HAUSDORFF DIMENSION OF SELF-SIMILAR SETS WITH OVERLAPS

2001 ◽  
Vol 63 (3) ◽  
pp. 655-672 ◽  
Author(s):  
SZE-MAN NGAI ◽  
YANG WANG
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar

The notion of ‘finite type’ iterated function systems of contractive similitudes is introduced, and a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition is described. This method extends a previous one by Lalley, and applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.

scholarly journals The quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

2011 ◽  
Vol 63 (3) ◽  
pp. 648-688 ◽  
Author(s):  
Sze-Man Ngai
Keyword(s):  
Golden Ratio ◽  
Iterated Function Systems ◽  
Spectral Dimension ◽  
Open Set Condition ◽  
Bernoulli Convolution ◽  
One Dimensional ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

AbstractWe set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

scholarly journals Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

2021 ◽  
Vol 9 (1) ◽  
pp. 90-119
Author(s):  
Matthew Badger ◽  
Vyron Vellis
Keyword(s):  
Metric Spaces ◽  
Optimal Parameter ◽  
Complete Metric Space ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Interesting Phenomenon ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Path Connected

Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

-SPECTRUM OF SELF-SIMILAR MEASURES WITH OVERLAPS IN THE ABSENCE OF SECOND-ORDER IDENTITIES

2018 ◽  
Vol 106 (1) ◽  
pp. 56-103 ◽  
Author(s):  
SZE-MAN NGAI ◽  
YUANYUAN XIE
Keyword(s):  
Hausdorff Dimension ◽  
Finite Type ◽  
Iterated Function Systems ◽  
Second Order ◽  
Higher Dimension ◽  
Closed Formula ◽  
Dimension Spectrum ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

For the class of self-similar measures in $\mathbb{R}^{d}$ with overlaps that are essentially of finite type, we set up a framework for deriving a closed formula for the $L^{q}$-spectrum of the measure for $q\geq 0$. This framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the $L^{q}$-spectrum have only been obtained earlier for measures satisfying Strichartz’s second-order identities. We illustrate how to use our results to prove the differentiability of the $L^{q}$-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

Continuity of packing measure functions of self-similar iterated function systems

2011 ◽  
Vol 32 (3) ◽  
pp. 1101-1115 ◽  
Author(s):  
HUA QIU
Keyword(s):  
Iterated Function Systems ◽  
Packing Measure ◽  
Open Set Condition ◽  
Open Set ◽  
Density Theorems ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Image Position

AbstractIn this paper, we focus on the packing measures of self-similar sets. Let K be a self-similar set whose Hausdorff dimension and packing dimension equal s. We state that if K satisfies the strong open set condition with an open set 𝒪, then for each open ball B(x,r)⊂𝒪 centred in K, where 𝒫s denotes the s-dimensional packing measure. We use this inequality to obtain some precise density theorems for the packing measures of self-similar sets. These theorems can be used to compute the exact value of the s-dimensional packing measure 𝒫s (K) of K. Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by Olsen in [15].

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

scholarly journals An analogue of Khintchine's theorem for self-conformal sets

2018 ◽  
Vol 167 (3) ◽  
pp. 567-597 ◽  
Author(s):  
SIMON BAKER
Keyword(s):  
Number Theory ◽  
General Setting ◽  
Iterated Function Systems ◽  
Open Set Condition ◽  
Metric Number Theory ◽  
Open Set ◽  
Naturally Occurring ◽  
Iterated Function ◽  
Badly Approximable ◽  
Insight Into

AbstractKhintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin–Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.Combining our results with the mass transference principle of Beresnevich and Velani [4], we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are “very well approximated”. As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.The ideas put forward in this paper are introduced in the general setting of iterated function systems that may contain overlaps. We believe that by viewing iterated function systems from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.

Dimension of the generalized 4-corner set and its projections

2011 ◽  
Vol 32 (4) ◽  
pp. 1190-1215 ◽  
Author(s):  
BALÁZS BÁRÁNY
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Iterated Function Systems ◽  
Common Fixed Points ◽  
New Method ◽  
Dimension Theory ◽  
Box Dimension ◽  
Extended Version ◽  
Iterated Function ◽  
Self Similar

AbstractIn the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.

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