Ian Melbourne's Publication List


Publications since 2001


  1. H. Bruin, I. Melbourne and D. Terhesiu. Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards.
    Annales Henri Lebesgue 4 (2021) 407-451. (abstract, pdf file)

  2. G. Gottwald and I. Melbourne. Simulation of non-Lipschitz α-stable stochastic differential equations: a method based on deterministic homogenisation.
    Multiscale Modeling and Simulation 19 (2021) 665-687. (abstract, pdf file)


  3. I. Chevyrev, P. K. Friz, A. Korepanov and I. Melbourne. Superdiffusive limits for deterministic fast-slow dynamical systems.
    Probab. Theory Related Fields 178 (2020) 735-770. (abstract, pdf file)

  4. M. Demers, I. Melbourne and M. Nicol. Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas.
    Nonlinearity 33 (2020) 4095-4113. (abstract, pdf file)

  5. W. Bahsoun, I. Melbourne and M. Ruziboev. Variance continuity for Lorenz flows.
    Annales Henri Poincaré 21 (2020) 1873-1892. (abstract, postscript file, pdf file, published version)

  6. I. Melbourne and P. Varandas. Convergence to a Lévy process in the Skorohod M1 and M2 topologies for nonuniformly hyperbolic systems, including billiards with cusps.
    Commun. Math. Phys. 375 (2020) 653-678. (abstract, postscript file, pdf file, View only of published version)

  7. I. Melbourne and D. Terhesiu. Renewal theorems and mixing for non Markov flows with infinite measure.
    Ann. Inst. H. Poincaré (B) Probab. Statist. 56 (2020) 449-476. (abstract, postscript file, pdf file, published version)


  8. M. Antoniou and I. Melbourne. Rate of convergence in the weak invariance principle for deterministic systems.
    Commun. Math. Phys. 369 (2019) 1147-1165. (abstract, postscript file, pdf file, View only of published version)

  9. I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne and H. Zhang. Multiscale systems, homogenization, and rough paths.
    Probability and Analysis in Interacting Physical Systems: In Honor of S.R.S. Varadhan, Berlin, August, 2016, eds. P. Friz et al. Springer Proc. in Maths. & Stat. 283 (2019) 17-42. (abstract, postscript file, pdf file)

  10. P. Bálint, O. Butterley and I. Melbourne. Polynomial decay of correlations for flows, including Lorentz gas examples.
    Commun. Math. Phys. 368 (2019) 55-111. (abstract, postscript file, pdf file, View only of published version)

  11. V. Araújo and I. Melbourne. Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation.
    Adv. Math. 349 (2019) 212-245. (abstract, postscript file, pdf file)

  12. A. Korepanov, Z. Kosloff and I. Melbourne. Explicit coupling argument for nonuniformly hyperbolic transformations.
    Proc. Roy. Soc. Edinburgh 149 (2019) 101-130. (abstract, postscript file, pdf file)

  13. H. Bruin, I. Melbourne and D. Terhesiu. Rates of mixing for nonMarkov infinite measure semiflows.
    Trans. Amer. Math. Soc. 371 (2019) 7343-7386. (abstract, postscript file, pdf file)


  14. I. Melbourne and D. Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure.
    J. Mod. Dyn. 12 (2018) 285-313. (abstract, postscript file, pdf file)

  15. I. Melbourne. Superpolynomial and polynomial mixing for semiflows and flows.
    Nonlinearity 31 (2018) R268-R316. (abstract, postscript file, pdf file)

  16. P. Bálint and I. Melbourne. Statistical properties for flows with unbounded roof function, including the Lorenz attractor.
    J. Stat. Phys. 172 (2018) 1101-1126. (abstract, postscript file, pdf file, View only of published version)

  17. A. Korepanov, Z. Kosloff and I. Melbourne. Martingale-coboundary decomposition for families of dynamical systems.
    Annales de l'Institut Henri Poincaré / Analyse Non Lineaire 35 (2018) 859-885. (abstract, postscript file, pdf file)


