相关论文: $L^q$-spectra of measures on planar non-conformal …
We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function…
We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak…
We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we…
In this paper a sponge in $\mathbb{R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced…
We study $L^q$-spectra of planar self-affine measures generated by diagonal systems with an emphasis on providing closed form expressions. We answer a question posed by Fraser in 2016 in the negative by proving that a certain natural closed…
We study the generalized q-dimensions of measures supported on non-autonomous conformal attractors, which are the generalizations of Moran sets and the attractors of iterated function systems. We first prove that the critical values of…
We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.…
We introduce and study a family of non-conformal and non-uniformly contracting iterated function systems. We refer to the attractors of such systems as parabolic carpets. Roughly speaking they may be thought of as nonlinear analogues of…
We study the multifractal analysis of a class of self-similar measures with overlaps. This class, for which we obtain explicit formulae for the L^q spectrum tau(q) as well as the singularity spectrum f(alpha), is sufficiently large to point…
We investigate certain spectral properties of the Bernoulli convolution measures on attractor sets arising from iterated function systems on the real line. In particular, we examine collections of orthogonal exponential functions in the…
Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this…
It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this…
We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class…
In the paper, we study the generalized $q$-dimensions of measures supported by nonautonomous attractors, which are the generalization of classic Moran sets and attractors of iterated function systems. First, we estimate the generalized…
We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the…
We study the $L^q$ dimension $D(\nu,q)\ (q>1)$ of stationary measures $\nu$ for M\"{o}bius iterated function systems on $\mathbb{R}$ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem…
We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we…
We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also…
We consider coupled linear parabolic systems and we establish estimates in $L^q$-norm for the sources in terms of observations on the corresponding solutions on a part of the boundary. The main tool is a family of Carleman estimates in…
For self-similar measures with overlaps, closed formulas of the $L^q$-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. \textbf{106} (2019), 56--103]. We extend the…