scholarly journals Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux–Yoccoz map

2017 ◽  
Vol 38 (7) ◽  
pp. 2537-2570 ◽  
Author(s):  
MILTON COBO ◽  
RODOLFO GUTIÉRREZ-ROMO ◽  
ALEJANDRO MAASS
Keyword(s):  
Sufficient Conditions ◽  
Interval Exchange ◽  
Self Similarity ◽  
Complex Eigenvalues ◽  
Algebraic Properties ◽  
Self Similar ◽  
Interval Exchange Maps

In this article, we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals that are semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux–Yoccoz interval exchange map satisfies these conditions.

WEAK SELF-SIMILAR SETS IN SEPARABLE COMPLETE METRIC SPACES

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750021
Author(s):  
R. K. ASWATHY ◽  
SUNIL MATHEW
Keyword(s):  
Metric Spaces ◽  
Complete Metric Space ◽  
Sufficient Conditions ◽  
Finite Union ◽  
Self Similarity ◽  
Complete Metric Spaces ◽  
Self Similar ◽  
Necessary And Sufficient ◽  
Physical Phenomena

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.

Stochastic properties of the linear multifractional stable motion

2004 ◽  
Vol 36 (04) ◽  
pp. 1085-1115 ◽  
Author(s):  
Stilian Stoev ◽  
Murad S. Taqqu
Keyword(s):  
Sufficient Conditions ◽  
Infinite Variance ◽  
Regularity Conditions ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Stable Motion ◽  
Linear Fractional Stable Motion ◽  
Motion Process ◽  
Self Similar ◽  
Fractional Stable Motion

We study a family of locally self-similar stochastic processes Y = {Y(t)} t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

scholarly journals Characterization of minimal sequences associated with self-similar interval exchange maps

Nonlinearity ◽  
2018 ◽  
Vol 31 (4) ◽  
pp. 1121-1154
Author(s):  
Milton Cobo ◽  
Rodolfo Gutiérrez-Romo ◽  
Alejandro Maass
Keyword(s):  
Interval Exchange ◽  
Self Similar ◽  
Minimal Sequences ◽  
Interval Exchange Maps

Stochastic properties of the linear multifractional stable motion

2004 ◽  
Vol 36 (4) ◽  
pp. 1085-1115 ◽  
Author(s):  
Stilian Stoev ◽  
Murad S. Taqqu
Keyword(s):  
Sufficient Conditions ◽  
Infinite Variance ◽  
Regularity Conditions ◽  
Self Similarity ◽  
Similarity Parameter ◽  
Stable Motion ◽  
Linear Fractional Stable Motion ◽  
Motion Process ◽  
Self Similar ◽  
Fractional Stable Motion

We study a family of locally self-similar stochastic processes Y = {Y(t)}t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

Self-similarity of Julia sets of the composition of polynomials

1997 ◽  
Vol 17 (6) ◽  
pp. 1289-1297 ◽  
Author(s):  
MATTHIAS BÜGER
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Self Similarity ◽  
Iteration Theory ◽  
Finite Case ◽  
Self Similar ◽  
Hyperbolic Domain

In the classical iteration theory we say that for a given polynomial $f$ a point $z_0\in\C$ belongs to the Julia set if the sequence of iterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In this paper, we look at the set of non-normality of $(F_n)$, $F_n:=f_n\circ\cdots\circ f_1$, where $(f_n)$ is a given sequence of polynomials of degree at least two. If we can find a hyperbolic domain $M$ which is invariant under all $f_n$, $n\in\N$, $\infty\in M$ and $F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then we ask whether these sets of non-normality, which we will also call Julia sets, have properties which we know from the classical case. We show that the Julia set is self-similar. Furthermore, the Julia set is perfect or finite. The finite case may actually occur. We will also give some sufficient conditions for the Julia set being perfect. In the last section we give some examples of sequences of polynomials (where no domain $M$ exists) which have a pathological behaviour in contrast to the classical case.

scholarly journals Equidistribution Results for Self-Similar Measures

2021 ◽  
Author(s):  
Simon Baker
Keyword(s):  
Sufficient Conditions ◽  
Cantor Set ◽  
Similar Measure ◽  
Natural Measure ◽  
Self Similar

Abstract A well-known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty }$ where $(f_n)_{n=1}^{\infty }$ is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a corollary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t,$ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

scholarly journals Depressurization-Induced Nucleation in the “Polylactide-Carbon Dioxide” System: Self-Similarity of the Bubble Embryos Expansion

Polymers ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1115
Author(s):  
Dmitry Zimnyakov ◽  
Marina Alonova ◽  
Ekaterina Ushakova
Keyword(s):  
Initial Rate ◽  
Initial Pressure ◽  
External Pressure ◽  
Self Similarity ◽  
Embryo Formation ◽  
Linear Behavior ◽  
Joint Influence ◽  
External Pressures ◽  
Adiabatic Cooling ◽  
Self Similar

Self-similar expansion of bubble embryos in a plasticized polymer under quasi-isothermal depressurization is examined using the experimental data on expansion rates of embryos in the CO2-plasticized d,l-polylactide and modeling the results. The CO2 initial pressure varied from 5 to 14 MPa, and the depressurization rate was 5 × 10−3 MPa/s. The constant temperature in experiments was in a range from 310 to 338 K. The initial rate of embryos expansion varied from ≈0.1 to ≈10 µm/s, with a decrease in the current external pressure. While modeling, a non-linear behavior of CO2 isotherms near the critical point was taken into account. The modeled data agree satisfactorily with the experimental results. The effect of a remarkable increase in the expansion rate at a decreasing external pressure is interpreted in terms of competing effects, including a decrease in the internal pressure, an increase in the polymer viscosity, and an increase in the embryo radius at the time of embryo formation. The vanishing probability of finding the steadily expanding embryos for external pressures around the CO2 critical pressure is interpreted in terms of a joint influence of the quasi-adiabatic cooling and high compressibility of CO2 in the embryos.

scholarly journals Self-similar resistive circuits as fractal-like structures

2017 ◽  
Vol 40 (1) ◽  
Author(s):  
Claudio Xavier Mendes dos Santos ◽  
Carlos Molina Mendes ◽  
Marcelo Ventura Freire
Keyword(s):  
Scale Invariance ◽  
Self Similarity ◽  
General Properties ◽  
Modern Physics ◽  
Resistive Networks ◽  
Self Similar ◽  
And Mathematics ◽  
Definition Of ◽  
The Individual

Fractals play a central role in several areas of modern physics and mathematics. In the present work we explore resistive circuits where the individual resistors are arranged in fractal-like patterns. These circuits have some of the characteristics typically found in geometric fractals, namely self-similarity and scale invariance. Considering resistive circuits as graphs, we propose a definition of self-similar circuits which mimics a self-similar fractal. General properties of the resistive circuits generated by this approach are investigated, and interesting examples are commented in detail. Specifically, we consider self-similar resistive series, tree-like resistive networks and Sierpinski’s configurations with resistors.

TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN
Keyword(s):  
The Self ◽  
Self Similarity ◽  
Complex Dimensions ◽  
Tubular Neighborhoods ◽  
Self Similar ◽  
Tube Formulas

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

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