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Probability, statistics and computation in dynamical systems

Published online by Cambridge University Press:  28 March 2014

STEFANO GALATOLO ,
ISAIA NISOLI  and
CRISTÓBAL ROJAS
STEFANO GALATOLO
Affiliation:
Dipartimento di Matematica Applicata, Università di Pisa, via Buonarroti 1, Pisa, Italy Email: galatolo@dm.unipi.it
ISAIA NISOLI
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática Cidade Universitària – Ilha do Fundão, Rio de Janeiro 21945-909, Brazil Email: nisoli@im.ufrj.br
CRISTÓBAL ROJAS
Affiliation:
Departamento de Matemáticas, Universidad Andres Bello, República 220, Santiago, Chile Email: cristobal.rojas@unab.cl
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Abstract

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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.

References

Avigad, J., Gerhardy, P. and Towsner, H. (2010) Local stability of ergodic averages. Transactions of the American Mathematical Society 362 261–288.CrossRefGoogle Scholar
Binder, I., Braverman, M. and Yampolsky, M. (2006) On computational complexity of Siegel Julia sets. Communications in Mathematical Physics 264 317–334.CrossRefGoogle Scholar
Binder, I., Braverman, M. and Yampolsky, M. (2007) Filled Julia sets with empty interior are computable. Journal FoCM 7 405–416.Google Scholar
Binder, I., Braverman, M., Rojas, C. and Yampolsky, M. (2011) Computability of Brolin–Lyubich measure. Communications in Mathematical Physics 308 3743–771.CrossRefGoogle Scholar
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 425–491.CrossRefGoogle Scholar
Brent, R. (1980) An Improved Monte Carlo Factorization Algorithm. BIT Numerical Mathematics 20 176–184CrossRefGoogle Scholar
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 17–30.CrossRefGoogle Scholar
Braverman, M. (2006) Parabolic Julia Sets are Polynomial Time Computable. Nonlinearity 19 (6)1383–1402.CrossRefGoogle Scholar
Braverman, M. and Yampolsky, M. (2006) Non-computable Julia sets. Journal of the American Mathematical Society 19 551–578.CrossRefGoogle Scholar
Braverman, M. and Yampolsky, M. (2008a) Computability of Julia sets, Algorithms and Computation in Mathematics 23, Springer-Verlag.Google Scholar
Braverman, M. and Yampolsky, M. (2008b) Computability of Julia sets. Moscow Mathematics Journal 8 185–231.CrossRefGoogle Scholar
Braverman, M., Grigo, A. and Rojas, C. (2012) Noise vs computational intractability in dynamics. arXiv:1201.0488.CrossRefGoogle Scholar
Brolin, H. (1965) Invariant sets under iteration of rational functions. Arkiv för matematik 6 103–144.CrossRefGoogle Scholar
Carleson, L. and Gamelin, T. W. (1993) Complex Dynamics, Springer.CrossRefGoogle Scholar
Dellnitz, M. and Junge, O. (1999) On the approximation of complicated dynamical behavior. SIAM Journal on Numerical Analysis 36 491–515.CrossRefGoogle Scholar
Dellnitz, M. and Junge, O. (2002) Set Oriented Numerical Methods for Dynamical Systems. In: Fiedler, B. (ed.) Handbook of dynamical systems 2, North Holland 221–264.Google Scholar
Froyland, G. (2001) Extracting dynamical behaviour via Markov models. In: Mees, A. (ed.) Proceedings Nonlinear Dynamics and Statistics: Newton Institute, Cambridge 1998, Birkhauser 283–324.Google Scholar
Froyland, G. (2007) On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete and Continuous Dynamical Systems 17 (3)203–221.CrossRefGoogle Scholar
Galatolo, S. and Nisoli, I. (2012) A simple approach to rigorous approximation of invariant measures. arXiv:1109.2342Google Scholar
Galatolo, S., Hoyrup, M. and Rojas, C. (2009) A constructive Borel–Cantelli lemma. Constructing orbits with required statistical properties. Theoretical Computer Science 410 2207–2222.CrossRefGoogle Scholar
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. arXiv.org/abs/1006.0392v1.CrossRefGoogle Scholar
Galatolo, S., Hoyrup, M. and Rojas, C. (2011a) Dynamics and abstract computability: computing invariant measures. Discrete and Continuous Dynamical Systems 29 (1)193–212.CrossRefGoogle Scholar
Galatolo, S., Hoyrup, M. and Rojas, C. (2011b) Statistical properties of dynamical systems - simulation and abstract computation. Chaos, Solitons and Fractals 45 (1)1–14.Google Scholar
Hasselblatt, B. and Katok, A. (1995) Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications 54, Cambridge University Press.Google Scholar
Hoyrup, M. and Rojas, C. (2009) Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation 207 830–847.CrossRefGoogle Scholar
Lanford, O. E. III (1998) Informal Remarks on the Orbit Structure of Discrete Approximations to Chaotic Maps. Experimental Mathematics 7 317–324CrossRefGoogle Scholar
Lasota, A. and Yorke, J. (1973) On the existence of invariant measures for piecewise monotonic transformations. Transactions of the American Mathematical Society 186 481–488.CrossRefGoogle Scholar
Liverani, C. (2001) Rigorous numerical investigations of the statistical properties of piecewise expanding maps – A feasibility study. Nonlinearity 14 463–490.CrossRefGoogle Scholar
Lyubich, M. (1982) The measure of maximal entropy of a rational endomorphism of a Riemann sphere. Funktsional'nyi Analiz i ego prilozheniya 16 78–79.Google Scholar
Mañé, R. (1987) Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 8, Springer-Verlag.CrossRefGoogle Scholar
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 222–252.CrossRefGoogle Scholar
Turing, A. (1936) On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society (2) 42 230–265.Google Scholar
Walters, P. (1982) An introduction to Ergodic Theory, Springer-Verlag Graduate Texts in Mathematics 79.CrossRefGoogle Scholar
Weihrauch, K. (2000) Computable Analysis. An Introduction, Springer-Verlag.CrossRefGoogle Scholar
Young, L.-S. (2002) What are SRB measures, and which dynamical systems have them? Journal of Statistical Physics 108 733–754.CrossRefGoogle Scholar