On angles determined by fractal subsets of the Euclidean space via Sobolev bounds for bi-linear operators (original) (raw)
Related papers
Harmonic analysis of fractal measures
Constructive Approximation, 1996
We consider affine systems inR n constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ bx:=R–1x+b; the corresponding measure μ onR n is a probability measure fixed by the self-similarity \(\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } \) . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR n such that the exponentials {eiλ·x}Λ form anorthogonal basis forL 2(μ); and (ii) the existence of a certaindual pair of representations of theC *-algebraO N wheren is the cardinality of the setB. (For eachN, theC *-algebraO N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL 2(μ) fail to span a dense subspace. Instead we show that theC *-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC *-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.
Dense analytic subspaces in fractalL 2-spaces
Journal D Analyse Mathematique, 1998
We consider self-similar measures mu\mu mu with support in the interval 0leqxleq10\leq x\leq 10leqxleq1 which have the analytic functions leftei2pinx:n=0,1,2,...right\left\{e^{i2\pi nx}:n=0,1,2,... \right\} leftei2pinx:n=0,1,2,...right span a dense subspace in L2(mu)L^{2}(\mu) L2(mu). Depending on the fractal dimension of mu\mu mu, we identify subsets PsubsetmathbbN0=0,1,2,...P\subset \mathbb{N}_{0}=\{0,1,2,... \} PsubsetmathbbN0=0,1,2,... such that the functions en:ninP\{e_{n}:n\in P\} en:ninP form an orthonormal basis for L2(mu)L^{2}(\mu) L2(mu). We also give a higher-dimensional affine construction leading to self-similar measures mu\mu mu with support in mathbbRnu\mathbb{R}^{\nu}mathbbRnu. It is obtained from a given expansive nu\nu nu-by-$\nu $ matrix and a finite set of translation vectors, and we show that the corresponding L2(mu)L^{2}(\mu) L2(mu) has an orthonormal basis of exponentials ei2pilambdacdotxe^{i2\pi \lambda \cdot x}ei2pilambdacdotx, indexed by vectors lambda\lambda lambda in mathbbRnu\mathbb{R}^{\nu}mathbbRnu, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.
Fourier duality for fractal measures with affine scales
Mathematics of Computation, 2012
For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in R d , and they both have the same matrix scaling. But the two use different translation vectors, one by a subset B in R d , and the other by a related subset L. Among other things, we show that there is then a pair of infinite discrete sets Γ(L) and Γ(B) in R d such that the Γ(L)-Fourier exponentials are orthogonal in L 2 (µB), and the Γ(B)-Fourier exponentials are orthogonal in L 2 (µL). These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line.
Multilinear generalized Radon transforms and point configurations
Forum Mathematicum, 2000
We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconertype problems involving (k + 1)-point configurations in geometric measure theory, with k ≥ 2, including the distribution of simplices, volumes and angles determined by the points of fractal subsets E ⊂ R d , d ≥ 2. If T k (E) denotes the set of noncongruent (k + 1)-point configurations determined by E, we show that if the Hausdorff dimension of E is greater than d − d−1 2k , then the k+1 2 -dimensional Lebesgue measure of T k (E) is positive. This compliments previous work on the Falconer conjecture ([5] and the references there), as well as work on finite point configurations . We also give applications to Erdös-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in .
Springer Proceedings in Mathematics & Statistics, 2012
In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of A · A + · · ·+ A · A, where A is a subset of the real line of a given Hausdorff dimension, A+A = {a+a ′ : a, a ′ ∈ A} and A · A = {a · a ′ : a, a ′ ∈ A}. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of R d is sufficiently large, then the k+1 2 -dimensional Lebesgue measure of the set of k-simplexes determined by this set is positive. The sharpness of these results and connection with number theoretic estimates is also discussed.
How projections affect the dimension spectrum of fractal measures
1997
We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension of the image of µ is min(m, D q (µ)); in particular, the q-dimension of µ is preserved provided m D q (µ). We also present results on the preservation of information dimension D 1 and pointwise dimension. Finally, for 0 q < 1 and q > 2 we give examples for which D q is not preserved by any linear transformation into R m . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.
Bulletin of the Brazilian Mathematical Society, New Series, 2021
We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions D ± µ (q), q ∈ R, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem [31] for the generalized fractal dimensions of the Bowen-Margulis measure associated with a C 1+α-Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like C 1-Axiom A systems), we show that the set of invariant measures such that D + µ (q) = 0 (q ≥ 1), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each s ∈ [0, 1), D + µ (s) is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in [25] for Lipschitz transformations which satisfy the specification property. Key words and phrases. Expansive homeomorphisms, Hausdorff dimension, packing dimension, invariant measures, generalized fractal dimensions, dynamical systems with specification * Work partially supported by CIENCIACTIVA C.G. 176-2015 † Work partially supported by FAPEMIG (a Brazilian government agency; Universal Project 001/17/CEX-APQ-00352-17) popular of all, the Hausdorff dimension, introduced in 1919 by Hausdorff, which gives a notion of size useful for distinguishing between sets of zero Lebesgue measure. Unfortunately, the Hausdorff dimension of relatively simple sets can be very hard to calculate; besides, the notion of Hausdorff dimension is not completely adapted to the dynamics per se (for instance, if Z is a periodic orbit, then its Hausdorff dimension is zero, regardless to whether the orbit is stable, unstable, or neutral). This fact led to the introduction of other characteristics for which it is possible to estimate the size of irregular sets. For this reason, some of these quantities were also branded as "dimensions" (although some of them lack some basic properties satisfied by Hausdorff dimension, such as σ-stability; see [12]). Several good candidates were proposed, such as the correlation, information, box counting and entropy dimensions, among others. Thus, in order to obtain relevant information about the dynamics, one should consider not only the geometry of the measurable set Z ⊂ X (where X is some Borel measurable space), but also the distribution of points on Z under f (which is assumed to be a measurable transformation). That is, one should be interested in how often a given point x ∈ Z visits a fixed subset Y ⊂ Z under f. If µ is an ergodic measure for which µ(Y) > 0, then for a typical point x ∈ Z, the average number of visits is equal to µ(Y). Thus, the orbit distribution is completely determined by the measure µ. On the other hand, the measure µ is completely specified by the distribution of a typical orbit. This fact is widely used in the numerical study of dynamical systems where the distributions are, in general, non-uniform and have a clearly visible fine-scaled interwoven structure of hot and cold spots, that is, regions where the frequency of visitations is either much greater than average or much less than average respectively.