scholarly journals Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

2020 ◽  
Author(s):  
Balázs Bárány ◽  
Thomas Jordan ◽  
Antti Käenmäki ◽  
Michał Rams
Keyword(s):  
Lyapunov Spectrum ◽  
Open Set Condition ◽  
Open Set ◽  
Lyapunov Spectra ◽  
Strongly Irreducible

Abstract Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

ON THE ASSOUAD DIMENSION OF GRAPH DIRECTED MORAN FRACTALS

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 221-226 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Direct Proof ◽  
Open Set Condition ◽  
Open Set ◽  
Assouad Dimension ◽  
Box Dimensions

We give a simple and direct proof of the fact that the Assouad dimension of a graph directed Moran fractal satisfying the Open Set Condition coincides with its Hausdorff and box dimensions.

scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

Open set condition for graph directed self-similar structure

2013 ◽  
Vol 276 (1-2) ◽  
pp. 243-260 ◽  
Author(s):  
Tian-jia Ni ◽  
Zhi-ying Wen
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

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 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.

Open set condition and post-critically finite self-similar sets

Nonlinearity ◽  
2008 ◽  
Vol 21 (6) ◽  
pp. 1227-1232 ◽  
Author(s):  
Qi-Rong Deng ◽  
Ka-Sing Lau
Keyword(s):  
Open Set Condition ◽  
Open Set ◽  
Self Similar

scholarly journals Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets

2001 ◽  
Vol 129 (9) ◽  
pp. 2689-2699 ◽  
Author(s):  
Yuval Peres ◽  
Michał Rams ◽  
Károly Simon ◽  
Boris Solomyak
Keyword(s):  
Hausdorff Measure ◽  
Open Set Condition ◽  
Open Set
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