Cramér distance and discretizations of circle expanding maps II: simulations

References

• P.M. Binder, Limit cycles in a quadratic discrete iteration, Phys. D 57(1–2) (1992), pp. 31–38.
   Google Scholar
• M. Blank, Discreteness and continuity in problems of chaotic dynamics, Amer. Math. Mon. 105(3) (1998), pp. 299 33299–(eng) .
   Google Scholar
• S. Bobkov and M. Ledoux, One-dimensional empirical measures, order statistics, and Kantorovich transport distances, Mem. Amer. Math. Soc. 261 (1259) (2019).
   Web of Science ®Google Scholar
• B. Bollobás, Random Graphs, 2nd ed., Cambridge University Press, 2001.
   Google Scholar
• A. Boyarsky, Computer orbits, Comput. Math. Appl. Ser. A 12(10) (1986), pp. 1057–1064.
   Google Scholar
• R.M. Corless, Continued fractions and chaos, Amer. Math. Monthly 99(3) (1992), pp. 203–215 MR 1216205.
   Web of Science ®Google Scholar
• R.M. Corless, G.W. Frank, and J.G. Monroe, Chaos and continued fractions, Phys. D 46(2) (1990), pp. 241–253 MR 1083721.
   Google Scholar
• J. Dedecker and F. Merlevède, The empirical distribution function for dependent variables: asymptotic and nonasymptotic results in Lpp, ESAIM Probab. Stat. 11 (2007), pp. 102–114.
   Google Scholar
• P. Diamond and I. Vladimirov, Asymptotic independence and uniform distribution of quantization errors for spatially discretized dynamical systems, Int. J. Bifurc. Chaos Appl. Sci. Eng. 8(7) (1998), pp. 1479–1490.
   Web of Science ®Google Scholar
• P. Diamond and I. Vladimirov, Branching processes and computational collapse of discretized unimodal mappings, Int. J. Bifurc. Chaos Appl. Sci. Eng. 12(12) (2002), pp. 2847–2867.
   Web of Science ®Google Scholar
• P. Diamond and I. Vladimirov, Set-valued Markov chains and negative semitrajectories of discretized dynamical systems, J. Nonlinear Sci. 12(2) (2002), pp. 113–141.
   Web of Science ®Google Scholar
• P. Diamond, P. Kloeden, and A.V. Pokrovskii, An invariant measure arising in computer simulation of a chaotic dynamical system, J. Nonlinear Sci. 4(1) (1994), pp. 59–68 MR 1258483 (94m:58136).
   Web of Science ®Google Scholar
• P. Diamond, P. Kloeden, A.V. Pokrovskii, and A. Vladimirov, Collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre, Phys. D 86(4) (1995), pp. 559–571 MR 1353178 (96e:58098).
   Google Scholar
• P. Diamond, A. Klemm, P. Kloeden, and A. Pokrovskii, Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures, J. Stat. Phys. 84(3–4) (1996), pp. 713–733 MR 1400185 (97i:58102).
   Web of Science ®Google Scholar
• P. Diamond, M. Suzuki, P. Kloeden, and A. Pokrovskii, Statistical properties of discretizations of a class of chaotic dynamical systems, Comput. Math. Appl. 31(11) (1996), pp. 83–95 MR 1392078 (96m:58164).
   Web of Science ®Google Scholar
• P. Diamond, V. Kozyakin, P. Kloeden, and A. Pokrovskii, A model for roundoff and collapse in computation of chaotic dynamical systems, Math. Comput. Simul. 44(2) (1997), pp. 163–185 MR 1481914 (99a:58114).
   Web of Science ®Google Scholar
• T. Erber, T.M. Rynne, W.F. Darsow, and M.J. Frank, The simulation of random processes on digital computers: unavoidable order, J. Comput. Phys. 49(3) (1983), pp. 394–419.
   Web of Science ®Google Scholar
• P.P. Flockermann, Discretizations of expanding maps, Ph.D. thesis, ETH (Zurich), 2002
   Google Scholar
• S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures, SIAM. J. Appl. Dyn. Syst. 13(2) (2014), pp. 958–985.
   Web of Science ®Google Scholar
• S. Galatolo and A. Sorrentino, Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle, Discrete Contin. Dyn. Syst. 42(2) (2022), pp. 815–839.
   Web of Science ®Google Scholar
• S. Galatolo, I. Nisoli, and C. Rojas, Probability, statistics and computation in dynamical systems, Math. Struct. Comput. Sci. 24(3) (2014), pp. e240304, 25.
   Web of Science ®Google Scholar
• S. Galatolo, M. Monge, I. Nisoli, and F. Poloni, A general framework for the rigorous computation of invariant densities and the coarse-fine strategy, Chaos Solit. Fractals 170 (2023), pp. 113329.
   Web of Science ®Google Scholar
• P.-A. Guihéneuf, Dynamical properties of spatial discretizations of a generic homeomorphism, Ergod. Theory Dyn. Syst. 35(5) (2015), pp. 1474–1523.
   Google Scholar
• P.-A. Guihéneuf, Discrétisations spatiales de systèmes dynamiques génériques, Ph.D. thesis, Université Paris-Sud, 2015
   Google Scholar
• P.-A. Guihéneuf, Physical measures of discretizations of generic diffeomorphisms, to appear in Ergodic Theory and Dynamical Systems, (2015). arXiv:1510.00720
   Google Scholar
• P.-A. Guihéneuf, Degree of recurrence of generic diffeomorphisms, Discrete Anal. (2019), Paper No. 1, 43. https://discreteanalysisjournal.com/article/7545.
   Web of Science ®Google Scholar
• P.-A. Guihéneuf and M. Monge, Cramér distance and discretizations of circle expanding maps I: theory, (2022). arXiv 2206.07991
   Google Scholar
• P. Góra and A. Boyarsky, Why computers like Lebesgue measure, Comput. Math. Appl. 16(4) (1988), pp. 321–329.
   Web of Science ®Google Scholar
• P. Kloeden, J. Mustard, and A.V. Pokrovskii, Statistical properties of some spatially discretized dynamical systems, Z. Angew. Math. Phys. 50(4) (1999), pp. 638–660 MR 1709708 (2000e:37039).
   Web of Science ®Google Scholar
• O.E. Lanford, Informal remarks on the orbit structure of discrete approximations to chaotic maps, Exp. Math. 7(4) (1998), pp. 317–324.
   Web of Science ®Google Scholar
• Y.-E. Levy, Some remarks about computer studies of dynamical systems, Phys. Lett. A 88(1) (1982), pp. 1–3.
   Web of Science ®Google Scholar
• T. Miernowski, Dynamique discrétisée et stochastique, géométrie conforme, Ph.D. thesis, ÉNS de Lyon, 2005
   Google Scholar
• T. Miernowski, Discrétisations des homéomorphismes du cercle, Ergod. Theory Dyn. Syst.26(6) (2006), pp. 1867–1903.
   Google Scholar
• M. Monge, I. Nisoli, and F. Poloni, CompInvMeas-Python, Computation of Invariant Measures in Python, (2015). Available at https://bitbucket.org/fph/compinvmeas-python/src/master/
   Google Scholar
• S.T. Rachev, Probability Metrics and the Stability of Stochastic Models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991.
   Google Scholar
• M. Scarowsky and A. Boyarsky, Long periodic orbits of the triangle map, Proc. Amer. Math. Soc.97(2) (1986), pp. 247–254.
   Web of Science ®Google Scholar
• The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 9.2), (2022). Available at https://www.sagemath.org
   Google Scholar
• F. Vivaldi and I. Vladimirov, Pseudo-randomness of round-off errors in discretized linear maps on the plane, Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(11) (2003), pp. 3373–3393.
   Web of Science ®Google Scholar
• I. Vladimirov, Quantized linear systems on integer lattices: a frequency-based approach, CADSEM Reports, 96–032, 96–033, Parts I,II. Center for Applied Dynamical Systems and Environmental Modeling, Deakin University, Geelong, Australia, 1996.
   Google Scholar
• I. Vladimirov, N. Kuznetsov, and P. Diamond, Frequency measurability, algebras of quasiperiodic sets and spatial discretizations of smooth dynamical systems, Math. Comput. Simul. 52(3–4) (2000), pp. 251–272.
   Web of Science ®Google Scholar
• V.V. Voevodin, The asymptotic distribution of round-off errors in linear transformations, Ž. Vyčisl. Mat I Mat. Fiz. 7 (1967), pp. 965–976.
   Google Scholar