American Institute of Mathematical Sciences

Advanced
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

Kengo Matsumoto 1,

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.
Keywords: Topological Markov shifts, K-group., full group.
Mathematics Subject Classification: Primary: 37B10; Secondary: 46L3.
Citation: Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
References:
[1]

M. Boyle, "Topological Orbit Equivalence and Factor Maps in Symbolic Dynamics,", Ph.D. thesis, (1983).   Google Scholar

[2]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology,, Israel J. Math., 95 (1996), 169.  doi: 10.1007/BF02761039.  Google Scholar

[3]

M. D. Choi, A simple $C^*$-algebras generated by two finite order unitaries,, Canad. J. Math., 31 (1979), 867.  doi: 10.4153/CJM-1979-082-4.  Google Scholar

[4]

J. Cuntz, Simple $C^*$-algebras generated by isometries,, Comm. Math. Phys., 57 (1977), 173.   Google Scholar

[5]

J. Cuntz, A class of $C^*$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^*$-algebras,, Invent. Math., 63 (1981), 25.  doi: 10.1007/BF01389192.  Google Scholar

[6]

J. Cuntz, K-theory for certain $C^*$-algebras,, Ann. Math. (2), 113 (1981), 181.  doi: 10.2307/1971137.  Google Scholar

[7]

J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains,, Invent. Math., 56 (1980), 251.  doi: 10.1007/BF01390048.  Google Scholar

[8]

T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products,, J. Reine Angew. Math., 469 (1995), 51.   Google Scholar

[9]

T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems,, Israel J. Math., 111 (1999), 285.  doi: 10.1007/BF02810689.  Google Scholar

[10]

T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\mathbb Z^d$-systems,, Invent. Math., 179 (2010), 119.  doi: 10.1007/s00222-009-0213-7.  Google Scholar

[11]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics,, Internat. J. Math., 3 (1992), 827.  doi: 10.1142/S0129167X92000382.  Google Scholar

[12]

W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory. Presentations of groups in terms of generators and relations,", Interscience Publishers [John Wiley & Sons, (1966).   Google Scholar

[13]

K. Matsumoto, On automorphisms of $C^*$-algebras associated with subshifts,, J. Operator Theory, 44 (2000), 91.   Google Scholar

[14]

K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras,, Pacific J. Math., 246 (2010), 199.  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]

H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces,, Proc. London Math. Soc. (3), 104 (2012), 27.  doi: 10.1112/plms/pdr029.  Google Scholar

[18]

I. F. Putnam, The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set,, Pacific J. Math., 136 (1989), 329.   Google Scholar

[19]

M. Rφrdam, Classification of Cuntz-Krieger algebras,, K-theory, 9 (1995), 31.  doi: 10.1007/BF00965458.  Google Scholar

[20]

M. Rφrdam, F. Larsen and N. J. Laustsen, "An Introduction to K-Theory for $C^*$-Algebras,", London Mathematical Society Student Texts, 49 (2000).   Google Scholar

[21]

J. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps,, Internat. J. Math., 2 (1991), 457.  doi: 10.1142/S0129167X91000260.  Google Scholar

show all references

References:
[1]

M. Boyle, "Topological Orbit Equivalence and Factor Maps in Symbolic Dynamics,", Ph.D. thesis, (1983).   Google Scholar

[2]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology,, Israel J. Math., 95 (1996), 169.  doi: 10.1007/BF02761039.  Google Scholar

[3]

M. D. Choi, A simple $C^*$-algebras generated by two finite order unitaries,, Canad. J. Math., 31 (1979), 867.  doi: 10.4153/CJM-1979-082-4.  Google Scholar

[4]

J. Cuntz, Simple $C^*$-algebras generated by isometries,, Comm. Math. Phys., 57 (1977), 173.   Google Scholar

[5]

J. Cuntz, A class of $C^*$-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for $C^*$-algebras,, Invent. Math., 63 (1981), 25.  doi: 10.1007/BF01389192.  Google Scholar

[6]

J. Cuntz, K-theory for certain $C^*$-algebras,, Ann. Math. (2), 113 (1981), 181.  doi: 10.2307/1971137.  Google Scholar

[7]

J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains,, Invent. Math., 56 (1980), 251.  doi: 10.1007/BF01390048.  Google Scholar

[8]

T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products,, J. Reine Angew. Math., 469 (1995), 51.   Google Scholar

[9]

T. Giordano, I. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems,, Israel J. Math., 111 (1999), 285.  doi: 10.1007/BF02810689.  Google Scholar

[10]

T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\mathbb Z^d$-systems,, Invent. Math., 179 (2010), 119.  doi: 10.1007/s00222-009-0213-7.  Google Scholar

[11]

R. H. Herman, I. F. Putnam and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics,, Internat. J. Math., 3 (1992), 827.  doi: 10.1142/S0129167X92000382.  Google Scholar

[12]

W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory. Presentations of groups in terms of generators and relations,", Interscience Publishers [John Wiley & Sons, (1966).   Google Scholar

[13]

K. Matsumoto, On automorphisms of $C^*$-algebras associated with subshifts,, J. Operator Theory, 44 (2000), 91.   Google Scholar

[14]

K. Matsumoto, Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras,, Pacific J. Math., 246 (2010), 199.  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]

H. Matui, Homology and topological full groups of étale groupoids on totally disconnected spaces,, Proc. London Math. Soc. (3), 104 (2012), 27.  doi: 10.1112/plms/pdr029.  Google Scholar

[18]

I. F. Putnam, The $C^*$-algebras associated with minimal homeomorphisms of the Cantor set,, Pacific J. Math., 136 (1989), 329.   Google Scholar

[19]

M. Rφrdam, Classification of Cuntz-Krieger algebras,, K-theory, 9 (1995), 31.  doi: 10.1007/BF00965458.  Google Scholar

[20]

M. Rφrdam, F. Larsen and N. J. Laustsen, "An Introduction to K-Theory for $C^*$-Algebras,", London Mathematical Society Student Texts, 49 (2000).   Google Scholar

[21]

J. Spielberg, Free-product groups, Cuntz-Krieger algebras, and covariant maps,, Internat. J. Math., 2 (1991), 457.  doi: 10.1142/S0129167X91000260.  Google Scholar

[1]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[2]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[3]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
[4]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[5]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020041
[6]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
[7]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[8]

Kengo Matsumoto. On the Markov-Dyck shifts of vertex type. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 403-422. doi: 10.3934/dcds.2016.36.403
[9]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[10]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
[11]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[12]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[13]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281
[14]

Daniele D'angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda. Schreier graphs of the Basilica group. Journal of Modern Dynamics, 2010, 4 (1) : 167-205. doi: 10.3934/jmd.2010.4.167
[15]

Van Cyr, John Franks, Bryna Kra, Samuel Petite. Distortion and the automorphism group of a shift. Journal of Modern Dynamics, 2018, 13: 147-161. doi: 10.3934/jmd.2018015
[16]

Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020130
[17]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451
[18]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[19]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
[20]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences