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Register a new userProbability and Fourier duality for affine iterated function systems
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Let $d$ be a positive integer, and let $mu$ be a finite measure on $br^d$. In this paper we ask when it is possible to find a subset $Lambda$ in $br^d$ such that the corresponding complex exponential functions $e_lambda$ indexed by $Lambda$ are orthogonal and total in $L^2(mu)$. If this happens, we say that $(mu, Lambda)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(mu)$-functions is one side in the duality, while the points in $Lambda$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $mu$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in $br^d$; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.
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In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a linear response formula at almost every parameter of the perturbation.
In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchines theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.
Let $Dinmathbb{N}$, $qin[2,infty)$ and $(mathbb{R}^D,|cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $dot{F}^0_{infty,,q}(mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $dot{F}^0_{infty,,q}(mathbb{R}^D) bigcap L^{infty}(mathbb{R}^D)$. To get such decomposition, {bf (i),} The authors introduce some auxiliary function space $mathrm{WE}^{1,,q}(mathbb R^D)$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ defined via wavelet expansions. The authors proved $dot{F}^{0}_{1,q} subsetneqq L^{1} bigcup dot{F}^{0}_{1,q}subset {rm WE}^{1,,q}subset L^{1} + dot{F}^{0}_{1,q}$ and $mathrm{WE}^{infty,,q}(mathbb{R}^D)$ is strictly contained in $dot{F}^0_{infty,,q}(mathbb{R}^D)$. {bf (ii),} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by function set $mathrm{WE}^{1,,q}(mathbb R^D)$. {bf (iii),} We also consider the dual of $mathrm{WE}^{1,,q}(mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $dot{F}^0_{1,,q}(mathbb{R}^D)$ by Banach space $L^{1} + dot{F}^{0}_{1,q}$. Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $Dge2$, which needs some new methodology.
We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces.
We investigate graph-directed iterated function systems in mixed Euclidean and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are defined, and an upper bound for the Hausdorff dimension is obtained. The relation between the Haar measure and the Hausdorff measure is clarified. Finally, we discus an example in ${Bbb R}times{Bbb Q}sb 2$ and calculate upper and lower bounds for its Hausdorff dimension.
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