2021

Melbourne, Ian, Paviato, Nicolo and Terhesiu, Dalia (2021) Nonexistence of spectral gaps in H\"older spaces for continuous time dynamical systems. Israel Journal of Mathematics . (In Press)

Gottwald, Georg A. and Melbourne, Ian (2021) Simulation of non-lipschitz stochastic differential 2 equations driven by α-stable noise : a method based on deterministic homogenisation. Multiscale Modeling and Simulation : A SIAM Interdisciplinary Journal . (In Press)

2020

Jung, Paul, Melbourne, Ian, Pene, Francoise, Varandas, Paulo and Zhang, Hong-Kun (2020) Necessary and sufficient condition for M2-convergence to a Lévy process for billiards with cusps at flat points. Stochastics and Dynamics . doi:10.1142/S0219493721500246 (In Press)

Chevyrev, Ilya, Friz, Peter K., Korepanov, Alexey and Melbourne, Ian (2020) Superdiffusive limits for deterministic fast–slow dynamical systems. Probability Theory and Related Fields . doi:10.1007/s00440-020-00988-5 (In Press)

Demers, Mark, Melbourne, Ian and Nicol, Matthew (2020) Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas. Nonlinearity, 33 (8). 4095. doi:10.1088/1361-6544/ab7d22

Bahsoun, Wael, Melbourne, Ian and Ruziboev, Marks (2020) Variance continuity for Lorenz flows. Annales Henri Poincare, 21 . pp. 1873-1892. doi:10.1007/s00023-020-00913-5

Melbourne, Ian and Varandas, Paulo (2020) Convergence to a Lévy process in the Skorohod M1 and M2 topologies for nonuniformly hyperbolic systems, including billiards with cusps. Communications in Mathematical Physics, 375 . pp. 653-678. doi:10.1007/s00220-019-03501-9

Melbourne, Ian and Terhesiu, Dalia (2020) Renewal theorems and mixing for non Markov flows with infinite measure. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 56 (1). pp. 449-476. doi:10.1214/19-AIHP968

2019

Antoniou, Marios and Melbourne, Ian (2019) Rate of Convergence in the Weak Invariance Principle for Deterministic Systems. Communications in Mathematical Physics, 369 (3). pp. 1147-1165. doi:10.1007/s00220-019-03334-6

Araújo, V. and Melbourne, Ian (2019) Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation. Advances in Mathematics, 349 . pp. 212-245. doi:10.1016/j.aim.2019.04.007

Bálint, Péter, Butterley, Oliver and Melbourne, Ian (2019) Polynomial decay of correlations for flows, including Lorentz gas examples. Communications in Mathematical Physics, 368 (1). pp. 55-111. doi:10.1007/s00220-019-03423-6

Korepanov, A., Kosloff, Z. and Melbourne, I. (2019) Explicit coupling argument for non-uniformly hyperbolic transformations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 149 (1). pp. 101-130. doi:10.1017/S0308210518000161

Korepanov, Alexey, Kosloff, Zemer and Melbourne, Ian (2019) Explicit coupling argument for nonuniformly hyperbolic transformations. Proceedings of the Royal Society of Edinburgh Section A : Mathematics, 149 (1). pp. 101-130. doi:10.1017/S0308210518000161

Antoniou, Marios and Melbourne, Ian (2019) Rate of convergence in the weak invariance principle for deterministic systems. Communications in Mathematical Physics, 369 . pp. 1147-1165. doi:10.1007/s00220-019-03334-6

Bruin, Henk, Melbourne, Ian and Terhesiu, Dalia (2019) Rates of mixing for non-Markov infinite measure semiflows. Transactions of the American Mathematical Society, 371 . pp. 7343-7386. doi:10.1090/tran/7582

Bruin, Henk, Melbourne, Ian and Terhesiu, Dalia (2019) Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards. Annales Henri Lebesgue . (In Press)

2018

Melbourne, Ian and Terhesiu, Dalia (2018) Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 12 . pp. 285-313. doi:10.3934/jmd.2018011

Bruin, Henk, Melbourne, Ian and Terhesiu, Dalia (2018) Rates of mixing for non-Markov infinite measure semiflows. Transactions of the American Mathematical Society, 371 (10). pp. 7343-7386. doi:10.1090/tran/7582

