Publications/Preprints of Henk Bruin

    Books

  • Topics from one-dimensional dynamics, (with Karen Brucks) Topics from one-dimensional dynamics,
    London Mathematical Society, Student Texts 62 Cambridge University Press 2004.
    Available at Cambridge University Press

  • Henk Bruin book: Topological and Ergodic Theory of Symbolic Dynamics Topological and Ergodic Theory of Symbolic Dynamics,
    AMS book series Graduate Studies in Matematics.
    The electronic version can be purchased on the AMS Website from
    November 2022. The printed version appeared in February 2023.

    List of corrections and remarks concerning this book.




      Research Papers

      1992

    1. Numerical determination of the continued fraction expansion of the rotation number, Physica D 59 (1992) 158-168.

      2. 1994

    2. Topological conditions for the existence of invariant measures for unimodal maps, Ergod. Th. and Dyn. Sys. 14 (1994) 433-451. [.pdf]

      3. 1995

    3. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Commun. Math. Phys. 168 (1995) 571-580. [.pdf] The original publication is available at [http://www.springerlink.com]
    4. Combinatorics of the kneading map, Int. Jour. of Bifur. and Chaos 5 (1995) 1339-1349. [.pdf]
    5. Non-monotonicity of entropy of interval maps, Phys. Lett. A 202 (1995) 359-362. Preprint version [.pdf]

      6. 1996

    6. (with Gerhard Keller, Tomasz Nowicki, Sebastian van Strien) Wild Cantor attractors exist, Annals of Math. 143 (1996) 97-130. [.pdf]
    7. Quasi-symmetry of conjugacies between interval maps, Nonlinearity 9 (1996) 1191-1207. [.pdf]
    8. An algorithm to compute the topological entropy of a unimodal map, Proceedings contribution for the conference in memory of W. Szlenk, Barcelona (1996) Internat. J. Bifur. Chaos 9 (1999) 1881-1882. [.pdf]

      1997

    9. (with Gerhard Keller and Matthias St. Pierre) Adding machines and wild attractors, Ergod. Th. and Dyn. Sys. 17 (1997) 1267-1287. [.pdf]

      10. 1998

    10. Topological conditions for the existence of Cantor attractors, Trans. Amer. Math. Soc. 350 (1998) 2229-2263. [.pdf]
    11. For almost every tent-map, the turning point is typical, Fund. Math. 155 (1998) 215-235; [.pdf]
    12. The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps, Ergod. Th.and Dyn. Sys. 18 (1998) 555-565. [.pdf]
    13. (with Gerhard Keller) Equilibrium states for unimodal maps, Ergod. Th. and Dyn. Sys. 18 (1998) 765-789. [.pdf]
    14. (with Jane Hawkins) Examples of expanding C1 maps having no sigma-finite invariant measure equivalent to Lebesgue, Isr. J. Math. 108 (1998) 83-107. [.pdf]

      15. 1999

    15. (with Karen M. Brucks) Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999) 219-246. [.pdf]
    16. Homeomorphic restrictions of unimodal maps, Contemp. Math. 246 (1999) 47-56; [.pdf]
    17. Planar embeddings of inverse limit spaces of unimodal maps. Topology and its Applications 96 (1999) 191-208. [.pdf]

      18. 2000

    18. Inverse limit spaces of post-critically finite tent maps. Fund. Math. 165 (2000) 125-138 [.pdf]
    19. Dimensions of recurrence times and minimal subshifts, Conference Proceedings: from Chrystal to Chaos, Luminy (2000), [.pdf]

      20. 2001

    20. (with Jane Hawkins) Exactness and maximal automorphic factors of unimodal maps, Ergod. Th. and Dyn. Sys. 21 (2001) 1009-1034; [.pdf]
    21. (with Sebastian van Strien) Existence of acips for multimodal maps, in Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday (2001) [.pdf]

      22. 2002

    22. (with Dierk Schleicher) Symbolic dynamics of quadratic polynomials. Preprint (2002) This is available on the web-pages of the Mittag-Leffler institute.

