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SYMMETRIC ITINERARY SETS: ALGORITHMS AND NONLINEAR EXAMPLES

Part of: Classical measure theory Communication, information Low-dimensional dynamical systems

Published online by Cambridge University Press:  20 March 2019

BRENDAN HARDING Open the ORCID record for BRENDAN HARDING [Opens in a new window]
BRENDAN HARDING*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, South Australia 5005, Australia email [email protected]
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Abstract

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We describe how to approximate fractal transformations generated by a one-parameter family of dynamical systems $W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms $W_{i}$ such that $W_{i}^{-1}:[0,1]\rightarrow [0,1]$ for $i=0,1$. An algorithm is provided for determining the unique parameter value such that the closure of the symbolic attractor $\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical. Several examples are given, one in which the $W_{i}$ are affine and two in which the $W_{i}$ are nonlinear. Applications to digital imaging are also discussed.

Keywords

expanding dynamical systemfractal transformationchaos game algorithm

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 28A80: Fractals 94A08: Image processing (compression, reconstruction, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 1 , August 2019 , pp. 109 - 118
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The author acknowledges support from an Australian Research Council Discovery Project (project number DP160102021) funded by the Australian Government.

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