  18. V. Araújo and I. Melbourne. Existence and smoothness of the stable foliation for sectional hyperbolic attractors.
    Bull. London Math. Soc. 49 (2017) 351-367. (abstract, pdf file)

  19. O. Butterley and I. Melbourne. Disintegration of invariant measures for hyperbolic skew products.
    Israel J. Math. 219 (2017) 171-188. (abstract, postscript file, pdf file, View only of published version)

  20. A. Korepanov, Z. Kosloff and I. Melbourne. Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems.
    Studia Math. 238 (2017) 59-89. (abstract, postscript file, Online First pdf file)

  21. D. Kelly and I. Melbourne. Deterministic homogenization for fast-slow systems with chaotic noise.
    J. Funct. Anal. 272 (2017) 4063-4102. (abstract, postscript file, pdf file)

  22. I. Melbourne and D. Terhesiu. Operator renewal theory for continuous time dynamical systems with finite and infinite measure.
    Monatsh. Math. 182 (2017) 377-431. (abstract, postscript file, pdf file, View only of published version)


  23. G. Gottwald and I. Melbourne. On the detection of superdiffusive behaviour in time series.
    J. Stat. Mech. (2016) 123205 (14 pages) (abstract, pdf file)

  24. V. Araújo and I. Melbourne. Exponential decay of correlations for nonuniformly hyperbolic flows with a C1+α stable foliation, including the classical Lorenz attractor.
    Annales Henri Poincaré. 17 (2016) 2975-3004. (abstract, postscript file, pdf file, View only of published version)

  25. G. Gottwald and I. Melbourne. Central limit theorems and suppression of anomalous diffusion for systems with symmetry.
    Nonlinearity 29 (2016) 2941-2960. (abstract, postscript file, pdf file)

  26. G. Gottwald and I. Melbourne. Broadband nature of power spectra for intermittent maps with summable and nonsummable decay of correlations.
    J. Phys. A 49 (2016) 174003 (17 pages) (abstract, postscript file, pdf file)

  27. I. Melbourne and P. Varandas. A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion.
    Stoch. Dyn. 16 (2016) 1660012 (13 pages) (abstract, postscript file, pdf file)

  28. D. Kelly and I. Melbourne. Smooth approximation of stochastic differential equations.
    Annals of Probability 44 (2016) 479-520. (abstract, postscript file, pdf file)


  29. V. Araújo, I. Melbourne and P. Varandas. Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 maps.
    Commun. Math. Phys. 340 (2015) 901-938. (abstract, pdf file)

  30. I. Melbourne and R. Zweimüller. Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems.
    Ann. Inst. H. Poincaré Probab. Statist. 51 (2015) 545-556. (abstract, postscript file, pdf file)

  31. I. Melbourne. Mixing for invertible dynamical systems with infinite measure.
    Stoch. Dyn. 15 (2015) 1550012 (25 pages). (abstract, postscript file, pdf file)


  32. S. Gouëzel and I. Melbourne. Moment bounds and concentration inequalities for slowly mixing dynamical systems.
    Electron.J. Probab. 19 (2014) 1-30. (abstract, or pdf file)

  33. I. Melbourne and D. Terhesiu. Decay of correlations for nonuniformly expanding systems with general return times.
    Ergodic Theory Dyn. Syst 34 (2014) 893-918 (abstract, postscript file, pdf file)

  34. G. Gottwald and I. Melbourne. A test for a conjecture on the nature of attractors for smooth dynamical systems.
    Chaos 24 (2014) 024403 (abstract, postscript file, pdf file)


  35. G. Gottwald and I. Melbourne. Homogenization for deterministic maps and multiplicative noise.
    Proc. Roy. Soc. London A 469 (2013) 20130201 (abstract, postscript file, pdf file, corrected version, April 2015)

