Michael Barnsley

Michael Barnsley
Emeritus Professor

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For more information about Michael and his research be sure to look at his SuperFractals website.



Supervised students


Room 3.48, Hanna Neumann Building 145


Below are listed some recent publications, but for more details see Michael's Google scholar citation page

Recent Journal Articles

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations, C Bandt, M Barnsley, M Hegland, A Vince, Chaos, Solitons & Fractals 91 (2016), 478-489

Approximation of rough functions, MF Barnsley, B Harding, A Vince, P Viswanathan, Journal of Approximation Theory, 209, (2016), 23–43

Chaos game for an IFS on topological spaces. Michael F Barnsley, Krzysztof Leśniak, Miroslav Rypka, J. Math.Anal and Appl. 435, (2016) 1458-1466.

Fast basins and branched fractal manifolds of attractors of iterated function systems. Michael F Barnsley, Andrew Vince. Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 11, (2015) ,084, 21 pages.

On the continuity of the Hutchinson operator. Michael F Barnsley and Krzysztof Leśniak. Symmetry 2015, 7(4), 1831-1840; doi:10.3390/sym7041831. (Invited contribution to a special issue devoted to fractal geometry).

The Chaos Game on a General Iterated Function System from a Topological Point of View: Michael Barnsley and Krzysztof Leśniak: International Journal of Bifurcation and Chaos, 24, No. 11 (2014) 1450139: DOI: 10.1142/S0218127414501399

Critical itineraries of maps with constant slope and one discontinuity. Michael Barnsley, Wolfgang Steiner and Andrew Vince. Mathematical Proceedings of the Cambridge Philosophical Society, 157 (2014) 547-565, doi:10.1017/S0305004114000486. 

Bilinear fractal interpolation and box dimension: Michael Barnsley and Peter Massopust: Journal of Approximation Theory doi:10.1016/j.jat.2014.10.014

Numerics and Fractals: Michael Barnsley, Markus Hegland and Peter Massopust.Bull. Inst. Math., Acad. Sin., 9(3), 2014, 389-439.

Fractal Tilings from iterated Function Systems: Michael Barnsley and Andrew Vince: Discrete & Computational Geometry: 51,(2014), 729-752 DOI 10.1007/s00454-014-9589-2

Fractal Continuation, Michael Barnsley and Andrew Vince, Constructive Approximation, 38 (2013) 311-337.

The Conley attractors of an iterated function system, Michael Barnsley and Andrew Vince, Bulletin of the Australian Mathematical Society. 88 (2013) doi:10.1017/S0004972713000348

Developments in fractal geometry: Michael Barnsley and Andrew Vince: Bulletin of Mathematical Sciences, 2013, Volume 3, pp 299-348,  DOI: 10.1007/s13373-013-0041-3

Fractal Homeomorphism for Bi-affine Iterated Function Systems: Michael Barnsley and Andrew Vince: Int. J. Applied Nonlinear Science, Vol. 1, No. 1, 2013 pp. 3-19

Overlapping Iterated Function Systems on a Segment, M. Barnsley and K.B. Igudesman :Russian Mathematics, Vol. 56, No. 12 (2012), pp. 1–12. Original Russian Text M. Barnsley and K.B. Igudesman, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, No. 12 (2012), pp. 3–15.

The Influence of Benoît B. Mandelbrot on Mathematics, Edited by Michael F. Barnsley and Michael Frame: Notices of the AMS, Vol 59 No 9 (2012) pp. 1208-1221

Glimpses of Benoît B. Mandelbrot,  Edited by Michael Barnsley and Michael Frame: Notices of the AMS, Vol 59 No 8 (2012) pp. 1056-1063

The Entropy of a Special Overlapping Dynamical System: Michael Barnsley, Brendan Harding, and Andrew Vince: Ergod. Th. & Dynam. Sys. (2012), 0, pp. 1–18 DOI:10.1017/etds.2012.140

Optimization on fractals and stability: Michael Barnsley, Davide LaTorre and Uta Freiberg: Journal of Nonlinear and Convex Analysis Vol. 13, No.4, 2012 pp. 695-708

Real Projective Iterated Function Systems, Michael Barnsley and Andrew Vince: J GeomAnal Vol 22, No 4, (2012) pp 1137-1172

The Eigenvalue Problem for Linear and Affine Iterated Function Systems. Michael Barnsley and Andrew Vince: Linear Algebra and Its Applications 435 (2011) pp. 3124-3138

Topological contractive systems,  Michael Barnsley and Konstantin Igudesman: Lobachevskii Journal of Mathematics Vol. 32, No. 3, 2011 pp. 220–223.

