Abstract
We prove an almost sure ergodic theorem for abstract quasistatic dynamical systems, as an attempt of taking steps toward an ergodic theory of such systems. The result at issue is meant to serve as a working counterpart of Birkhoff’s ergodic theorem which fails in the quasistatic setup. It is formulated so that the conditions, which essentially require sufficiently good memory-loss properties, could be verified in a straightforward way in physical applications. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then illustrate the use of the theorem by examples.
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References
Aimino, R., Hu, H., Nicol, M., Török, A., Sandro, V.: Polynomial loss of memory for maps of the interval with a neutral fixed point. arXiv:1402.4399
Chernov, N., Markarian, R.: Chaotic billiards, volume 127 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2006)
Dobbs, N., Stenlund, M.: Quasistatic dynamical systems. Submitted. Available from: arXiv:1504.01926
Gallavotti, G.: Irreversibility time scale. Chaos 16(2), 023120 (2006). 7, doi:10.1063/1.2205387
Gallavotti, G., Presutti, E.: Nonequilibrium, thermostats, and thermodynamic limit. J. Math. Phys. 51(1), 015202 (2010). 32, doi:10.1063/1.3257618
Gupta, C., Ott, W., Török, A.: Memory loss for time-dependent piecewise expanding systems in higher dimension. Math. Res. Lett. 20(1), 141–161 (2013). doi:10.4310/MRL.2013.v20.n1.a12
Leppänen, J., Stenlund, M.: Quasistatic dynamics with intermittency. Mathematical Physics. Math. Phys. Anal. Geom. 19(2), 1–23 (2016). doi:10.1007/s11040-016-9212-2
Mohapatra, A., Ott, W.: Memory loss for nonequilibrium open dynamical systems. Discrete Contin. Dyn. Syst. 34(9), 3747–3759 (2014). doi:10.3934/dcds.2014.34.3747
Ott, W., Stenlund, M., Young, L.-S.: Memory loss for time-dependent dynamical systems. Math. Res. Lett. 16(3), 463–475 (2009). doi:10.4310/MRL.2009.v16.n3.a7
Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Statist. Phys. 95(1-2), 393–468 (1999). doi:10.1023/A:1004593915069
Stenlund, M.: Non-stationary compositions of Anosov diffeomorphisms. Nonlinearity 24, 2991–3018 (2011). doi:10.1088/0951-7715/24/10/016
Stenlund, M.: Commun. Math. Phys. 325, 879–916 (2014). Available from. doi:10.1007/s00220-013-1870-3
Stenlund, M., Young, L.-S., Zhang, H.: Dispersing billiards with moving scatterers. Commun. Math. Phys. 322(3), 909–955 (2013). doi:10.1007/s00220-013-1746-6
Young, L.-S.: What are SRB measures, and which dynamical systems have them? J. Statist. Phys. 108(5–6), 733–754 (2002). doi:10.1023/A:1019762724717
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Stenlund, M. An Almost Sure Ergodic Theorem for Quasistatic Dynamical Systems. Math Phys Anal Geom 19, 14 (2016). https://doi.org/10.1007/s11040-016-9217-x
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DOI: https://doi.org/10.1007/s11040-016-9217-x