Skip to main content Accessibility help

Probability, statistics and computation in dynamical systems


We discuss some recent results related to the deduction of a suitable probabilistic model for the description of the statistical features of a given deterministic dynamics. More precisely, we motivate and investigate the computability of invariant measures and some related concepts. We also present some experiments investigating the limits of naive simulations in dynamics.

Hide All
Avigad, J., Gerhardy, P. and Towsner, H. (2010) Local stability of ergodic averages. Transactions of the American Mathematical Society 362 261288.
Binder, I., Braverman, M. and Yampolsky, M. (2006) On computational complexity of Siegel Julia sets. Communications in Mathematical Physics 264 317334.
Binder, I., Braverman, M. and Yampolsky, M. (2007) Filled Julia sets with empty interior are computable. Journal FoCM 7 405416.
Binder, I., Braverman, M., Rojas, C. and Yampolsky, M. (2011) Computability of Brolin–Lyubich measure. Communications in Mathematical Physics 308 3743771.
Brattka, V., Hertling, P. and Weihrauch, K. (2008) A Tutorial on Computable Analysis. In: Cooper, S. B., Lowe, B. and Sorbi, A.New Computational Paradigms: Changing Conceptions of What is Computable 425491.
Brent, R. (1980) An Improved Monte Carlo Factorization Algorithm. BIT Numerical Mathematics 20 176184
Braverman, M. (2004) Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are Poly-Time Computable. In: Brattka, V., Staiger, L. and Weihrauch, K. (eds.) Proceedings of the 6th Workshop on Computability and Complexity in Analysis. Electronic Notes in Theoretical Computer Science 120 1730.
Braverman, M. (2006) Parabolic Julia Sets are Polynomial Time Computable. Nonlinearity 19 (6)13831402.
Braverman, M. and Yampolsky, M. (2006) Non-computable Julia sets. Journal of the American Mathematical Society 19 551578.
Braverman, M. and Yampolsky, M. (2008a) Computability of Julia sets, Algorithms and Computation in Mathematics 23, Springer-Verlag.
Braverman, M. and Yampolsky, M. (2008b) Computability of Julia sets. Moscow Mathematics Journal 8 185231.
Braverman, M., Grigo, A. and Rojas, C. (2012) Noise vs computational intractability in dynamics. arXiv:1201.0488.
Brolin, H. (1965) Invariant sets under iteration of rational functions. Arkiv för matematik 6 103144.
Carleson, L. and Gamelin, T. W. (1993) Complex Dynamics, Springer.
Dellnitz, M. and Junge, O. (1999) On the approximation of complicated dynamical behavior. SIAM Journal on Numerical Analysis 36 491515.
Dellnitz, M. and Junge, O. (2002) Set Oriented Numerical Methods for Dynamical Systems. In: Fiedler, B. (ed.) Handbook of dynamical systems 2, North Holland 221264.
Froyland, G. (2001) Extracting dynamical behaviour via Markov models. In: Mees, A. (ed.) Proceedings Nonlinear Dynamics and Statistics: Newton Institute, Cambridge 1998, Birkhauser 283324.
Froyland, G. (2007) On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete and Continuous Dynamical Systems 17 (3)203221.
Galatolo, S. and Nisoli, I. (2012) A simple approach to rigorous approximation of invariant measures. arXiv:1109.2342
Galatolo, S., Hoyrup, M. and Rojas, C. (2009) A constructive Borel–Cantelli lemma. Constructing orbits with required statistical properties. Theoretical Computer Science 410 22072222.
Galatolo, S., Hoyrup, M. and Rojas, C. (2010) Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems. In: Proceedings 7th International Conference on Computability and Complexity in Analysis.
Galatolo, S., Hoyrup, M. and Rojas, C. (2011a) Dynamics and abstract computability: computing invariant measures. Discrete and Continuous Dynamical Systems 29 (1)193212.
Galatolo, S., Hoyrup, M. and Rojas, C. (2011b) Statistical properties of dynamical systems - simulation and abstract computation. Chaos, Solitons and Fractals 45 (1)114.
Hasselblatt, B. and Katok, A. (1995) Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press.
Hoyrup, M. and Rojas, C. (2009) Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation 207 830847.
Lanford, O. E. III (1998) Informal Remarks on the Orbit Structure of Discrete Approximations to Chaotic Maps. Experimental Mathematics 7 317324
Lasota, A. and Yorke, J. (1973) On the existence of invariant measures for piecewise monotonic transformations. Transactions of the American Mathematical Society 186 481488.
Liverani, C. (2001) Rigorous numerical investigations of the statistical properties of piecewise expanding maps – A feasibility study. Nonlinearity 14 463490.
Lyubich, M. (1982) The measure of maximal entropy of a rational endomorphism of a Riemann sphere. Funktsional'nyi Analiz i ego prilozheniya 16 7879.
Mañé, R. (1987) Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 8, Springer-Verlag.
Müller, N. (2001) The iRRAM: Exact Arithmetic in C++. In: Blanck, J., Brattka, V. and Hertling, P. (eds.) Computability and Complexity in Analysis. Springer-Verlag Lecture Notes in Computer Science 2064 222252.
Turing, A. (1936) On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society (2) 42 230265.
Walters, P. (1982) An introduction to Ergodic Theory, Springer-Verlag Graduate Texts in Mathematics 79.
Weihrauch, K. (2000) Computable Analysis. An Introduction, Springer-Verlag.
Young, L.-S. (2002) What are SRB measures, and which dynamical systems have them? Journal of Statistical Physics 108 733754.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed