doi: 10.3934/dcdss.2019146

Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations

1. 

Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA

2. 

Department of Mathematics, Hampton University, Hampton, VA 23668, USA

* Corresponding author: Palle Jorgensen

Received  November 2016 Revised  July 2017 Published  January 2019

We study positive transfer operators $R$ in the setting of general measure spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated path-space probability spaces $\left(Ω,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(Ω,\mathbb{P}\right)$ where the path-space $Ω$ may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-resolutions. Applications are given to topological dynamics, ergodic theory, and spectral theory, in general; to iterated function systems (IFSs), and to Markov chains in particular.

Citation: Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019146
References:
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show all references

References:
[1]

N. Agram and B. Oksendal, Malliavin calculus and optimal control of stochastic Volterra equations, J. Optim. Theory Appl., 167 (2015), 1070-1094. doi: 10.1007/s10957-015-0753-5. Google Scholar

[2]

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[3]

D. Alpay and A. Kipnis, Wiener chaos approach to optimal prediction, Numer. Funct. Anal. Optim., 36 (2015), 1286-1306. doi: 10.1080/01630563.2015.1065273. Google Scholar

[4]

D. B. Applebaum and R. L. Hudson, Fermion Itô's formula and stochastic evolutions, Comm. Math. Phys., 96 (1984), 473-496. doi: 10.1007/BF01212531. Google Scholar

[5]

W. B. Arveson, Subalgebras of $C^{*} $-algebras, Acta Math., 123 (1969), 141-224. doi: 10.1007/BF02392388. Google Scholar

[6]

E. Au-Yeung and J. J. Benedetto, Generalized Fourier frames in terms of balayage, J. Fourier Anal. Appl., 21 (2015), 472-508. doi: 10.1007/s00041-014-9369-7. Google Scholar

[7]

L. BaggettP. JorgensenK. Merrill and J. Packer, A non-MRA $C^r$ frame wavelet with rapid decay, Acta Appl. Math., 89 (2005), 251-270 (2006). doi: 10.1007/s10440-005-9011-4. Google Scholar

[8]

L. W. BaggettK. D. MerrillJ. A. Packer and A. B. Ramsay, Probability measures on solenoids corresponding to fractal wavelets, Trans. Amer. Math. Soc., 364 (2012), 2723-2748. doi: 10.1090/S0002-9947-2012-05584-X. Google Scholar

[9]

K. BandaraT. Rüberg and F. Cirak, Shape optimisation with multiresolution subdivision surfaces and immersed finite elements, Comput. Methods Appl. Mech. Engrg., 300 (2016), 510-539. doi: 10.1016/j.cma.2015.11.015. Google Scholar

[10]

O. E. Barndorff-NielsenF. E. Benth and B. Szozda, On stochastic integration for volatility modulated Brownian-driven Volterra processes via white noise analysis, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 1450011, 28pp. doi: 10.1142/S0219025714500118. Google Scholar

[11]

V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062. Google Scholar

[12]

O. Bratteli and P. Jorgensen, Wavelets Through a Looking Glass, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2002, The world of the spectrum. doi: 10.1007/978-0-8176-8144-9. Google Scholar

[13]

O. Bratteli and P. E. T. Jorgensen, Wavelet filters and infinite-dimensional unitary groups, in Wavelet analysis and applications (Guangzhou, 1999), vol. 25 of AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2002, 35-65. Google Scholar

[14]

M. E. Cates and T. A. Witten, Diffusion near absorbing fractals: Harmonic measure exponents for polymers, Phys. Rev. A (3), 35 (1987), 1809-1824. doi: 10.1103/PhysRevA.35.1809. Google Scholar

[15]

S. Chen and R. Hudson, Some properties of quantum Lévy area in Fock and non-Fock quantum stochastic calculus, Probab. Math. Statist., 33 (2013), 425-434. Google Scholar

[16]

Q. S. Cheng, Singularity and spectral representation of the Wold decomposition for multivariate stationary sequences, Acta Math. Sinica, 23 (1980), 684-694. Google Scholar

[17]

A. Colojoarǎ, On the Wold decomposition of some periodical stochastic processes, in Proceedings of the Sixth Congress of Romanian Mathematicians, Ed. Acad. Române, Bucharest, 1 2009,453-460. Google Scholar

[18]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, vol. 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1982, Translated from the Russian by A. B. Sosinskiĭ. doi: 10.1007/978-1-4615-6927-5. Google Scholar

[19]

C. D. Cutler, A general approach to predictive and fractal scaling dimensions in discrete-index time series, in Nonlinear dynamics and time series (Montreal, PQ, 1995), vol. 11 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 1997, 29-48. Google Scholar

[20]

I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970104. Google Scholar

[21]

S. Degenfeld-SchonburgE. Kaniuth and R. Lasser, Spectral synthesis in Fourier algebras of ultrapherical hypergroups, J. Fourier Anal. Appl., 20 (2014), 258-281. doi: 10.1007/s00041-013-9311-4. Google Scholar

[22]

Q. -R. Deng, K. -S. Lau and S. -M. Ngai, Separation conditions for iterated function systems with overlaps, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics, vol. 600 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, 1-20. doi: 10.1090/conm/600/11928. Google Scholar

[23]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446. Google Scholar

[24]

D. E. Dutkay and P. E. T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoam., 22 (2006), 131-180. doi: 10.4171/RMI/452. Google Scholar

[25]

D. E. Dutkay and P. E. T. Jorgensen, Wavelet constructions in non-linear dynamics, Electron. Res. Announc. Amer. Math. Soc., 11 (2005), 21-33. doi: 10.1090/S1079-6762-05-00143-5. Google Scholar

[26]

D. E. Dutkay and P. E. T. Jorgensen, Iterated function systems, Ruelle operators, and invariant projective measures, Math. Comp., 75 (2006), 1931-1970 (electronic). doi: 10.1090/S0025-5718-06-01861-8. Google Scholar

[27]

D. E. Dutkay and P. E. T. Jorgensen, The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative, in Geometry and Analysis of Fractals, vol. 88 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014, 65-95. doi: 10.1007/978-3-662-43920-3_3. Google Scholar

[28]

D. E. Dutkay and P. E. T. Jorgensen, Representations of Cuntz algebras associated to quasi-stationary Markov measures, Ergodic Theory Dynam. Systems, 35 (2015), 2080-2093. doi: 10.1017/etds.2014.37. Google Scholar

[29]

D. E. DutkayG. Picioroaga and M.-S. Song, Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 409 (2014), 1128-1139. doi: 10.1016/j.jmaa.2013.07.012. Google Scholar

[30]

D. E. Dutkay and K. Roysland, The algebra of harmonic functions for a matrix-valued transfer operator, J. Funct. Anal., 252 (2007), 734-762. doi: 10.1016/j.jfa.2007.04.014. Google Scholar

[31]

D. E. Dutkay and K. Roysland, Covariant representations for matrix-valued transfer operators, Integral Equations Operator Theory, 62 (2008), 383-410. doi: 10.1007/s00020-008-1623-4. Google Scholar

[32]

J. FageotE. Bostan and M. Unser, Wavelet statistics of sparse and self-similar images, SIAM J. Imaging Sci., 8 (2015), 2951-2975. doi: 10.1137/151003015. Google Scholar

[33]

C. FarsiE. GillaspyS. Kang and J. A. Packer, Separable representations, KMS states, and wavelets for higher-rank graphs, J. Math. Anal. Appl., 434 (2016), 241-270. doi: 10.1016/j.jmaa.2015.09.003. Google Scholar

[34]

