Overview

Prof Mark Holland is an Associate Professor in Applied Mathematics. He has a PhD from Warwick University in the subject of Ergodic Theory and Dynamical Systems. Before arriving at Exeter he was a Research Fellow first at the University of Manchester and then at the University of Surrey where he studied the statistical properties of symmetric chaotic attractors. He has published work on the mixing properties of intermittent systems and discrete time maps with singularities, and more recently has produced work on statistical properties of the Lorenz equations.

Research Interests

My main interest is the study of ergodic theory and its applications. Below an overview of the basic ideas in ergodic theory is given, followed by some specific research interests. See also my papers/preprints for formal details of results.

• Dynamical Systems and Ergodic Theory

• Statistical Properties of Lorenz Attractors

• Ergodic Theory of Strange Attractors

Other Relevant Information

Qualifications  : PhD (Mathematics), University of Warwick, awarded 2002.

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Publications

_{Copyright Notice: Any articles made available for download are for personal use only. Any other use requires prior permission of the author and the copyright holder.}

| 2024 | 2021 | 2018 | 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2009 | 2007 | 2006 | 2005 | 2004 |

2024

• Carney M, Holland M, Nicol M, Tran P. (2024) Runs of extremes of observables on dynamical systems and applications, Physica D Nonlinear Phenomena, volume 460, DOI:10.1016/j.physd.2024.134093.

2021

• Holland MP, Sterk AE. (2021) On max-semistable laws and extremes for dynamical systems, Entropy, volume 23, no. 9, DOI:10.3390/e23091192.
• Carney M, Holland M, Nicol M. (2021) Extremes and extremal indices for level set observables on hyperbolic systems, NONLINEARITY, volume 34, no. 2, pages 1136-1167, DOI:10.1088/1361-6544/abd85f. [PDF]
• Coates D, Holland M, Terhesiu D. (2021) Limit theorems for wobbly interval intermittent maps, Studia Mathematica, DOI:10.4064/sm200427-21-11.

2018

• Sterk AE, Holland MP. (2018) Extreme value laws and mean squared error growth in dynamical systems, Dynamics and Statistics of the Climate System, volume 3, no. 1, DOI:10.1093/climsys/dzy007.

2017

• Holland MP, Todd M. (2017) WEAK CONVERGENCE TO EXTREMAL PROCESSES AND RECORD EVENTS FOR NON-UNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS, Ergodic Theory and Dynamical Systems. [PDF]

2016

• Lucarini V, Faranda D, de Freitas ACGMM, de Freitas JMM, Holland M, Kuna T, Nicol M, Todd M, Vaienti S. (2016) Extremes and Recurrence in Dynamical Systems, Wiley, DOI:10.1002/9781118632321.
• Lucarini V, Faranda D, Freitas ACM, Freitas JM, Kuna T, Holland M, Nicol M, Todd M, Vaienti S. (2016) Extremes and Recurrence in Dynamical Systems, DOI:10.48550/arxiv.1605.07006.
• Holland MP, Rabassa P, Sterk AE. (2016) Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems, Nonlinearity, volume 146, pages 12-43, DOI:10.1088/0951-7715/29/8/2355. [PDF]
• Holland MP, Nicol M, Török A. (2016) Almost sure convergence of maxima for chaotic dynamical systems, Stochastic Processes and their Applications, volume 126, no. 10, pages 3145-3170, DOI:10.1016/j.spa.2016.04.023. [PDF]

2015

• Carvalho M, Freitas ACM, Freitas JM, Holland M, Nicol M. (2015) Extremal dichotomy for uniformly hyperbolic systems, DOI:10.48550/arxiv.1501.05023.
• Holland M, Nicol M. (2015) Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems, STOCHASTICS AND DYNAMICS, volume 15, no. 4, article no. ARTN 1550028, DOI:10.1142/S0219493715500288. [PDF]
• Hunter A, Stephenson DB, Economou T, Holland M, Cook I. (2015) New perspectives on the collective risk of extratropical cyclones, Quarterly Journal of the Royal Meteorological Society, volume 142, no. 694, pages 243-256, DOI:10.1002/qj.2649. [PDF]
• Carvalho M, Moreira Freitas AC, Freitas JM, Holland M, Nicol M. (2015) Extremal dichotomy for uniformly hyperbolic systems, DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, volume 30, no. 4, pages 383-403, DOI:10.1080/14689367.2015.1056722. [PDF]

2014

• Holland M, Rabassa P, Sterk A. (2014) Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems, DOI:10.48550/arxiv.1404.3941.

