scholarly journals Self-embeddings of Bedford–McMullen carpets

2017 ◽  
Vol 39 (3) ◽  
pp. 577-603 ◽  
Author(s):  
AMIR ALGOM ◽  
MICHAEL HOCHMAN
Keyword(s):  
Tangent Sets ◽  
Self Similar ◽  
Projection Theorems

Let $F\subseteq \mathbb{R}^{2}$ be a Bedford–McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between zero and one. We prove that any similarity $g$ such that $g(F)\subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by ‘zooming in’ on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self-similar sets in the line.

scholarly journals Local structure of self-affine sets

2012 ◽  
Vol 33 (5) ◽  
pp. 1326-1337 ◽  
Author(s):  
CHRISTOPH BANDT ◽  
ANTTI KÄENMÄKI
Keyword(s):  
Cantor Sets ◽  
Address Space ◽  
Open Set Condition ◽  
Open Set ◽  
The Family ◽  
Tangent Sets ◽  
Self Similar ◽  
Almost All ◽  
Product Sets ◽  
Totally Disconnected

AbstractThe structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that in small square $\varepsilon $-neighborhoods $N$ of almost each point $x$ in $E,$ with respect to many Bernoulli measures on the address space, $E\cap N$ is well approximated by product sets $[0,1]\times C$, where $C$ is a Cantor set. Even though $E$ is totally disconnected, all tangent sets have a product structure with interval fibers, reminiscent of the view of attractors of chaotic differentiable dynamical systems. We also prove that $E$has uniformly scaling scenery in the sense of Furstenberg, Gavish and Hochman: the family of tangent sets is the same at almost all points$x.$

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar

PREDICTING CHARACTERISTICS OF SELF SIMILAR TRAFFIC BY SPLINE-EXTRAPOLATION

2019 ◽  
Author(s):  
Irina Strelkovskay ◽  
Irina Solovskaya ◽  
Anastasija Makoganjuk ◽  
Nikolaj Severin
Keyword(s):  
Quality Characteristics ◽  
Traffic Prediction ◽  
Normative Values ◽  
Spline Functions ◽  
Network Nodes ◽  
Traffic Forecasting ◽  
Network Equipment ◽  
Self Similar ◽  
Accuracy Of Prediction

The problem of forecasting self-similar traffic, which is characterized by a considerable number of ripples and the property of long-term dependence, is considered. It is proposed to use the method of spline extrapolation using linear and cubic splines. The results of self-similar traffic prediction were obtained, which will allow to predict the necessary size of the buffer devices of the network nodes in order to avoid congestion in the network and exceed the normative values ​​of QoS quality characteristics. The solution of the problem of self-similar traffic forecasting obtained with the Simulink software package in Matlab environment is considered. A method of extrapolation based on spline functions is developed. The proposed method has several advantages over the known methods, first of all, it is sufficient ease of implementation, low resource intensity and accuracy of prediction, which can be enhanced by the use of quadratic or cubic interpolation spline functions. Using the method of spline extrapolation, the results of self-similar traffic prediction were obtained, which will allow to predict the required volume of buffer devices, thereby avoiding network congestion and exceeding the normative values ​​of QoS quality characteristics. Given that self-similar traffic is characterized by the presence of "bursts" and a long-term dependence between the moments of receipt of applications in this study, given predetermined data to improve the prediction accuracy, it is possible to use extrapolation based on wavelet functions, the so-called wavelet-extrapolation method. Based on the results of traffic forecasting, taking into account the maximum values ​​of network node traffic, you can give practical guidance on how traffic is redistributed across the network. This will balance the load of network objects and increase the efficiency of network equipment.

Self-similar traffic discrimination and generating methods based on fractal Brown motion

2013 ◽  
Vol 33 (4) ◽  
pp. 947-949
Author(s):  
Xueyuan ZHANG ◽  
Yonggang WANG ◽  
Qiong ZHANG
Keyword(s):  
Self Similar ◽  
Brown Motion
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