How to transform and filter images using iterated function systems,  Michael Barnsley, Konstantin Igudesman and Brendan Harding: SIAM J. Imaging Sci., 4 (2011) , pp. 1001-1028

The Life and Survival of Mathematical Ideas, Michael Barnsley: Notices of the AMS, Vol 57 Issue 1 (2010) 10-22

V-variable fractals: dimension results, Michael Barnsley, John E. Hutchinson and Örjan Stenflo: Forum Math. 24 (3), (2012) 445-470, DOI 10.1515/FORM.2011.075

The chaos game on a general iterated function system : Michael Barnsley and Andrew Vince: Ergodic Theory and Dynamical Systems Vol 1 (1),pp 1-7, DOI: 10.1017/S0143385710000428

Transformations between self-referential sets,  Michael Barnsley: Amer. Math. Monthly 116 (2009) 291-304

Transformations between Fractals,  Michael Barnsley: Series Progress in Probability, Vol 61, Fractal Geometry and Stochastics IV, pp 227-250, Bandt, Christoph; Mörters, Peter; Zähle, Martina (Eds.) Birkhauser 2009

Measure-Valued Images, Associated Fractal Transforms, and the Affine Self-Similarity of Images Michael Barnsley,  E.R.Vrscay,  M.Ebrahimi and David Latorre: SIAM Journal on Imaging Sciences (2009) - 2, 2, 470-507

V-variable fractals: Fractals with partial self similarity,  Michael F. Barnsley, John E. Hutchinson and Örjan Stenflo : Advances in Mathematics Volume 218, Issue 6, 20 August 2008, Pages 2051-2088

A fractal valued random iteration algorithm and fractal hierarchy, Michael Barnsley, John Hutchinson, Örjan Stenflo:  Fractals 13 (2005) no. 2, 111–146.

Recent Books

Fractals Everywhere – New Edition, Dover Publications, Mineola, NY 2012

SuperFractals,  Cambridge University Press, Cambridge 2006

Recent Book Chapters

Some Recent Progress Concerning Topology of Fractals,  Michael F. Barnsley, David C. Wilson and Krzysztof Leśniak, Recent Progress in Topology III, Atlantis Press (2014) ISBN 978-94-6239-023-2, DOI 10.2991/978-94-6239-024-9

Three-Dimensional Fractal Homeomorphisms: Michael Barnsley and Brendan Harding. Benoit Mandelbrot: A Life in Many Dimensions

Fractal Transformations,  Michael Barnsley and Louisa Barnsley: The Colours of Infinity - The Beauty and Power of Fractals, pp. 66-81. London Clear Books, 2004

Recent Conference Proceedings

Michael F Barnsley, Brendan Harding and Miroslav Rypka, Measure preserving Fractal Homeomorphisms, in Fractals, Wavelets, and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets, Springer Proceedings in Mathematics and Statistics 92, 2014

A Characterization of Hyperbolic Affine Iterated Function Systems, Ross Atkins, Michael F. Barnsley, Andrew Vince, David C. Wilson: Topology Proceedings, Vol 36 (2010) Pages 189-211 E-Published on April 2, 2010

Fractal transformations of harmonic functions, Michael Barnsley and Uta Freiberg: Complexity and Nonlinear Dynamics. Edited by Bender, Axel. Proceedings of the SPIE, Volume 6417, pp. 64170C (2007).

New Methods in Fractal Imaging, Michael Barnsley and John Hutchinson: Proceedings of the International Conference on Computer Graphics, Imaging and Visualisation, (July 26-28, 2006) IEEE Society, Washington DC 296-301

Theory and Application of Fractal Tops, Michael Barnsley: Fractals in Engineering: New Trends in Theory and Applications , Springer-Verlag (2005) Edited by Jacques Levy-Vehel and Evelyne Lutton. pages 3-20. 

Recent Reviews

Review of Differential Equations on Fractals: A Tutorial.By Robert S. Strichartz.  SIAM REVIEW 2007 Society for Industrial and Applied Mathematics: Vol. 49,No . 4,pp . 700–702 

Recent ARXIV Preprints

Symmetric Itinerary Sets, Michael F. Barnsley and Nicolae Mihalache