SHRINKING TARGET PROBLEM FOR RANDOM IFS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850085
Author(s):  
ZHIHUI YUAN
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Main Idea ◽  
Iterated Function Systems ◽  
Target Problem ◽  
Iterated Function ◽  
Poincare Recurrence ◽  
Poincaré Recurrence

We describe the shrinking target problem for random iterated function systems which are semi-conjugate to random subshifts. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to more general targets and the sets based on the quantitative Poincaré recurrence properties.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals The quasi-Assouad dimension of stochastically self-similar sets

2019 ◽  
Vol 150 (1) ◽  
pp. 261-275 ◽  
Author(s):  
Sascha Troscheit
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Random Sets ◽  
Open Set Condition ◽  
Fractal Percolation ◽  
Open Set ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar

AbstractThe class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

THE BOUNDARY OF THE ATTRACTOR OF A RECURRENT ITERATED FUNCTION SYSTEM

Fractals ◽  
2002 ◽  
Vol 10 (01) ◽  
pp. 77-89 ◽  
Author(s):  
F. M. DEKKING ◽  
P. VAN DER WAL
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Iterated Function Systems ◽  
Function System ◽  
Iterated Function

We prove for a subclass of recurrent iterated function systems (also called graph-directed iterated function systems) that the boundary of their attractor is again the attractor of a recurrent IFS. Our method is constructive and permits computation of the Hausdorff dimension of the attractor and its boundary.

scholarly journals An investigation of machine learning method based on fractal compression

2019 ◽  
pp. 204-208
Author(s):  
E Y Minaev
Keyword(s):  
Machine Learning ◽  
Autonomous Robots ◽  
Main Idea ◽  
Iterated Function Systems ◽  
Mobile Platforms ◽  
Fractal Coding ◽  
Fractal Compression ◽  
Iterated Function ◽  
Fractal Images ◽  
Domain Block

In this article the method of machine learning with cyclic fractal coding and the use of domain block dictionary, adapted for use on mobile platforms, with optimization of performance and volume of stored fractal images is investigated. The main idea of the method is to use the fractal compression method based on iterated function systems to reduce the dimension of the original images, and to use cyclic fractal coding to represent the class of images. As a result of research of the method it was found that the share of correctly recognized objects on MSTAR averages 0.892, the recognition time averages 254 ms. The achieved results are acceptable for use in mobile platforms, including UAVs and ground autonomous robots.

Infinite iterated function systems with overlaps

2014 ◽  
Vol 36 (3) ◽  
pp. 890-907 ◽  
Author(s):  
SZE-MAN NGAI ◽  
JI-XI TONG
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function Systems ◽  
Topological Pressure ◽  
Separation Condition ◽  
Limit Sets ◽  
Weak Separation Condition ◽  
Iterated Function ◽  
Weak Separation ◽  
Separation Conditions

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

Weakly divergent partial quotients

2019 ◽  
Vol 13 (01) ◽  
pp. 2050158
Author(s):  
D. Dyussekenov ◽  
S. Kadyrov
Keyword(s):  
Hausdorff Dimension ◽  
Studia Math ◽  
Iterated Function Systems ◽  
Growth Speed ◽  
Positive Density ◽  
Real Numbers ◽  
Dimensional Theory ◽  
Cambridge Philos ◽  
Iterated Function ◽  
Partial Quotients

We study the real numbers with partial quotients diverging to infinity in a subsequence. We show that if the subsequence has positive density then such sets have Hausdorff dimension equal to 1/2. This generalizes one of the results obtained in [C. Y. Cao, B. W. Wang and J. Wu, The growth speed of digits in infinite iterated function systems, Studia. Math. 217(2) (2013) 139–158; I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941) 199–228].

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