Ian Melbourne's Publication List

Publications since 2001

  1. A. C. Freitas, J. Freitas, I. Melbourne and M. Todd. Convergence to decorated Lévy processes in non-Skorohod topologies for dynamical systems.
    Electron. J. Probab. 29 paper no. 170 (2024) 1-24. (abstract, pdf file)

  2. G. Gottwald and I. Melbourne. Time-reversibility and nonvanishing Lévy area.
    Nonlinearity 37 (2024) 075018 (12 pages at 1.5 months per page). (abstract, pdf file, journal version)

  3. Y. Lima, C. Matheus and I. Melbourne. Polynomial decay of correlations for nonpositively curved surfaces.
    Trans. Amer. Math. Soc. 377 (2024) 6043-6095. (abstract, pdf file)

  4. I. Melbourne and P. Varandas. Good inducing schemes for uniformly hyperbolic flows, and applications to exponential decay of correlations.
    Ann. Henri Poincaré. Appeared online April 2024. (abstract, pdf file, link to journal version)

  5. I. Melbourne, F. Pène and D. Terhesiu. Local large deviations for periodic infinite horizon Lorentz gases.
    J. d'Analyse Math. 152 (2024) 283-316. (abstract, pdf file, link to journal version)

  6. I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne and H. Zhang. Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 2.
    Ann. Inst. H. Poincaré (B) Probab. Statist. 58 (2022) 1328-1350. (abstract, pdf file, published version)

  7. A. Korepanov, Z. Kosloff and I. Melbourne. Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1.
    Ann. Inst. H. Poincaré (B) Probab. Statist. 58 (2022) 1305-1327. (abstract, pdf file, published version)

  8. I. Melbourne, N. Paviato and D. Terhesiu. Decay in norm of transfer operators for semiflows.
    Studia Math. 266 (2022) 149-166. (abstract, pdf file, OnlineFirst version)

  9. I. Melbourne, N. Paviato and D. Terhesiu. Nonexistence of spectral gaps in Hölder spaces for continuous time dynamical systems.
    Israel. J. Math. 247 (2022) 987-991. (abstract, pdf file, View only of published version)

  10. M. Galton and I. Melbourne. Iterated invariance principle for slowly mixing dynamical systems.
    Ann. Inst. H. Poincaré (B) Probab. Statist. 58 (2022) 1284-1304. (abstract, pdf file, published version)

  11. I. Melbourne and D. Terhesiu. Analytic proof of multivariate stable local large deviations and application to deterministic dynamical systems.
    Electron. J. Probab. 27, paper no. 21 (2022) 1-17. (abstract, pdf file, link to journal version)

  12. P. Jung, I. Melbourne, F. Pène, P. Varandas and H.-K. Zhang. Necessary and sufficient condition for M2-convergence to a Lévy process for billiards with cusps at flat points.
    Stoch. Dyn. 21 (2021) 2150024, 8 pages. (abstract, pdf file)

  13. P. Eslami, I. Melbourne and S. Vaienti. Sharp statistical properties for a family of multidimensional nonMarkovian nonconformal intermittent maps.
    Adv. Math. 388 (2021) 107853. (abstract, pdf file)

  14. 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)

  15. 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)

  16. 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)

  17. 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)

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

  19. 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, pdf file, View only of published version)

  20. 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, pdf file, published version)

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

  22. 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, pdf file)

  23. 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, pdf file, View only of published version)

  24. 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, pdf file)

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

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

  27. 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, pdf file)

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

  29. 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, pdf file, View only of published version)

  30. 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, pdf file)

  31. 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)

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

  33. 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, Online First pdf file)

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

  35. 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, pdf file, View only of published version)

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

  37. 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, pdf file, View only of published version)

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

  39. 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, pdf file)

  40. 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, pdf file)

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

  42. 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)

  43. 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, pdf file)

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

  45. 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)

  46. 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, pdf file)

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

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

  49. 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, pdf file) Supplementary information 1 (pdf file) Supplementary information 2 (pdf file)

  50. 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, pdf file)

  51. 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, pdf file)

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

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

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

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

  56. 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, pdf file)

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

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

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

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

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

  62. 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, pdf file)

  63. 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, pdf file)

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

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

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

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

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

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

  70. 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, pdf file)

  71. 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, pdf file)

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

  73. 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, pdf file)

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

  75. 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)

  76. 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, pdf file)

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

  78. 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, pdf file)

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

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

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

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

  83. 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, pdf file)

  84. 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, pdf file . The final two figures are in a separate postscript file.)

  85. 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, pdf file)

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

  87. 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, pdf file)

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

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

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

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

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

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

  94. 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, pdf file)

  95. 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)

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

  97. 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, pdf file)

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

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

  100. 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)

  101. 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)