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摘要
In the preceding paper [S. Scott, C. Redd, L. Kutznetsov, I. Mezic, and C. Jones, Phys. D, 238 (2009), pp. 1668-1679] a technique for assessing the extent to which a given dynamical system falls short of being ergodic is introduced, and in [I. Rypina, S. Scott, L. J. Pratt, and M. G. Brown, Nonlinear Process. Geophys., 18 (2011), pp. 977-987], the technique is applied to ocean flows. The quantity measured with this technique is called the ergodicity defect of a map T at scale s, and it is denoted as d(s, T). The approach is aimed at capturing both deviation from ergodicity and its dependence on scale. In order to build intuition, the d(s, T) discussed in the former paper uses the Haar scaling functions (dilations and translations of the indicator function on the phase space) as analyzing functions and is subsequently called the Haar ergodicity defect. In this paper, this Haar d(s, T) is further developed and alternate and recursive formulas for d(s, T) are given. With these formulas, the Haar d(s, T) is expressed in terms of time averages of the Haar mother wavelets, and this mother wavelet perspective allows for easier identification and computation of subregions which are ergodic. The Haar wavelet is a standard example of a general class of wavelets called multiresolution analysis (MRA) wavelets, and while the Haar wavelets are ideal for gaining insight, they lack properties often desired in applications, e. g., continuity. With this in mind, the ergodicity defect is expressed and considered in a general MRA wavelet framework (i.e., using a general MRA scaling function and mother wavelet to obtain analyzing functions) and conditions for the existence of a recursive formula for this general defect are discussed. Specifically it is shown that a recursive (and alternate) formula can only be obtained in the case of the Haar defect.
- 出版日期2013