Melbourne, Ian (2018) Superpolynomial and polynomial mixing for semiflows and flows. Nonlinearity, 31 (10). R268. doi:10.1088/1361-6544/aad309

Bálint, Péter and Melbourne, Ian (2018) Statistical properties for flows with unbounded roof function, including the Lorenz attractor. Journal of Statistical Physics, 172 (4). pp. 1101-1126. doi:10.1007/s10955-018-2093-y

Korepanov, Alexey, Kosloff, Zemer and Melbourne, Ian (2018) Martingale-coboundary decomposition for families of dynamical systems. Annales de l'Institut Henri Poincare (C) Non Linear Analysis , 35 (4). pp. 859-885. doi:10.1016/j.anihpc.2017.08.005

2017

Kelly, David and Melbourne, Ian (2017) Deterministic homogenization for fast-slow systems with chaotic noise. Journal of Functional Analysis, 272 (10). pp. 4063-4102. doi:10.1016/j.jfa.2017.01.015

Korepanov, Alexey, Kosloff, Zemer and Melbourne, Ian (2017) Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems. Studia Mathematica, 238 . pp. 59-89. doi:10.4064/sm8540-1-2017

Araújo, Vitor and Melbourne, Ian (2017) Existence and smoothness of the stable foliation for sectional hyperbolic attractors. Bulletin of the London Mathematical Society, 49 (2). pp. 351-367. doi:10.1112/blms.12037

Melbourne, Ian and Terhesiu, Dalia (2017) Operator renewal theory for continuous time dynamical systems with finite and infinite measure. Monatshefte fur Mathematik, 182 (2). pp. 377-431. doi:10.1007/s00605-016-0922-0

Butterley, Oliver and Melbourne, Ian (2017) Disintegration of invariant measures for hyperbolic skew products. Israel Journal of Mathematics, 219 (1). pp. 171-188. doi:10.1007/s11856-017-1477-z

2016

Gottwald, Georg A. and Melbourne, Ian (2016) On the detection of superdiffusive behaviour in time series. Journal of Statistical Mechanics: Theory and Experiment, 2016 . 123205. doi:10.1088/1742-5468/aa4f0f

Araújo, Vitor and Melbourne, Ian (2016) Exponential decay of correlations for nonuniformly hyperbolic flows with a C1+αC1+α stable foliation, including the classical Lorenz attractor. Annales Henri Poincaré, 17 (11). pp. 2975-3004. doi:10.1007/s00023-016-0482-9

Gottwald, Georg A. and Melbourne, Ian (2016) Central limit theorems and suppression of anomalous diffusion for systems with symmetry. Nonlinearity, 29 (10). 2941. doi:10.1088/0951-7715/29/10/2941

Gottwald, Georg A. and Melbourne, Ian (2016) Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations. Journal of Physics A: Mathematical and Theoretical, 49 (17). 174003. doi:10.1088/1751-8113/49/17/174003

Melbourne, Ian and Varandas, Paulo (2016) A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion. Stochastics and Dynamics, 16 (03). 1660012. doi:10.1142/S0219493716600121

Kelly, David and Melbourne, Ian (2016) Smooth approximation of stochastic differential equations. The Annals of Probability, 44 (1). pp. 479-520. doi:10.1214/14-AOP979

2014

Melbourne, Ian and Terhesiu, Dalia (2014) Decay of correlations for non-uniformly expanding systems with general return times. Ergodic Theory and Dynamical Systems, Volume 34 (Number 6). pp. 893-918. doi:10.1017/etds.2012.158

Gouëzel, Sébastien and Melbourne, Ian (2014) Moment bounds and concentration inequalities for slowly mixing dynamical systems. Electronic Journal of Probability, 19 . pp. 1-30. doi:10.1214/EJP.v19-3427

Gottwald, Georg A. and Melbourne, Ian (2014) A test for a conjecture on the nature of attractors for smooth dynamical systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 24 (Number 2). Article number 024403. doi:10.1063/1.4868984

2013

Gottwald, G. A. and Melbourne, Ian (2013) Homogenization for deterministic maps and multiplicative noise. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 469 (Number 2156). Article number 20130201. doi:10.1098/rspa.2013.0201

Melbourne, Ian and Terhesiu, Dalia (2013) First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure. Israel Journal of Mathematics, Volume 194 (Number 2). pp. 793-830. doi:10.1007/s11856-012-0154-5

