scholarly journals Self-similar and self-affine sets: measure of the intersection of two copies

2009 ◽  
Vol 30 (2) ◽  
pp. 399-440 ◽  
Author(s):  
MÁRTON ELEKES ◽  
TAMÁS KELETI ◽  
ANDRÁS MÁTHÉ
Keyword(s):  
List Type ◽  
Small Perturbations ◽  
Invariant Extension ◽  
Affine Maps ◽  
And Topology ◽  
Self Similar ◽  
Affine Set ◽  
Separation Conditions

AbstractLetK⊂ℝdbe a self-similar or self-affine set and letμbe a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or thatKis a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.•(Non-stability)There exists a constantc<1 such that for everyg∈𝒢 we have eitherμ(K∩g(K))<c⋅μ(K) orK⊂g(K).•(Measure and topology)For everyg∈𝒢 we haveμ(K∩g(K))>0⟺∫K(K∩g(K))≠0̸ (where ∫Kis interior relative toK).•(Extension)The measureμhas a 𝒢-invariant extension to ℝd.Moreover, in many situations we characterize thosegfor whichμ(K∩g(K))>0. We also obtain results about thosegfor whichg(K)⊂Korg(K)⊃K.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

Canonical number systems, counting automata and fractals

2002 ◽  
Vol 133 (1) ◽  
pp. 163-182 ◽  
Author(s):  
KLAUS SCHEICHER ◽  
JÖRG M. THUSWALDNER
Keyword(s):  
Direct Methods ◽  
Number Fields ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Box Counting ◽  
Number Systems ◽  
Rings Of Integers ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Affine Set

In this paper we study properties of the fundamental domain [Fscr ]β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ]β defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ]β. It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ]β. Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

scholarly journals Bars in Cusps

1996 ◽  
Vol 171 ◽  
pp. 357-357 ◽  
Author(s):  
Walter Dehnen
Keyword(s):  
Elliptical Galaxies ◽  
Distribution Functions ◽  
List Type ◽  
Type Number ◽  
Initial Model ◽  
Pattern Speed ◽  
Self Similar ◽  
The Stability ◽  
Inside Out ◽  
Insta Bility

In order to investigate the stability properties of galaxy models with central density cusps, TV-body simulations of oblate models with density ρ ∝ m–1 (m+a)–3 where m2=R2+[z/q]2 and distribution functions f(E, Lz) (computed as in Dehnen, 1995) have been performed with the following results. 1.An E7 model with identical amounts of stars of either sense of rotation was stable over 30 tdyn(r=a). This is interesting for the bending instability has been argued to set in at about this flattening and be responsible for the absence of flatter elliptical galaxies (Merritt & Sellwood, 1994).2.Rapidly rotating E≳E5 models quickly form weak bars inside the cusp, which are stronger for the more flattened, faster rotating initial configurations. The bars grow in a self similar fashion from inside out: the pattern speed decreases with increasing bar length and time. This process is initiated at the origin, where, because of finite AT, the actual density no longer follows the power law, and stops when the edge of the cusp is reached. A typical example is given in the figure showing the z-y-coordinates of particles with |z|<0.1a after ≃20tdyn(r=a) for an initially rapidly rotating E7-model. The bar has axis ratios of about 5:3:1, and extends almost to corotation. However, it has no sharp edge, but an inhomogenous density with a cusp steeper than the initial model. No sign of a buckling insta-bility has been observerd.

On the group of isometries of planar IFS fractals*

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 445-469
Author(s):  
Qi-Rong Deng ◽  
Yong-Hua Yao
Keyword(s):  
Iterated Function System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Function System ◽  
Finite Case ◽  
Iterated Function ◽  
Self Similar ◽  
Group Of Isometries ◽  
Necessary And Sufficient ◽  
Affine Set

Abstract For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Renying Zeng
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Lagrangian Multipliers ◽  
Weakly Efficient Solutions ◽  
Affine Maps ◽  
Convex Vector Optimization ◽  
Affine Set

Abstract In this paper, we introduce some definitions of generalized affine set-valued maps: affinelike, preaffinelike, nearaffinelike, and prenearaffinelike maps. We present examples to explain that our definitions of generalized affine maps are different from each other. We derive a theorem of alternative of Farkas–Minkowski type, discuss Lagrangian multipliers for constrained set-valued optimization problems, and obtain some optimality conditions for weakly efficient solutions.

Dynamics and Stability of Surfactant Coated thin Spreading Films

1996 ◽  
Vol 464 ◽  
Author(s):  
Omar K. Matar ◽  
Sandra M. Troian
Keyword(s):  
Surfactant Concentration ◽  
Evolution Equations ◽  
Third Order ◽  
Small Perturbations ◽  
Insoluble Surfactant ◽  
State Approximation ◽  
Steady State Approximation ◽  
Self Similar ◽  
Similar Solutions ◽  
Air Liquid Interface

ABSTRACTWithin lubrication theory, we investigate the hydrodynamic stability of a thin surfactant coated liquid film spreading strictly by Marangoni stresses. These stresses are generated along the air-liquid interface because of local variations in surfactant concentration. The evolution equations governing the unperturbed film thickness and surface surfactant concentration admit simple self-similar solutions for rectilinear geometry and global conservation of insoluble surfactant. A linear stability analysis of these self-similar flows within a quasi steady-state approximation (QSSA) yields an eigenvalue problem for a single third-order nonlinear differential equation. The analysis indicates that a thin film driven purely by Marangoni stresses is linearly stable to small perturbations of all wavenumbers. The insights gained from this calculation suggest a flow mechanism that can potentially destabilize the spreading process.

Dimensions associated with recurrent self-similar sets

1991 ◽  
Vol 110 (2) ◽  
pp. 327-336 ◽  
Author(s):  
Anca Deliu ◽  
J. S. Geronimo ◽  
R. Shonkwiler ◽  
D. Hardin
Keyword(s):  
Box Dimension ◽  
Two Dimensional ◽  
Affine Maps ◽  
Self Similar ◽  
Box Dimensions

AbstractThe Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

EXPERIMENTAL VERIFICATION FOR FRACTAL TRANSITION USING A FORCED DAMPED OSCILLATOR

Fractals ◽  
2000 ◽  
Vol 08 (01) ◽  
pp. 67-72 ◽  
Author(s):  
KAZUTOSHI GOHARA ◽  
HIROSHI SAKURAI ◽  
SHOZO SATO
Keyword(s):  
Correlation Dimension ◽  
Contraction Mappings ◽  
Fractal Sets ◽  
Self Similar ◽  
Initial States ◽  
Affine Set ◽  
Damped Oscillator ◽  
Set Of Initial States ◽  
Similarity Dimension ◽  
Good Agreement

A damped oscillator stochastically driven by temporal forces is experimentally investigated. The dynamics is characterized by a set Γ(C) of trajectories in a cylindrical space, where C is a set of initial states on the Poincaré section. Two sets, Γ(C) and C, are attractive and unique invariant fractal sets that approximately satisfy specific equations derived previously by the authors. The correlation dimension of the set C is in good agreement with the similarity dimension obtained for a strictly self-similar set constructed by contraction mappings while C is a self-affine set constructed by non-contraction mappings.

The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

2016 ◽  
Vol 38 (4) ◽  
pp. 1353-1368 ◽  
Author(s):  
QI-RONG DENG ◽  
XIANG-YANG WANG
Keyword(s):  
Iterated Function System ◽  
The Self ◽  
Function System ◽  
Separation Condition ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Affine Set ◽  
Totally Disconnected

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.

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