  36. G. Gottwald and I. Melbourne. A Huygens principle for diffusion and anomalous diffusion in spatially extended systems.
    Proc. Natl. Acad. Sci. USA 110 (2013) 8411-8416. (abstract, postscript file, pdf file) Supplementary information 1 (pdf file) Supplementary information 2 (pdf file)
  37. S. Luzzatto and I. Melbourne. Statistical properties and decay of correlations for interval maps with critical points and singularities.
    Commun. Math. Phys. 320 (2013) 21-35. (abstract, postscript file, pdf file)

  38. I. Melbourne and D. Terhesiu. First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure.
    Israel J. Math. 194 (2013) 793-830. (abstract, postscript file, pdf file)


  39. I. Melbourne and D. Terhesiu. Operator renewal theory and mixing rates for dynamical systems with infinite measure.
    Inventiones Mathematicae 189 (2012) 61-110 (abstract, postscript file, pdf file, corrected version, March 2015)

  40. I. Melbourne and A. Török. Convergence of moments for Axiom A and nonuniformly hyperbolic flows.
    Ergodic Theory Dyn. Syst. 32 (2012) 1091-1100 (abstract, postscript file, pdf file)

  41. I. Melbourne, V. Niţică and A. Török. Transitivity of Heisenberg group extensions of hyperbolic systems.
    Ergodic Theory Dyn. Syst. 32 (2012) 223-235 (abstract, postscript file, pdf file)


  42. I. Melbourne and A. M. Stuart. A note on diffusion limits of chaotic skew product flows.
    Nonlinearity 24 (2011) 1361-1367. (abstract, postscript file, pdf file, correction, April 2015, corrected version, April 2015)


  43. H. Bruin, M. Demers and I. Melbourne. Existence and convergence properties of physical measures for certain dynamical systems with holes.
    Ergodic Theory Dyn. Syst. 30 (2010) 687-728. (abstract, postscript file, pdf file)


  44. I. Melbourne, V. Niţică and A. Török. Transitivity of Euclidean-type extensions of hyperbolic systems.
    Ergodic Theory Dyn. Syst. 29 (2009) 1585-1602. (abstract, postscript file, pdf file)

  45. G. Gottwald and I. Melbourne. On the validity of the 0-1 test for chaos.
    Nonlinearity 22 (2009) 1367-1382. (abstract, postscript file, pdf file)

  46. I. Melbourne and M. Nicol. A vector-valued almost sure invariance principle for hyperbolic dynamical systems.
    Annals of Probability 37 (2009) 478-505. (abstract, postscript file, pdf file)

  47. I. Melbourne. Large and moderate deviations for slowly mixing dynamical systems.
    Proc. Amer. Math. Soc. 137 (2009) 1735-1741. (abstract, postscript file, pdf file)

  48. I. Melbourne. Decay of correlations for slowly mixing flows.
    Proc. London Math. Soc. 98 (2009) 163-190. (abstract, postscript file, pdf file)

  49. G. Gottwald and I. Melbourne. On the implementation of the 0-1 test for chaos.
    SIAM J. Appl. Dyn. Sys. 8 (2009) 129-145 (abstract, postscript file, pdf file)


  50. P. Bálint and I. Melbourne. Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows.
    J. Stat. Phys. 133 (2008) 435-447. (abstract, postscript file, pdf file)

  51. I. Melbourne and M. Nicol. Large deviations for nonuniformly hyperbolic systems.
    Trans. Amer. Math. Soc. 360 (2008) 6661-6676. (abstract, postscript file, pdf file)

  52. G. Gottwald and I. Melbourne. Comment on ``Reliability of the 0-1 test for chaos''.
    Phys. Rev. E 77 (2008) 028201. (abstract, postscript file, pdf file)

  53. I. Melbourne and G. Gottwald. Power spectra for deterministic chaotic dynamical systems.
    Nonlinearity 21 (2008) 179-189. (abstract, postscript file, pdf file)


