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.

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.

More Variations on the Sierpiński Sieve

2010 ◽  
Vol 62 (3) ◽  
pp. 543-562
Author(s):  
Kevin G. Hare
Keyword(s):  
Iterated Function Systems ◽  
Open Set Condition ◽  
Open Set ◽  
Iterated Function

AbstractThis paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.

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].

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 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.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

Exact Hausdorff and packing measures of Cantor sets with overlaps

2014 ◽  
Vol 35 (8) ◽  
pp. 2632-2668 ◽  
Author(s):  
HUA QIU
Keyword(s):  
Cantor Sets ◽  
Packing Measure ◽  
Type Condition ◽  
Elementary Functions ◽  
Open Set Condition ◽  
Open Set ◽  
Density Theorems ◽  
Iterated Function ◽  
Finite Set ◽  
Image Position

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that $$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$, and $$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$, where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

ITERATED FUNCTION SYSTEMS FOR OBJECT GENERATION AND RENDERING

2001 ◽  
Vol 11 (02) ◽  
pp. 259-289
Author(s):  
HUW JONES
Keyword(s):  
Linear Equations ◽  
Complete Information ◽  
Iterated Function Systems ◽  
Data Sets ◽  
Affine Transformations ◽  
3D Objects ◽  
Naturally Occurring ◽  
Iterated Function ◽  
Definition Of ◽  
Smooth Objects

Iterated function systems (IFS) have been used since the mid 1980s in generating fractal forms and in image compression. They have the property of encoding complex structures as relatively simple and concise algorithms and data sets. The complete information for generation of an IFS structure, known as an "attractor", is held in the definition of a few transformation functions, typically affine transformations described by simple linear equations. These transform 2D or 3D objects by combinations of translation, scaling and rotation, preserving parallel lines. The process of generating an attractor using either of the two standard methods is straightforward, but the inverse problem of identifying the IFS for a particular attractor is more difficult. The paper introduces and reviews the uses of Iterated Function Systems, particularly for 2D image and 3D object generation. As well as a general exposition of the method, results developed by the author for creation of fractal objects are shown. Some are based on regular geometric structures, others mimic a number of naturally occurring forms. These deviate from normal practice in using nonaffine transformations in the IFS definition. A new method for synthesising images of smooth objects using IFS is developed and illustrated as an alternative to standard 3D rendering methods (see the cover picture), which are briefly described.

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