August  2013, 33(8): 3753-3765. doi: 10.3934/dcds.2013.33.3753

K-groups of the full group actions on one-sided topological Markov shifts

1. 

Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan

Received  May 2012 Revised  July 2012 Published  January 2013

Let $(X_A,\sigma_A)$ be the one-sided topological Markov shift for an $N\times N$ irreducible matrix $A$ with entries in $\{ 0,1 \}$. We will first show that the continuous full group $\Gamma_A$ of $(X_A,\sigma_A)$ is non amenable as a discrete group and contains all finite groups and free groups. We will second introduce the K-group $K^0(X_A,\Gamma_A)$ for the action of $\Gamma_A$ on $X_A$, and show that the group $K^0(X_A,\Gamma_A)$ with the position of the constant $1$ function is a complete invariant for the topological conjugacy class of the action of $\Gamma_A$ on $X_A$ under some conditions.
Citation: Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
References:
[1]

Ph.D. thesis, University of Washington, 1983.  Google Scholar

[2]

Israel J. Math., 95 (1996), 169-210. doi: 10.1007/BF02761039.  Google Scholar

[3]

Canad. J. Math., 31 (1979), 867-880. doi: 10.4153/CJM-1979-082-4.  Google Scholar

[4]

Comm. Math. Phys., 57 (1977), 173-185.  Google Scholar

[5]

Invent. Math., 63 (1981), 25-40. doi: 10.1007/BF01389192.  Google Scholar

[6]

Ann. Math. (2), 113 (1981), 181-197. doi: 10.2307/1971137.  Google Scholar

[7]

Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048.  Google Scholar

[8]

J. Reine Angew. Math., 469 (1995), 51-111.  Google Scholar

[9]

Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689.  Google Scholar

[10]

Invent. Math., 179 (2010), 119-158. doi: 10.1007/s00222-009-0213-7.  Google Scholar

[11]

Internat. J. Math., 3 (1992), 827-864. doi: 10.1142/S0129167X92000382.  Google Scholar

[12]

Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966.  Google Scholar

[13]

J. Operator Theory, 44 (2000), 91-112.  Google Scholar

[14]

Pacific J. Math., 246 (2010), 199-225. doi: 10.2140/pjm.2010.246.199.  Google Scholar

[15]

K. Matsumoto, Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts,, to appear in Proc. Amer. Math. Soc., ().   Google Scholar

[16]

K. Matsumoto, Full groups of one-sided topological Markov shifts,, preprint, ().   Google Scholar

[17]

Proc. London Math. Soc. (3), 104 (2012), 27-56. doi: 10.1112/plms/pdr029.  Google Scholar

[18]

Pacific J. Math., 136 (1989), 329-353.  Google Scholar

[19]

K-theory, 9 (1995), 31-58. doi: 10.1007/BF00965458.  Google Scholar

[20]

London Mathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000.  Google Scholar

[21]

Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.  Google Scholar

show all references

References:
[1]

Ph.D. thesis, University of Washington, 1983.  Google Scholar

[2]

Israel J. Math., 95 (1996), 169-210. doi: 10.1007/BF02761039.  Google Scholar

[3]

Canad. J. Math., 31 (1979), 867-880. doi: 10.4153/CJM-1979-082-4.  Google Scholar

[4]

Comm. Math. Phys., 57 (1977), 173-185.  Google Scholar

[5]

Invent. Math., 63 (1981), 25-40. doi: 10.1007/BF01389192.  Google Scholar

[6]

Ann. Math. (2), 113 (1981), 181-197. doi: 10.2307/1971137.  Google Scholar

[7]

Invent. Math., 56 (1980), 251-268. doi: 10.1007/BF01390048.  Google Scholar

[8]

J. Reine Angew. Math., 469 (1995), 51-111.  Google Scholar

[9]

Israel J. Math., 111 (1999), 285-320. doi: 10.1007/BF02810689.  Google Scholar

[10]

Invent. Math., 179 (2010), 119-158. doi: 10.1007/s00222-009-0213-7.  Google Scholar

[11]

Internat. J. Math., 3 (1992), 827-864. doi: 10.1142/S0129167X92000382.  Google Scholar

[12]

Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966.  Google Scholar

[13]

J. Operator Theory, 44 (2000), 91-112.  Google Scholar

[14]

Pacific J. Math., 246 (2010), 199-225. doi: 10.2140/pjm.2010.246.199.  Google Scholar

[15]

K. Matsumoto, Classification of Cuntz-Krieger algebras by orbit equivalence of topological Markov shifts,, to appear in Proc. Amer. Math. Soc., ().   Google Scholar

[16]

K. Matsumoto, Full groups of one-sided topological Markov shifts,, preprint, ().   Google Scholar

[17]

Proc. London Math. Soc. (3), 104 (2012), 27-56. doi: 10.1112/plms/pdr029.  Google Scholar

[18]

Pacific J. Math., 136 (1989), 329-353.  Google Scholar

[19]

K-theory, 9 (1995), 31-58. doi: 10.1007/BF00965458.  Google Scholar

[20]

London Mathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000.  Google Scholar

[21]

Internat. J. Math., 2 (1991), 457-476. doi: 10.1142/S0129167X91000260.  Google Scholar

[1]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008

[2]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[3]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062

[4]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077

[5]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[6]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

[7]

Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004

[8]

Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021046

[9]

Roberto Civino, Riccardo Longo. Formal security proof for a scheme on a topological network. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021009

[10]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056

[11]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074

[12]

Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082

[13]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]