Measure preserving fractal homeomorphisms

Michael F. Barnsley; Brendan Harding; Miroslav Rypka

doi:10.1007/978-3-319-08105-2__5

Measure preserving fractal homeomorphisms

Michael F. Barnsley^*, Brendan Harding, Miroslav Rypka

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

2 Citations (Scopus)

Abstract

The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve n-dimensional Lebesque measure, where n is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged.

Original language	English
Title of host publication	Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets
Editors	V. Kannan, Michael F. Barnsley, Robert L. Devaney, Vinod Kumar P.B., Kenneth J. Devaney, Christoph Bandt
Publisher	Springer New York LLC
Pages	79-102
Number of pages	24
ISBN (Electronic)	9783319081045
DOIs	https://doi.org/10.1007/978-3-319-08105-2__5
Publication status	Published - 2014
Event	1st International Conference and Workshop on Fractals and Wavelets, ICFW India - Kochi, India Duration: 13 Nov 2013 → 16 Nov 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	92
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	1st International Conference and Workshop on Fractals and Wavelets, ICFW India
Country/Territory	India
City	Kochi
Period	13/11/13 → 16/11/13

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10.1007/978-3-319-08105-2__5

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Barnsley, M. F., Harding, B., & Rypka, M. (2014). Measure preserving fractal homeomorphisms. In V. Kannan, M. F. Barnsley, R. L. Devaney, V. K. P.B., K. J. Devaney, & C. Bandt (Eds.), Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets (pp. 79-102). (Springer Proceedings in Mathematics and Statistics; Vol. 92). Springer New York LLC. https://doi.org/10.1007/978-3-319-08105-2__5

Barnsley, Michael F. ; Harding, Brendan ; Rypka, Miroslav. / Measure preserving fractal homeomorphisms. Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets. editor / V. Kannan ; Michael F. Barnsley ; Robert L. Devaney ; Vinod Kumar P.B. ; Kenneth J. Devaney ; Christoph Bandt. Springer New York LLC, 2014. pp. 79-102 (Springer Proceedings in Mathematics and Statistics).

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abstract = "The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve n-dimensional Lebesque measure, where n is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged.",

keywords = "Fractal, Iterated function system",

author = "Barnsley, {Michael F.} and Brendan Harding and Miroslav Rypka",

note = "Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2014.; 1st International Conference and Workshop on Fractals and Wavelets, ICFW India ; Conference date: 13-11-2013 Through 16-11-2013",

year = "2014",

doi = "10.1007/978-3-319-08105-2__5",

language = "English",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

pages = "79--102",

editor = "V. Kannan and Barnsley, {Michael F.} and Devaney, {Robert L.} and P.B., {Vinod Kumar} and Devaney, {Kenneth J.} and Christoph Bandt",

booktitle = "Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets",

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Barnsley, MF, Harding, B & Rypka, M 2014, Measure preserving fractal homeomorphisms. in V Kannan, MF Barnsley, RL Devaney, VK P.B., KJ Devaney & C Bandt (eds), Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets. Springer Proceedings in Mathematics and Statistics, vol. 92, Springer New York LLC, pp. 79-102, 1st International Conference and Workshop on Fractals and Wavelets, ICFW India, Kochi, India, 13/11/13. https://doi.org/10.1007/978-3-319-08105-2__5

Measure preserving fractal homeomorphisms. / Barnsley, Michael F.; Harding, Brendan; Rypka, Miroslav.
Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets. ed. / V. Kannan; Michael F. Barnsley; Robert L. Devaney; Vinod Kumar P.B.; Kenneth J. Devaney; Christoph Bandt. Springer New York LLC, 2014. p. 79-102 (Springer Proceedings in Mathematics and Statistics; Vol. 92).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Measure preserving fractal homeomorphisms

AU - Barnsley, Michael F.

AU - Harding, Brendan

AU - Rypka, Miroslav

N1 - Publisher Copyright: © Springer International Publishing Switzerland 2014.

PY - 2014

Y1 - 2014

N2 - The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve n-dimensional Lebesque measure, where n is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged.

AB - The basic theory of fractal transformations is recalled. For a fractal homeomorphism generated by a pair of affine iterated function systems (IFSs), a condition under which the transformation is measure (i.e. area, volume, etc.) preserving is established. Then three families of fractal homeomorphisms, two of them entirely new, generated by pairs of affine IFSs, are introduced. It is proved that they admit subfamilies that preserve n-dimensional Lebesque measure, where n is 2 or 3. Several examples are illustrated and applications to computer aided design and manufacture, via three-dimensional printing, are envisaged.

KW - Fractal

KW - Iterated function system

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U2 - 10.1007/978-3-319-08105-2__5

DO - 10.1007/978-3-319-08105-2__5

M3 - Conference contribution

T3 - Springer Proceedings in Mathematics and Statistics

SP - 79

EP - 102

BT - Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets

A2 - Kannan, V.

A2 - Barnsley, Michael F.

A2 - Devaney, Robert L.

A2 - P.B., Vinod Kumar

A2 - Devaney, Kenneth J.

A2 - Bandt, Christoph

PB - Springer New York LLC

T2 - 1st International Conference and Workshop on Fractals and Wavelets, ICFW India

Y2 - 13 November 2013 through 16 November 2013

ER -

Barnsley MF, Harding B, Rypka M. Measure preserving fractal homeomorphisms. In Kannan V, Barnsley MF, Devaney RL, P.B. VK, Devaney KJ, Bandt C, editors, Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets. Springer New York LLC. 2014. p. 79-102. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-319-08105-2__5

Measure preserving fractal homeomorphisms

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