      23. 2003

    23. (with Stefano Luzzatto and Sebastian van Strien) Decay of correlations in one-dimensional dynamics, Preprint (1999) and (2001), Ann. Sci. Ec. Norm. Sup. 36 (2003) 621-646. [.pdf]
    24. (with Benoit Saussol, Serge Troubetzkoy and Sandro Vaienti) Return time statistics via inducing, Ergod. Th. and Dyn. Sys. 23 (2003) 991-1013; [.pdf]
    25. (with Sebastian van Strien) Expansion of derivatives in one-dimensional dynamics, Isr. J. Math. 137 (2003) 223-263. [.pdf]
    26. (with Sergiy Kolyada and L'ubomir Snoha) Minimal nonhomogeneous continua, Colloq. Math. 95 (2003) 123-132. [.pdf]
    27. (with Sandro Vaienti) Return time statistics for unimodal maps. Fund. Math. 176 (2003) 77--94. [.pdf]
    28. Minimal Cantor systems and unimodal maps. J. Difference Eq. and Appl. 9 (2003) 305--318. [.pdf]
    29. (with Serge Troubetzkoy) The Gauss map on a class of interval translation mappings. Isr. J. Math. 137 (2003) 125-148. [.ps]
    30. (with Andre Lambert, Guillaume Poggiaspalla and Sandro Vaienti) Numerical analysis for a discontinuous rotation of the torus. Chaos 13 (2003) 558-571. The original publication is available at [http://www.springerlink.com]
    31. (with Weixiao Shen and Sebastian van Strien) Invariant measures exist without a growth condition. Commun. Math. Phys. 241 (2003) 287-306 [.pdf].
      The original publication is available at [http://www.springerlink.com]

      32. 2004

    32. (with Karen Brucks) Topics from one-dimensional dynamics,
      London Mathematical Society, Student Texts 62 Cambridge University Press 2004.
      Available at Cambridge University Press
    33. (with Martijn van Noort) Nonconformal perturbations of z -> z 2 + c: the 1:3 resonance, Nonlinearity 17 (2004) 765 - 789 [.pdf]

      34. 2005

    34. Asymptotic arc-components of unimodal inverse limit spaces, Top. & Appl. 152 (2005) 182 - 200. [.pdf]
    35. (with Mark Holland and Ian Melbourne) Subexponential decay of correlations for compact group extension of nonuniformly expanding systems, Ergod. Th. and Dyn. Sys. 25 (2005) 1719-1738. [.pdf]
    36. (with Mark Holland and Matt Nicol) Livsic regularity for Markov systems, Ergod. Th. and Dyn. Sys. 25 (2005) 1739-1766. [.pdf]
    37. (with Oksana Volkova) The complexity of Fibonacci-like kneading sequences, Theo. Comp. Science. 377 (2005) 379-389. [.pdf]

      38. 2006

    38. (with Weixiao Shen and Sebastian van Strien) Existence of unique SRB-measures is typical for unimodal families, Annales Scientifiques de l'ENS, 4e serie, 39 (2006) 381-414. [.pdf]
    39. (with Stefano Luzzatto) Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps, Proc. Amer. Math. Soc. 134 (2006), no. 1, 265-272 (electronic). In Preprint 2004 version: [.pdf]
    40. (with Mike Todd) Markov extensions and lifting measures for complex polynomials, [.pdf] Ergod. Th. and Dyn. Sys. 27 (2007) 743-768.
    41. Renormalisation in a class of interval translation maps of d branches, Dynamical Systems, an international journal 22 (2007) 11 - 24. Preprint version 2006. [.pdf]
    42. Subcontinua of Fibonacci-like unimodal inverse limit spaces. Topology Proceedings 31 (2007) no. 1, 37-50. [.pdf]
    43. (with Mike Todd) Complex maps without invariant densities, Nonlinearity 19 (2006) 2929-2945. Preprint (Version October 2006). [.pdf]

      44. 2007

    44. (with Mike Todd) Equilibrium states for interval maps: the potential -t log |Df|. Ann. Sci. Ecol Norm. Sup. 42 2009, 559-600. Preprint 2007, Revised version of January 2009: [.pdf]
    45. (with Juan Rivera-Letelier, Weixiao Shen and Sebastian van Strien) Large derivatives, backward contraction and invariant densities for interval maps, Preprint version 2007. [.pdf].
      Inventiones Mathematicae 172 (2008), 509-533. The original publication is available at [http://www.springerlink.com].
    46. (with Jane Hawkins) Rigidity of smooth one-sided Bernoulli endomorphisms, Preprint 2007 - revised 2009, [.pdf] New York Journal of Maths. 15 (2009), 451-483.
    47. Combinatorics of (Fibonacci-like) unimodal maps. Notes used for Spring School at Luminy, April 2006, "Séminaires et Congrès" 20 17-37 (2010). [.pdf]

      48. 2008

    48. (with Jawad Al-Khal and Michael Jakobson) New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure, Discrete and Continuous Dynamical Systems, 22 no. 1-2, (2008) 35 - 61, see [here] for journal's version. Earlier preprint version in [.pdf]
    49. (with Mike Todd) Equilibrium states for interval maps: potentials with sup φ - inf φ < h top(f), Commun. Math. Phys. 283 (2008) 579-611. The original publication is available at this Springer site
      Updated version December 2010, [.pdf]
    50. (with Mike Todd) Return time statistics for invariant measures for interval maps with positive Lyapunov exponent. Version August 2008, [.pdf]. Stochastics and Dynamics 9 (2009), 81-100.
    51. (with Michael Jakobson and appendix by Genadi Levin) New examples of topologically equivalent S-unimodal maps with different metric properties, Contemp. Math. 469 2008 119-139. [.pdf]
    52. (with Jonathan Deane) Piecewise contractions are asymptotically periodic, Proc. Amer. Math. Soc. 137 (2009), 1389-1395. [.pdf]
    53. (with Dierk Schleicher) Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials. Journal Lond. Math. Soc. 78 (2008), no. 2, 502--522. [.pdf]
    54. (Non)invertibility of Fibonacci-like unimodal maps restricted to their critical omega-limit sets. Preprint 2008. [.pdf], Topology Proceedings 37 (2011) 459-480.
    55. (with Alexandra Kaffl and Dierk Schleicher) Existence of quadratic Hubbard trees. Fund. Math. 202 (2009) 251-279, [.pdf]
    56. (with Mark Demers and Ian Melbourne) Existence and convergence properties of physical measures for certain dynamical systems with holes, Preprint 2008 (revised and extended after the 2007 preprint) [.pdf]
      Ergod. Th. and Dyn. Sys. 30 2010, 687-728 (Published Online by Cambridge University Press 24 Nov 2009 doi:10.1017/S0143385709000200)

      57. 2009

    57. (with Matt Nicol and Dalia Terhesiu) On Young towers associated with infinite measure preserving transformations, Stoch. and Dynamics, 9 (2009), 635 - 655, in [.pdf], see also World Scientific Publishing Company
    58. (with Sebastian van Strien) Monotonicity of entropy for real multimodal maps Preprint 2009, revised version of December 2013 [.pdf]. J. Amer. Math. Soc. 28 (2015), no. 1, 1-61.
    59. (with Marcy Barge and Sonja Štimac) The Ingram conjecture Updated preprint version July 2011 [.pdf] Geom. Topol. 16 (2012), 2481-2516.
    60. (with Víctor Jiménez López) On the Lebesgue measure of Li-Yorke pairs for interval maps Preprint 2009, [.pdf] Commun. Math. Phys. 299 (2010), 523--560. The final publication is available at http://www.springerlink.com/content/fu01lpu5u4305572/. (DOI: 10.1007/s00220-010-1085-9)

      61. 2010

    61. (with Marcy Barge, Leslie Jones and Lorenzo Sadun) Homological Pisot substitutions and exact regularity Preprint 2010, [.pdf] Isr. J. Math. 188 2012, 281-300 (DOI) 10.1007/s11856-011-0123-4
    62. (with Sonja Štimac) On isotopy and unimodal inverse limit spaces Preprint 2011, [.pdf], Discrete and Continuous Dynamical Systems 32 (2012), no. 4, 1245-1253.
    63. (with Renaud Leplaideur) Renormalization, thermodynamic formalism and quasi-crystals in subshifts. Preprint 2010 (updated 2012) [.pdf] Commun. Math. Phys. 321 (2013), 209-247. DOI: 10.1007/s00220-012-1651-4

      64. 2011

    64. (with Jozef Bobok) The topological entropy of Banach spaces Preprint 2011 [.pdf] J. Difference Equ. Appl. 18 (2012), no. 4, 569-578.
    65. (with Mike Todd) Transience and thermodynamic formalism for infinitely branched interval maps Preprint 2011 [.pdf]. Journal of the London Mathematical Society 86 2012, 171-194 DOI: 10.1112/jlms/jdr081
    66. (with Greg Clack) Inducing and unique ergodicity of double rotations Preprint 2011 [.pdf], Discrete and Continuous Dynamical Systems 32, no. 12 2012, 4133-4147. Click here for the electronic version of the journal.

      67. 2012

    67. (with Mike Todd) Wild attractors and thermodynamic formalism, Preprint 2012 [.pdf] Monatshefte für Mathematik, 178 (2015), 39-83. DOI: 10.1007/s00605-015-0747-2
    68. (with Dierk Schleicher) Bernoulli measure of complex admissible kneading sequences, Preprint 2012 [.pdf] Ergod. Th. Dyn. Sys. 33 2013, 821-830. DOI: 10.1017/edts.2012.134
    69. (with Dierk Schleicher) Hausdorff dimension of biaccessible angles for quadratic polynomials, Preprint 2012 and revised 2016 [.pdf] Fund. Math. 238 (2017), 201-240 DOI: 10.4064/fm276-6-2016
    70. On double spirals in Fibonacci-like unimodal inverse limit spaces, [.pdf] Topology Proc. 43 (2014), 83-97. and Erratum Topology Proc. 49 (2017), 251-252.
    71. Lebesgue ergodicity of a dissipative subtractive algorithm, [.pdf] Springer Proceedings in Mathematics 70 (2014), 105-117.
    72. (with Sonja Štimac) Entropy of homeomorphisms on unimodal inverse limit spaces, [.pdf] Nonlinearity 26 (2013) 991-1000, and Corrigendum, Nonlinearity 31 (2018), no. 8, C1-C3. See also corrigendum and corrected preprint version.

      73. 2013

    73. (with Sebastian van Strien) On the structure of isentropes of polynomial maps, [.pdf] Dynamical Systems: an international journal. 28 (2013) 381-392.
    74. (with Robbert Fokkink and Cor Kraaikamp) The convergence of the generalised Selmer algorithm, Preprint 2013 revised 2014 [.pdf] Israel J. Math. 209 (2015), no. 2, 803-823.
      Erratum: Israel Journal of Mathematics, (), 1 DOI 10.1007/s11856-019-1861-y
    75. (with Sonja Štimac) Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture, Preprint 2013 and revised 2014 [.pdf]. Topological Methods in Nonlinear Analysis (TMNA) 47 (2016), 147--185. doi: 10.12775/TMNA.2016.001
    76. (with Charlene Kalle) Natural extensions for piecewise affine maps via Hofbauer towers, Preprint 2013 [.pdf] Monatsh. Math. 175 (2014), no. 1, 65-88. Springer publication: (DOI) 10.1007/s00605-014-0644-0
    77. (with Robbert Fokkink, Guoyong Gu and Cees Roos) On the chaotic behavior of the primal-dual affine scaling algorithm for linear optimization, Preprint 2013 [.pdf]. Chaos 24 , 043132 (2014); DOI: 10.1063/1.4902900
    78. (with Renaud Leplaideur) Renormalization, freezing phase transitions and Fibonacci quasicrystals, Preprint 2013 [.pdf]. Ann. Sc. Ecole Normale Supérieur 48 (2015), no. 3, 739-763.

      2014

    79. (with Piotr Oprocha) On "observable" Li-Yorke tuples for interval maps, Preprint 2014. [.pdf] Nonlinearity 28 (2015) 1675-1694. (click for the published article)
    80. The work of Y. Sinai; on the occasion of him receiving the Abel Prize 2014, [.pdf] Internationale Mathematische Nachrichten (IMN) 226 (2014), 21-26.
    81. (with Dalia Terhesiu) A renewal scheme for non-uniformly hyperbolic semiflows Preprint 2014 and updated January 2015 (in this last update, the paper was split into this one, and the one below. [.pdf]

      82. 2015

    82. (with Dalia Terhesiu) Verifying abstract hypotheses used in obtaining mixing rates for some non-uniformly hyperbolic semiflows Preprint 2015 [.pdf].
    83. (with Dalia Terhesiu) Upper and lower bounds for the correlation function via inducing with general return times Preprint 2015 [.pdf]. Ergod. Th. and Dyn. Sys. 38 (2018) 34-62. DOI: http://dx.doi.org/10.1017/etds.2016.20
    84. (with Ana Anušić and Jernej Činč) The Core Ingram Conjecture for non-recurrent critical points Preprint 2015, revision 2016 [.pdf] Fund. Math. 241 (2018), no. 3, 209-235.

      85. 2016

    85. (with Jozef Bobok) Constant slope maps and the Vere-Jones classification. Preprint 2016 [.pdf]. Entropy 18(6) , 234, 2016. doi:10.3390/e18060234
    86. (with Ana Anušić and Jernej Činč) Uncountably many planar embeddings of unimodal inverse limit spaces Preprint 2016 [.pdf]. Discrete Contin. Dyn. Syst. 37 (2017), no. 5, 2285-2300.
    87. (with Dalia Terhesiu) The Dolgopyat inequality in BV for non-Markov maps [.pdf] Stochastics and Dynamics, 18 (2018), no. 2, 1850006, 44 pp. doi.org/10.1142/S0219493718500065
    88. (with Ian Melbourne and Dalia Terhesiu) Rates of mixing for nonMarkov infinite measure semiflows Preprint 2016 [.pdf]. Trans. AMS. 371 (2019), no. 10, 7343-7386.
    89. (with Mark Demers and Mike Todd) Hitting and escaping statistics: mixing, targets and holes, Preprint 2016 (updated December 2017) [.pdf] Adv. Math. 328 (2018), 1263-1298.
    90. (with Carlo Carminati and Charlene Kalle) Matching for generalised β-transformations, Preprint 2016 [.pdf], arXiv version. Indagationes Mathematicae, 28 (2017), no. 1, 55-73. DOI: 10.1016/j.indag.2016.11.005.

      91. 2017

    91. (with Sonja Štimac) On isotopy of self-homeomorphisms of quadratic inverse limit spaces Preprint July 217 [.pdf]. arXiv version, Topology Proceedings 52 (2018), 167-177
    92. (with Carlo Carminati, Stefano Marmi and Alessandro Profeti) Matching in a family of piecewise affine interval maps, Preprint July 2016 [.pdf], arXiv version. Nonlinearity 32 (2018), 172-208.
    93. (with Dalia Terhesiu) Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms Preprint 2017 (updated Jan 2018) [.pdf] arXiv version. Ergod. Th. and Dyn. Sys. 40 2020, 663-698, doi.org/10.1017/etds.2018.58).
    94. (with Dalia Terhesiu and Mike Todd) The pressure function for infinite equilibrium measures Preprint 2017 [.pdf] arXiv version. Israel J. Math. 232 (2019), 775-826. Doi: s11856-019-1887-1

      95. 2018

    95. (with Ana Anušić and Jernej Činč) Problems on planar embeddings of chainable continua and accessibility. In: Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali, Topology Proceedings 52 2018, 283-285 in [pdf] and the Auburn source.
    96. (with Ana Anušić and Jernej Činč) Planar embeddings of chainable continua. Preprint 2018 [.pdf], Topology Proceedings 56 (2020), 263-296.
    97. (with Ian Melbourne and Dalia Terhesiu) Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards. Preprint 2018 [.pdf] and arXiv version (updated November 15 2019). Ann. Henri Lebesgue 4 (2021), 407-451
    98. (with Dalia Terhesiu and Mike Todd) Pressure function and limit theorems for almost Anosov flows. Preprint 2018 [.pdf] and arXiv version. Commun. Math. Phys. 382(1) (2021) 1-47 DOI: 10.1007/s00220-021-03962-x
    99. An overview of unimodal inverse limit spaces inverse limit spaces. [.pdf] Conference proceeding and minicourse notes for the conference ``Interval dynamics and Inverse limit spaces'', at IWCTA: International Workshop and Conference on Topology & Applications, Rajagiri School of Engineering & Technology, Kochi, December 5-8 2018.

