American Institute of Mathematical Sciences

Advanced
February  2006, 15(1): 353-366. doi: 10.3934/dcds.2006.15.353

Asymptotic orbit complexity of infinite measure preserving transformations

Roland Zweimüller 1,

1. 

Faculty of Mathematics, University of Vienna, Nordbergstraβe 15, 1090 Vienna, Austria

Received  December 2004 Revised  September 2005 Published  February 2006

We determine the asymptotics of the Kolmogorov complexity of symbolic orbits of certain infinite measure preserving transformations. Specifically, we prove that the Brudno - White individual ergodic theorem for the complexity generalizes to a ratio ergodic theorem analogous to previously established extensions of the Shannon - McMillan - Breiman theorem.
Keywords: Kolmogorov complexity, infinite invariant measure, indifferent orbits, algorithmic information content, ratio ergodic theorem., intermittency.
Mathematics Subject Classification: Primary: 37A40, 03D15; Secondary: 37A35, 37E0.
Citation: Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[1]

C. Bonanno. The algorithmic information content for randomly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 921-934. doi: 10.3934/dcdsb.2004.4.921
[2]

Mahendra Piraveenan, Mikhail Prokopenko, Albert Y. Zomaya. On congruity of nodes and assortative information content in complex networks. Networks and Heterogeneous Media, 2012, 7 (3) : 441-461. doi: 10.3934/nhm.2012.7.441
[3]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[4]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[5]

H. T. Banks, John E. Banks, R. A. Everett, John D. Stark. An adaptive feedback methodology for determining information content in stable population studies. Mathematical Biosciences & Engineering, 2016, 13 (4) : 653-671. doi: 10.3934/mbe.2016013
[6]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[7]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1
[8]

Manfred Denker, Samuel Senti, Xuan Zhang. Fluctuations of ergodic sums on periodic orbits under specification. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4665-4687. doi: 10.3934/dcds.2020197
[9]

Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005
[10]

Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[11]

Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042
[12]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223
[13]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[14]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[15]

Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149
[16]

Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250
[17]

Tomasz Downarowicz, Benjamin Weiss. Pure strictly uniform models of non-ergodic measure automorphisms. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 863-884. doi: 10.3934/dcds.2021140
[18]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[19]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[20]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy