scholarly journals On the Ledrappier–Young formula for self-affine measures

2015 ◽  
Vol 159 (3) ◽  
pp. 405-432 ◽  
Author(s):  
BALÁZS BÁRÁNY
Keyword(s):  
Iterated Function System ◽  
Sufficient Conditions ◽  
Lyapunov Dimension ◽  
Dominated Splitting ◽  
Triangular Matrices ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition ◽  
Lower Triangular Matrices ◽  
Splitting Condition

AbstractLedrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier–Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter–Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.

scholarly journals Dimension approximation of attractors of graph directed IFSs by self-similar sets

2018 ◽  
Vol 167 (01) ◽  
pp. 193-207 ◽  
Author(s):  
ÁBEL FARKAS
Keyword(s):  
Iterated Function System ◽  
Black Box ◽  
Function System ◽  
Separation Condition ◽  
Iterated Function ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Distance Sets

AbstractWe show that for the attractor (K1, . . ., Kq) of a graph directed iterated function system, for each 1 ⩽ j ⩽ q and ϵ > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dimHKj − ϵ < dimHK. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property as a ‘black box’ we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

Regularity of Hausdorff measure function for conformal dynamical systems

2016 ◽  
Vol 160 (3) ◽  
pp. 537-563 ◽  
Author(s):  
MARIUSZ URBAŃSKI ◽  
ANNA ZDUNIK
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Measure ◽  
Hölder Continuity ◽  
Iterated Function Systems ◽  
Separation Condition ◽  
Holder Continuity ◽  
Measure Function ◽  
Iterated Function ◽  
Strong Separation ◽  
Strong Separation Condition

AbstractWe deal with the question of continuity of numerical values of Hausdorff measures in parametrised families of linear (similarity) and conformal dynamical systems by developing the pioneering work of Lars Olsen and the work [SUZ]. We prove Hölder continuity of the function ascribing to a parameter the numerical value of the Hausdorff measure of either the corresponding limit set or the corresponding Julia set. We consider three cases. Firstly, we consider the case of parametrised families of conformal iterated function systems in $\mathbb{R}$k with k ⩾ 3. Secondly, we consider all linear iterated function systems consisting of similarities in $\mathbb{R}$k with k ⩾ 1. In either of these two cases, the strong separation condition is assumed. In the latter case the Hölder exponent obtained is equal to 1/2. Thirdly, we prove such Hölder continuity for analytic families of conformal expanding repellers in the complex plane $\mathbb{C}$. Furthermore, we prove the Hausdorff measure function to be piecewise real–analytic for families of naturally parametrised linear IFSs in $\mathbb{R}$ satisfying the strong separation condition. On the other hand, we also give an example of a family of linear IFSs in $\mathbb{R}$ for which this function is not even differentiable at some parameters.

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.

Attractors of Infinite Iterated Function Systems Containing Con

2013 ◽  
Vol 59 (2) ◽  
pp. 281-298
Author(s):  
Dan Dumitru
Keyword(s):  
Metric Space ◽  
Existence And Uniqueness ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

Abstract We consider a complete ε-chainable metric space (X, d) and an infinite iterated function system (IIFS) formed by an infinite family of (ε, φ)-functions on X. The aim of this paper is to prove the existence and uniqueness of the attractors of such infinite iterated systems (IIFS) and to give some sufficient conditions for these attractors to be connected. Similar results are obtained in the case when the IIFS is formed by an infinite family of uniformly ε-locally strong Meir-Keeler functions.

The (b,c)-inverse for products and lower triangular matrices

2017 ◽  
Vol 16 (12) ◽  
pp. 1750222 ◽  
Author(s):  
Yuanyuan Ke ◽  
Zhou Wang ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Drazin Inverse ◽  
Triangular Matrices ◽  
Core Inverse ◽  
Special Cases ◽  
Lower Triangular ◽  
Necessary And Sufficient ◽  
Lower Triangular Matrices ◽  
Dual Core

Let [Formula: see text] be a semigroup and [Formula: see text]. The concept of [Formula: see text]-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the [Formula: see text]-inverse. In this paper, a new relationship between the [Formula: see text]-inverse and the Bott–Duffin [Formula: see text]-inverse is established. The relations between the [Formula: see text]-inverse of [Formula: see text] and certain classes of generalized inverses of [Formula: see text] and [Formula: see text], and the [Formula: see text]-inverse of [Formula: see text] are characterized for some [Formula: see text], where [Formula: see text]. Necessary and sufficient conditions for the existence of the [Formula: see text]-inverse of a lower triangular matrix over an associative ring [Formula: see text] are also given, and its expression is derived, where [Formula: see text] are regular triangular matrices.

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].

Recursive Solution to a Type of Finite Field Matrix Equation and its Application in Information Security

2013 ◽  
Vol 846-847 ◽  
pp. 934-938
Author(s):  
Shen Hai Yan ◽  
Xian Tong Huang
Keyword(s):  
Matrix Equation ◽  
Symmetric Matrix ◽  
Recursive Algorithm ◽  
Sufficient Conditions ◽  
Finite Domain ◽  
Multiplicative Inverse ◽  
Triangular Matrices ◽  
Recursive Solution ◽  
Necessary And Sufficient ◽  
Lower Triangular Matrices

On the question of solving lower triangular matrices or symmetric matrix by a type of matrix equation, we obtain the necessary and sufficient conditions for its only solution and design a recursive algorithm to compute the solution. With finite domain multiplicative inverse technology and Differ-Hellman protocol, a new cryptosystem is designed based on finite fields matrix equations. Numerical experiments demonstrate the correctness of the new cryptosystem and feasibility.

scholarly journals Some subordination involving polynomials induced by lower triangular matrices

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saiful R. Mondal ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Unit Disk ◽  
Triangular Matrices ◽  
Cesàro Mean ◽  
Cesaro Mean ◽  
Lower Triangular ◽  
Lower Triangular Matrices

Abstract The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works.

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