2013

• Holland M, Zhang Y. (2013) Dimension results for inhomogeneous Moran set constructions, Dynamical Systems, volume 28, no. 2, pages 222-250, DOI:10.1080/14689367.2013.800470.
• Byott NP, Holland M, Zhang Y. (2013) On the mixing properties of piecewise expanding maps under composition with permutations, Discrete and Continuous Dynamical Systems- Series A, volume 33, no. 8, pages 3365-3390, DOI:10.3934/dcds.2013.33.3365.

2012

• Sterk AE, Holland MP, Rabassa P, Broer HW, Vitolo R. (2012) Predictability of extreme values in geophysical models, Nonlinear Processes in Geophysics, volume 19, no. 5, pages 529-539, DOI:10.5194/npg-19-529-2012.
• Holland MP, Vitolo R, Rabassa P, Sterk AE, Broer HW. (2012) Extreme value laws in dynamical systems under physical observables, Physica D: Nonlinear Phenomena, volume 241, no. 5, pages 497-513, DOI:10.1016/j.physd.2011.11.005.
• Holland M, Nicol M, Török A. (2012) Extreme value theory for non-uniformly expanding dynamical systems, Transactions of the American Mathematical Society, volume 364, no. 2, pages 661-688, DOI:10.1090/S0002-9947-2011-05271-2.

2011

• Holland MP, Vitolo R, Rabassa P, Sterk AE, Broer HW. (2011) Extreme value laws in dynamical systems under physical observables, DOI:10.48550/arxiv.1107.5673.
• Gupta C, Holland M, Nicol M. (2011) Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps, Ergodic Theory and Dynamical Systems, volume 31, no. 5, pages 1363-1390, DOI:10.1017/S014338571000057X.

2009

• Vitolo R, Holland MP, Ferro CAT. (2009) Robust extremes in chaotic deterministic systems, CHAOS, volume 19, no. 4, article no. ARTN 043127, DOI:10.1063/1.3270389. [PDF]

2007

• Díaz-Ordaz K, Holland MP, Luzzatto S. (2007) Statistical properties of one-dimensional maps with critical points and singularities, DOI:10.48550/arxiv.0709.1723.
• Bhansali RJ, Holland MP, Kokoszka PS. (2007) Intermittency, long memory and financial returns, Long Memory in Economics, pages 39-68.
• Bhansali RJ, Holland MP. (2007) Frequency analysis of chaotic intermittency maps with slowly decaying correlations, Statistica Sinica, volume 17, no. 1, pages 15-41.
• Holland M, Melbourne I. (2007) Central limit theorems and invariance principles for Lorenz attractors, Journal of the London Mathematical Society, volume 76, no. 2, pages 345-364, DOI:10.1112/jlms/jdm060.

2006

• Díaz-Ordaz K, Holland MP, Luzzatto S. (2006) Statistical properties of one-dimensional maps with critical points and singularities, Stochastics and Dynamics, volume 6, no. 4, pages 423-458, DOI:10.1142/S0219493706001852.
• Holland MP, Luzzatto S. (2006) Stable manifolds under very weak hyperbolicity conditions, Journal of Differential Equations, volume 221, no. 2, pages 444-469, DOI:10.1016/j.jde.2005.07.013.

2005

• Holland M, Luzzatto S. (2005) Stable manifolds under very weak hyperbolicity conditions, DOI:10.48550/arxiv.math/0502236.
• Holland M, Luzzatto S. (2005) A new proof of the stable manifold theorem for hyperbolic fixed points on surfaces, Journal of Difference Equations and Applications, volume 11, no. 6, pages 535-551, DOI:10.1080/10236190500127554.
• Holland M. (2005) Slowly mixing systems and intermittency maps, ERGODIC THEORY AND DYNAMICAL SYSTEMS, volume 25, pages 133-159, DOI:10.1017/S0143385704000343. [PDF]
• Bruin H, Holland MP, Nicol M. (2005) Livsic regularity for Markov systems, Ergodic Theory and Dynamical Systems, volume 25, no. 6, pages 1739-1765, DOI:10.1017/S0143385705000179.
• Bruin, H., Holland MP, Melbourne, I.. (2005) Subexponential decay of correlations for compact group extensions of non-uniformly expanding systems, Ergodic Theory and Dynamical Systems, volume 25, no. 6, pages 1719-1738.

2004

• Bhansali RJ, Holland MP, Kokoszka PS. (2004) Chaotic maps with slowly decaying correlations and intermittency, ASYMPTOTIC METHODS IN STOCHASTICS, volume 44, pages 99-126. [PDF]

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