-5100.  doi: 10.1090/S0002-9947-08-04227-X.  Google Scholar

[8]

S. Lim, Entropy rigidity for metric spaces, The Pure and Applied Mathematics of Korea Society of Mathematical Education, 19 (2012), 73-86.  doi: 10.7468/jksmeb.2012.19.1.73.  Google Scholar

[9]

A. Manning, Topological entropy for geodesic flows, Annals of Mathematics (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[10]

C. T. McMullen, Entropy and the clique polynomial, Journal of Topology, 8 (2015), 184-212.  doi: 10.1112/jtopol/jtu022.  Google Scholar

[11]

M. Pollicott, Asymptotic vertex growth for graphs, Spectrum and Dynamics, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 52 (2010), 137-145.   Google Scholar

[12]

W. X. Sun, Topological entropy and the complete invariant for expansive maps, Nonlinearity, 13 (2000), 663-673.  doi: 10.1088/0951-7715/13/3/309.  Google Scholar

[13]

Z. H. Xia and P. F. Zhang, Exponential growth rate of paths and its connection with dynamics, Progress in Variational Methods, Nankai Ser. Pure Appl. Math. Theoret. Phys., World Sci. Publ., Hackensack, NJ, 7 (2011), 212-224.   Google Scholar

show all references

References:
[1]

F. Balacheff, Volume entropy, weighted girths and stable balls on graphs, Journal of Graph Theory, 55 (2007), 291-305.  doi: 10.1002/jgt.20236.  Google Scholar

[2]

G. BessonG. Courtois and S. Gallot, Volume et entropie minimale des espaces localement symétriques, Inventiones Mathematicae, 103 (1991), 417-445.  doi: 10.1007/BF01239520.  Google Scholar

[3]

A. Broise-Alamichel, J. Parkkonen and F. Paulin, Equidistribution and Counting under Equilibrium States in Negatively Curved Spaces and Graphs of Groups, Progress Mathematics, 329. Birkhäuser, 2019. Google Scholar

[4]

G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308.  doi: 10.1090/S0273-0979-09-01249-X.  Google Scholar

[5]

S. Karam, Growth of balls in the universal cover of surfaces and graphs, Trans. Amer. Math. Soc., 367 (2015), 5355-5373.  doi: 10.1090/S0002-9947-2015-06189-3.  Google Scholar

[6]

H. Lee, E. Kim, H. Kang, Y. Huh, Y. Lee, S. Lim and D. S. Lee, Volume entropy and information flow in a brain graph, Sci. Rep., 9 (2019), 256. doi: 10.1038/s41598-018-36339-7.  Google Scholar

[7]

S. Lim, Minimal volume entropy for graphs, Trans. Amer. Math. Soc., 360 (2008), 5089-5100.  doi: 10.1090/S0002-9947-08-04227-X.  Google Scholar

[8]

S. Lim, Entropy rigidity for metric spaces, The Pure and Applied Mathematics of Korea Society of Mathematical Education, 19 (2012), 73-86.  doi: 10.7468/jksmeb.2012.19.1.73.  Google Scholar

[9]

A. Manning, Topological entropy for geodesic flows, Annals of Mathematics (2), 110 (1979), 567-573.  doi: 10.2307/1971239.  Google Scholar

[10]

C. T. McMullen, Entropy and the clique polynomial, Journal of Topology, 8 (2015), 184-212.  doi: 10.1112/jtopol/jtu022.  Google Scholar

[11]

M. Pollicott, Asymptotic vertex growth for graphs, Spectrum and Dynamics, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 52 (2010), 137-145.   Google Scholar

[12]

W. X. Sun, Topological entropy and the complete invariant for expansive maps, Nonlinearity, 13 (2000), 663-673.  doi: 10.1088/0951-7715/13/3/309.  Google Scholar

[13]

Z. H. Xia and P. F. Zhang, Exponential growth rate of paths and its connection with dynamics, Progress in Variational Methods, Nankai Ser. Pure Appl. Math. Theoret. Phys., World Sci. Publ., Hackensack, NJ, 7 (2011), 212-224.   Google Scholar

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