2011, 18: 22-30. doi: 10.3934/era.2011.18.22

Realization of joint spectral radius via Ergodic theory

1. 

Department of Mathematics, Nanjing University, Nanjing, 210093

2. 

Department of Mathematics, Zhongshan University, Guangzhou 510275

3. 

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901

Received  August 2010 Revised  March 2011 Published  June 2011

Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.
Citation: Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
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show all references

References:
[1]

Springer Monographs in Math., Springer-Verlag, Berlin, Heidelberg, New York, 1998.  Google Scholar

[2]

Autom. Remote Control, 49 (1988), 152-157, 283-287, 558-565. Google Scholar

[3]

Linear Algebra Appl., 402 (2005), 101-110. doi: 10.1016/j.laa.2004.12.007.  Google Scholar

[4]

Linear Algebra Appl., 166 (1992), 21-27. doi: 10.1016/0024-3795(92)90267-E.  Google Scholar

[5]

SIAM J. Matrix Anal. Appl., 24 (2003), 963-970. doi: 10.1137/S0895479801397846.  Google Scholar

[6]

SIAM J. Matrix Anal. Appl., 31 (2009), 865-876. doi: 10.1137/080723764.  Google Scholar

[7]

IEEE Trans. Inform. Theory, 52 (2006), 5122-5127. doi: 10.1109/TIT.2006.883615.  Google Scholar

[8]

Systems Control Lett., 41 (2000), 135-140. doi: 10.1016/S0167-6911(00)00049-9.  Google Scholar

[9]

Linear Algebra Appl., 368 (2003), 71-81. doi: 10.1016/S0024-3795(02)00658-4.  Google Scholar

[10]

J. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2.  Google Scholar

[11]

IEEE Trans. Inform. Theory, 38 (1992), 876-881. doi: 10.1109/18.119744.  Google Scholar

[12]

J. Math. Anal. Appl., 379 (2011), 827-833. doi: 10.1016/j.jmaa.2010.12.059.  Google Scholar

[13]

J. Differential Equations, 250 (2011), 3584-3629. doi: 10.1016/j.jde.2011.01.029.  Google Scholar

[14]

Nonlinearity, 24 (2011), 1565-1573. doi: 10.1088/0951-7715/24/5/009.  Google Scholar

[15]

SIAM J. Control Optim., 47 (2008), 2137-2156. doi: 10.1137/070699676.  Google Scholar

[16]

Automatica, 47 (2011), 1512-1519. doi: doi: 10.1016/j.automatica.2011.02.034.  Google Scholar

[17]

SIAM J. Math. Anal., 22 (1991), 1388-1410. doi: 10.1137/0522089.  Google Scholar

[18]

SIAM J. Math. Anal., 23 (1992), 1031-1079. doi: 10.1137/0523059.  Google Scholar

[19]

Linear Algebra Appl., 161 (1992), 227-263. doi: 10.1016/0024-3795(92)90012-Y.  Google Scholar

[20]

Linear Algebra Appl., 161 (1992), 227-263, Corrigendum/addendum, 327 (2001), 69-83.  Google Scholar

[21]

Linear Algebra Appl., 220 (1995), 151-159. doi: 10.1016/0024-3795(93)00320-Y.  Google Scholar

[22]

IEEE Trans. Automat. Control, 54 (2009), 337-341. doi: 10.1109/TAC.2008.2007177.  Google Scholar

[23]

Adv. Math., 226 (2011), 4667-4701. doi: 10.1016/j.aim.2010.12.012.  Google Scholar

[24]

in "Linear Algebra for Signal Processing," IMA Vol. Math. Appl. 69, Springer-Verlag, New York, (1995), 51-61.  Google Scholar

[25]

Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3.  Google Scholar

[26]

Ann. Math. Statist., 31 (1960), 457-469. doi: 10.1214/aoms/1177705909.  Google Scholar

[27]

Linear Algebra Appl., 428 (2008), 2296-2311. doi: 10.1016/j.laa.2007.08.001.  Google Scholar

[28]

Autom. Remote Control, 68 (2007), 174-209. doi: 10.1134/S0005117906040171.  Google Scholar

[29]

Linear Algebra Appl., 214 (1995), 17-42. doi: 10.1016/0024-3795(93)00052-2.  Google Scholar

[30]

IEEE Trans. Inform. Theory, 47 (2001), 433-442. doi: 10.1109/18.904557.  Google Scholar

[31]

Princeton University Press, Princeton, New Jersey, 1960.  Google Scholar

[32]

Trudy Mosk Mat. Obšč., 19 (1968), 179-210.  Google Scholar

[33]

Linear Algebra Appl., 428 (2008), 2368-2384. doi: 10.1016/j.laa.2007.12.009.  Google Scholar

[34]

Automat. Remote Control, 52 (1991), 1379-1387.  Google Scholar

[35]

Indag. Math., 22 (1960), 379-381.  Google Scholar

[36]

J. Differential Equations, 148 (1998), 334-350. doi: 10.1006/jdeq.1998.3471.  Google Scholar

[37]

Linear Algebra Appl., 252 (1997), 61-70. doi: 10.1016/0024-3795(95)00592-7.  Google Scholar

[38]

SIAM Rev., 49 (2007), 545-592. doi: 10.1137/05063516X.  Google Scholar

[39]

Nonlinearity, 13 (2000), 113-143. doi: 10.1088/0951-7715/13/1/306.  Google Scholar

[40]

Ph.D thesis, Université Catholique de Louvain, 2005. Google Scholar

[41]

Math. Control Signals Systems, 10 (1997), 31-40. doi: 10.1007/BF01219774.  Google Scholar

[42]

G.T.M., 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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