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Figure 3.1.  The function G, see (3.8)
Figure 4.1.  The endomorphism $\sigma^{\left(u\right)}$ from (4.4)
Figure 4.2.  The Markov-move $\pi_{0}\rightarrow\pi_{1}$, see (4.18)
Figure 5.1.  The subspaces of a resolution
Figure 5.2.  Incremental Detail
Figure 5.3.  A coarser resolution in three directions in the plane, filtering in directions, $x,y$, and diagonal; — corresponding dyadic scaling in each coordinate direction. (Image cited from M.-S. Song, "Wavelet Image Compression" in [39].)
Figure 7.1.  $\sigma\left(x\right) = 2x$ mod 1
Figure 7.2.  Implications and containments. The containments and intersections hold for the sets of measures associated to $\left(X,\mathscr{B},\sigma,R\right)$. Note that in Example 7.2, $d\lambda = $ Lebesgue measure, $\sigma\left(x\right) = 2x$ mod 1; $\lambda\in Fix\left(\sigma\right)\cap\mathscr{L}\left(R\right)$, but $\lambda \notin\mathscr{K}_{1}$. For the various sets referenced in the figure, we refer to Definition 3.11 and Lemma 3.4 above
Figure 13.1.  Multiresolution expansion
Table 8.1.  Illustration by Examples. The set of measures itemized in the first two lines of the table refer to the operator $R$ as given in the two examples, Examples 7.1 (line 3), and 7.2 (line 4.) The verification of the respective properties is left to the reader
Meas. $\mathscr{L}\left(R\right)$ $\mathscr{L}_{1}\left(R\right)$ $Fix\left(\sigma\right)$ $\mathscr{K}_{1}=M_{1}R$ $\sqrt{\lambda}\in\mathscr{H}_{\infty}$ $\underset{{\displaystyle \cap_{i}\mathscr{H}\left(\lambda R^{i}\right)}}{\sqrt{\lambda}\in}$
Defn. $\lambda R\ll\lambda$ $\lambda R=\lambda$ $\lambda=\lambda\circ\sigma^{-1}$ $\lambda=\nu R$ $\widehat{S}\sqrt{\lambda}=\sqrt{\lambda}$
Ex 7.1 all $\lambda$ s.t.
$\lambda\ll dx$
$\lambda_{1}=dx$
(1)
$\lambda_{1}=dx$
$\lambda_{1}=dx$ Ex 7.1
$\lambda_{1}=dx$
$\lambda=\lambda R$
$\lambda_{1}=dx$ Ex 7.1
$\lambda_{1}=dx$
Ex 7.2 $\delta_{0}$, $\lambda_{1}=dx$ (2) $\delta_{0}$,
singletons
$\delta_{0}$, $\lambda=dx$ $\delta_{0}$
Ex 7.2
$\lambda\notin\mathscr{K}_{1}$
$\delta_{0}$ Ex 7.2
If $\lambda=dx$,
then
$\cap_{i}\mathscr{H}\left(\lambda R^{i}\right)\\=0$
Meas. $\mathscr{L}\left(R\right)$ $\mathscr{L}_{1}\left(R\right)$ $Fix\left(\sigma\right)$ $\mathscr{K}_{1}=M_{1}R$ $\sqrt{\lambda}\in\mathscr{H}_{\infty}$ $\underset{{\displaystyle \cap_{i}\mathscr{H}\left(\lambda R^{i}\right)}}{\sqrt{\lambda}\in}$
Defn. $\lambda R\ll\lambda$ $\lambda R=\lambda$ $\lambda=\lambda\circ\sigma^{-1}$ $\lambda=\nu R$ $\widehat{S}\sqrt{\lambda}=\sqrt{\lambda}$
Ex 7.1 all $\lambda$ s.t.
$\lambda\ll dx$
$\lambda_{1}=dx$
(1)
$\lambda_{1}=dx$
$\lambda_{1}=dx$ Ex 7.1
$\lambda_{1}=dx$
$\lambda=\lambda R$
$\lambda_{1}=dx$ Ex 7.1
$\lambda_{1}=dx$
Ex 7.2 $\delta_{0}$, $\lambda_{1}=dx$ (2) $\delta_{0}$,
singletons
$\delta_{0}$, $\lambda=dx$ $\delta_{0}$
Ex 7.2
$\lambda\notin\mathscr{K}_{1}$
$\delta_{0}$ Ex 7.2
If $\lambda=dx$,
then
$\cap_{i}\mathscr{H}\left(\lambda R^{i}\right)\\=0$
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