  54. M. Holland and I. Melbourne. Central limit theorems and invariance principles for Lorenz attractors.
    J London Math Soc 76 (2007) 345-364. (abstract, postscript file, pdf file)

  55. M. Field, I. Melbourne and A. Török. Stability of mixing and rapid mixing for hyperbolic flows.
    Annals of Mathematics 166 (2007) 269-291. (abstract, postscript file, pdf file)

  56. D. Chan and I. Melbourne. A geometric characterisation of resonance in Hopf bifurcation from relative equilibria.
    Physica D 234 (2007) 98-104. (abstract, postscript file, pdf file)

  57. I. Falconer, G. Gottwald, I. Melbourne and K. Wormnes. Application of the 0-1 Test for Chaos to Experimental Data.
    SIAM J. Appl. Dyn. Sys. 6 (2007) 395-402. (abstract, postscript file, pdf file)

  58. J. S. W. Lamb and I. Melbourne. Normal form theory for relative equilibria and relative periodic solutions.
    Trans. Amer. Math. Soc. 359 (2007) 4537-4556. (abstract, postscript file, pdf file)

  59. I. Melbourne. Rapid decay of correlations for nonuniformly hyperbolic flows.
    Trans. Amer. Math. Soc. 359 (2007) 2421-2441. (abstract, postscript file, pdf file)


  60. J. S. W. Lamb, I. Melbourne and C. Wulff. Hopf bifurcation from relative periodic solutions; Secondary bifurcations from meandering spirals.
    J. Difference Equ. Appl. 12 (2006) 1127-1145. (abstract, postscript file, pdf file)

  61. P. Aston and I. Melbourne. Lyapunov exponents of symmetric attractors.
    Nonlinearity 19 (2006) 2455-2466. (abstract, postscript file, pdf file)

  62. I. Melbourne. Symmetry and symmetry breaking in dynamical systems.
    Encycl. Math. Phys. (eds. J.-P. Francoise, G.L. Naber and S.T. Tsou). Oxford: Elsevier, 2006, vol. 5, pp. 184-190. (abstract, pdf file)

  63. I. Melbourne, V. Niţică and A. Török. A note about stable transitivity of noncompact extensions of hyperbolic systems.
    Disc. Cont. Dyn. Syst. 14 (2006) 355-363. (abstract, postscript file, pdf file)


  64. I. Melbourne and A. Windsor. A C diffeomorphism with infinitely many intermingled basins.
    Ergodic Theory Dyn. Syst. 25 (2005) 1951-1959. (abstract, postscript file, pdf file)

  65. H. Bruin, M. Holland and I. Melbourne. Subexponential decay of correlations for compact group extensions of nonuniformly expanding systems.
    Ergodic Theory Dyn. Syst. 25 (2005) 1719-1738. (abstract, postscript file, pdf file)

  66. G. Gottwald and I. Melbourne. Testing for chaos in deterministic systems with noise.
    Physica D 212 (2005) 100-110. (abstract, postscript file, pdf file)

  67. S. Luzzatto, I. Melbourne and F. Paccaut. The Lorenz attractor is mixing.
    Commun. Math. Phys. 260 (2005) 393-401. (abstract, postscript file, pdf file)

  68. I. Melbourne and M. Nicol. Almost sure invariance principle for nonuniformly hyperbolic systems.
    Commun. Math. Phys. 260 (2005) 131-146. (abstract, postscript file, pdf file)

  69. I. Melbourne, V. Niţică and A. Török. Stable transitivity of certain noncompact extensions of hyperbolic systems.
    Annales Henri Poincaré 6 (2005) 725-746. (abstract, postscript file, pdf file)

  70. M. Field, I. Melbourne and A. Török. Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets.
    Ergodic Theory Dyn. Syst. 25 (2005) 517-551. (abstract, postscript file, pdf file)

  71. S. Abreu, P. Aston and I. Melbourne. Symmetric chaos in a local codimension two bifurcation with the symmetry group of a square.
    SIAM J Appl. Dyn. Sys 4 (2005) 32-52. (abstract, postscript file, pdf file . The final two figures are in a separate postscript file.)