      100. 2019

    100. (with Lori Alvin, Ana Anušić and Jernej Činč) Folding points of unimodal inverse limit spaces. Preprint 2019 [.pdf], Nonlinearity 33 (2020), 224-248. Doi: 10.1088/1361-6544/ab4e31
    101. On volume preserving almost Anosov flows. Preprint 2019 updated 2022 [.pdf] and arXiv version. Monatsh. Math. 201 2023 1003-1026 DOI:10.1007/s00605-022-01807-w
    102. (with Ana Anušić and Jernej Činč) Topological properties of Lorenz maps derived from unimodal maps Preprint 2019 [.pdf], Journal of Difference Equations and Dynamics, 26 Issue 8, 1174-1191.

      103. 2020

    103. On Sinai billiards on flat surfaces with horns. Preprint 2020 (revised Jan. 2021) [.pdf] and arXiv version Journ. Stat. Phys. 183 (18) (2021). DOI:10.1007/s10955-021-02746-w
    104. (with Benjamin Vejnar) Classification of one dimensional dynamical systems by countable structures. Preprint 2020 [.pdf], and arXiv version. Journ. Symb. Logic. 88 (2023), no. 2, 562-578. DOI:10.1017/jsl.2022.67

      105. 2021

    105. (with Olga Lukina) Rotated odometers. Preprint 2021, updated in August 2022: (updated, and arXiv version, Journ. London. Math. Soc. 107 (2023), no. 6, 1983-2024. (DOI:10.1112/jlms.12731)
    106. (with Olga Lukina) Rotated odometers and actions on rooted trees. Preprint 2021 (updated 2022) [.pdf], and arXiv version. Fund. Math. 260 (2023) 233-249. (DOI: 10.4064/fm74-10-2022)
    107. (with Gabrielle Keszthelyi) On co-σ-porosity of the parameters with dense critical orbits for skew tent maps and matching on generalized β-transformations. Preprint 2021 [.pdf], and arXiv version. Indagationes Mathematicae 33 (2022), no. 3, 533-545.
    108. (with Péter Bálint and Dalia Terhesiu) Lorentz gas with small scatterers, Preprint 2021 (updated November 2022) [.pdf], and arXiv version, Prob. Th. and Rel. Fields 186 (2023), 159-219, DOI:10.1007/s00440-023-01197-6

      109. 2022

    109. Topological and Ergodic Theory of Symbolic Dynamics, AMS book series Graduate Studies in Matematics. The electronic version can be purchased on the AMS Website

      2023

    110. (with Homero Canales) Mixing rates of the geometrical neutral Lorenz model, Preprint 2023 [.pdf], and arXiv version Appeared electronically in Journ. Stat. Phys., Dec 2023 (DOI 10.1007/s10955-023-03212-5)
    111. (with Jan Boroński and Przemek Kucharski) Translation algorithms for graph covers, Preprint 2023 [.pdf], and arXiv version,
    112. (with Silvia Radinger) Interval translation maps with weakly mixing attractors, Preprint 2023 [.pdf], and arXiv version,

      113. 2024

    113. (with Charles Fougeron, Davide Ravotti and Dalia Terhesiu) On asymptotic expansions of ergodic integrals for Zd-extensions of translation flows, Preprint 2024 [.pdf], and the arXiv version.
    114. (with Olga Lukina) Skew-product systems over infinite interval exchange transformations. Preprint 2024, pdf, and arXiv version.
    115. (with Niels Kolenbrander en Dalia Terhesiu) On some random billiards in a tube with superdiffusion. Preprint 2024, pdf, and arXiv version.
    116. (with Andrzej Biś) On translocal entropy functions. Preprint 2024, pdf, and arXiv version.
See also MathSciNet and the arXiv
henk.bruin@univie.ac.at
Updated November 2024