Gottwald, G. A. and Melbourne, Ian (2013) A Huygens principle for diffusion and anomalous diffusion in spatially extended systems. Proceedings of the National Academy of Sciences of the United States of America, Volume 110 (Number 21). pp. 8411-8416. doi:10.1073/pnas.1217926110

Luzzatto, Stefano and Melbourne, Ian (2013) Statistical properties and decay of correlations for interval maps with critical points and singularities. Communications in Mathematical Physics, Volume 320 (Number 1). pp. 21-35. doi:10.1007/s00220-013-1709-y

2012

Melbourne, Ian and Torok, Andrei (2012) Convergence of moments for Axiom A and non-uniformly hyperbolic flows. Ergodic Theory and Dynamical Systems, Vol.32 (No.3). pp. 1091-1100. doi:10.1017/S0143385711000174

Melbourne, Ian and Terhesiu, Dalia (2012) Operator renewal theory and mixing rates for dynamical systems with infinite measure. Inventiones mathematicae, Vol. 189 (No. 1). pp. 61-110. doi:10.1007/s00222-011-0361-4

Melbourne, Ian, Nijica, Viorel and Torok, Andrei (2012) Transitivity of Heisenberg group extensions of hyperbolic systems. Ergodic Theory and Dynamical Systems, Vol.32 (No.1). pp. 223-235. doi:10.1017/S014338571000091X

2011

Melbourne, Ian and Stuart, A. M. (2011) A note on diffusion limits of chaotic skew-product flows. Nonlinearity, Volume 24 (Number 4). pp. 1361-1367. doi:10.1088/0951-7715/24/4/018

2010

Bruin, Henk, Demers, Mark and Melbourne, Ian (2010) Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergodic Theory and Dynamical Systems, Vol.30 (No.3). pp. 687-728. doi:10.1017/S0143385709000200

2009

Gottwald, Georg A. and Melbourne, Ian (2009) On the implementation of the 0–1 test for chaos. SIAM Journal on Applied Dynamical Systems, Vol.8 (No.1). pp. 129-145. doi:10.1137/080718851

Gottwald, Georg A. and Melbourne, Ian (2009) On the validity of the 0–1 test for chaos. Nonlinearity, Vol.22 (No.6). pp. 1367-1382. doi:10.1088/0951-7715/22/6/006

Melbourne, Ian, Nitica, V. and Torok, Andrei (2009) Transitivity of Euclidean-type extensions of hyperbolic systems. Ergodic Theory and Dynamical Systems, Vol.29 (No.5). pp. 1585-1602. doi:10.1017/S0143385708000886

Melbourne, Ian and Nicol, Matthew (2009) A vector-valued almost sure invariance principle for hyperbolic dynamical systems. The Annals of Probability, Vol.37 (No.2). pp. 478-505. doi:10.1214/08-AOP410

2008

Gottwald, Georg A. and Melbourne, Ian (2008) Comment on “Reliability of the 0-1 test for chaos”. Physical Review E, Vol.77 (No.2). Article no. 028201 . doi:10.1103/PhysRevE.77.028201

Bálint, Péter and Melbourne, Ian (2008) Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows. Journal of Statistical Physics, Vol.133 (No.3). pp. 435-447. doi:10.1007/s10955-008-9623-y

Melbourne, Ian (2008) Decay of correlations for slowly mixing flows. Proceedings of the London Mathematical Society, Vol. 98 (No. 1). pp. 163-190. doi:10.1112/plms/pdn028

Melbourne, Ian (2008) Large and moderate deviations for slowly mixing dynamical systems. Proceedings of the American Mathematical Society, Vol.137 (No.5). pp. 1735-1741. doi:10.1090/S0002-9939-08-09751-7

Melbourne, Ian and Nicol, Matthew (2008) Large deviations for nonuniformly hyperbolic systems. Transactions of the American Mathematical Society, Vol.360 (No.12). pp. 6661-6676. doi:10.1090/S0002-9947-08-04520-0

Melbourne, Ian and Gottwald, Georg A (2008) Power spectra for deterministic chaotic dynamical systems. Nonlinearity, vol.21 (No.1). pp. 179-189. doi:10.1088/0951-7715/21/1/010