  72. M. Field, I. Melbourne, M. Nicol and A. Török. Statistical properties of compact group extensions of hyperbolic flows and their time one maps.
    Disc. Cont. Dyn. Syst. 12 (2005) 79-96. (abstract, postscript file, pdf file)


  73. I. Melbourne and A. Török. Statistical limit theorems for suspension flows.
    Israel Journal of Math. 144 (2004) 191-209. (abstract, postscript file, pdf file)

  74. M. Krupa and I. Melbourne. Asymptotic stability of heteroclinic cycles in systems with symmetry, II.
    Proc. Roy. Soc. Edinburgh A 134A (2004) 1177-1197. (abstract, postscript file, pdf file)

  75. I. Melbourne and M. Nicol. Statistical properties of endomorphisms and compact group extensions.
    J. London Math. Soc. 70 (2004) 427-446. (abstract, postscript file, pdf file)

  76. I. Melbourne and M. Nicol. Statistical limit laws for equivariant observations.
    Stoch. Dyn. 4 (2004) 1-13. (abstract, postscript file, pdf file)

  77. G. Gottwald and I. Melbourne. A new test for chaos in deterministic systems.
    Proc. Roy. Soc. London A 460 (2004) 603-611. (abstract, postscript file, pdf file)

  78. I. Melbourne and G. Schneider. Phase dynamics in the complex Ginzburg-Landau equation.
    J. Diff. Eqns. 199 (2004) 22-46. (abstract, postscript file, pdf file)

  79. I. Melbourne and G. Schneider. Phase dynamics in the real Ginzburg-Landau equation.
    Math. Nachr. 263-264 (2004) 171-180. (abstract, postscript file, pdf file)


  80. M. Ådahl, I. Melbourne and M. Nicol. Random iteration of Euclidean isometries.
    Nonlinearity 16 (2003) 977-987. (abstract, postscript file, pdf file)

  81. M. Field, I. Melbourne and A. Török. Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions.
    Ergodic Theory Dyn. Syst. 23 (2003) 87-110. (abstract, postscript file, pdf file)

  82. J. S. W. Lamb, I. Melbourne and C. Wulff. Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions.
    J. Diff. Eq. 191 (2003) 377-407. (abstract, paper)

  83. I. Melbourne and M. Nicol. Stable transitivity of Euclidean group extensions.
    Ergodic Theory Dyn. Syst. 23 (2003) 611-619. (abstract, postscript file, pdf file)


  84. I. Melbourne and A. Török. Central limit theorems and invariance principles for time-one maps of hyperbolic flows.
    Commun. Math. Phys. 229 (2002) 57-71. (abstract, postscript file, pdf file)


  85. P. Ashwin, I. Melbourne and M. Nicol. Hypermeander of spirals; local bifurcations and statistical properties.
    Physica D 156 (2001) 364-382. (abstract, postscript file, pdf file)

  86. M. Nicol, I. Melbourne and P. Ashwin. Euclidean extensions of dynamical systems.
    Nonlinearity. 14 (2001) 275-300. (abstract, postscript file, pdf file)

  87. C. Wulff, J. S. W. Lamb and I. Melbourne. Bifurcation from relative periodic solutions.
    Ergodic Theory and Dynamical Systems. 21 (2001) 605-635. (abstract, paper)

  88. I. Melbourne, M. R. E. Proctor and A. M. Rucklidge. A heteroclinic model of geodynamo reversals and excursions.
    Dynamo and Dynamics, a Mathematical Challenge (P. Chossat et al. eds.) Kluwer, Netherlands, 2001, 363-370. (abstract, paper)