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arxiv: 2606.17857 · v2 · pith:E2HFBKPZnew · submitted 2026-06-16 · 🧮 math.DS

From Ergodic Theory and Probability to Fractal Geometry and Dynamics: Themes in the Work of Manfred Denker

Suzanne Boyd , Herold Dehling , Martin Schmoll , Manuel Stadlbauer , Christian Wolf This is my paper

Pith reviewed 2026-06-26 22:40 UTC · model grok-4.3

classification 🧮 math.DS
keywords ergodic theorydynamical systemsfractal geometrylimit theoremsthermodynamic formalismconformal measuresstatistical propertiesdependent processes
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The pith

Manfred Denker developed limit theorems for dependent processes in dynamical systems and applied thermodynamic formalism to connect their geometric and measure-theoretic properties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This survey examines Manfred Denker's contributions across ergodic theory, probability, dynamical systems, fractal geometry, and statistics. It highlights the emergence of probabilistic behavior such as central limit theorems and invariance principles in deterministic systems under weak dependence or infinite measure. Denker used thermodynamic formalism to relate geometric and measure-theoretic properties through equilibrium states and conformal measures. The work also covers asymptotic theory for statistical procedures on dependent data like rank statistics and U-statistics, along with links between rigorous analysis and computational methods. These elements present an integrated approach where ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

Core claim

Denker's work establishes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties, with particular emphasis on probabilistic behavior in deterministic systems including central limit theorems, invariance principles or local limit theorems under weak dependence assumptions or in infinite measure, equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data.

What carries the argument

Thermodynamic formalism, which uses equilibrium states and transfer operators to connect geometric properties of fractals with the statistical behavior of dynamical systems.

If this is right

  • Central limit theorems and invariance principles apply to dynamical systems under weak dependence assumptions or infinite measure.
  • Conformal measures determine geometric features such as fractal dimensions through their connection to equilibrium states.
  • Asymptotic results for rank statistics and U-statistics extend to data generated by dependent dynamical processes.
  • Transfer operator methods provide a bridge between measure-theoretic and geometric analysis in these systems.
  • Computational methods integrate with the theoretical framework to analyze statistical properties of dynamical systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The methods for handling infinite measure cases could extend to other infinite ergodic systems in related areas like number theory.
  • The asymptotic theory for dependent data might support new statistical tests for time series exhibiting chaotic behavior.
  • Connections between conformal measures and fractal geometry could inform dimension calculations in applied settings such as image analysis.
  • The unifying perspective might suggest hybrid models that combine thermodynamic formalism with machine learning for system identification.

Load-bearing premise

The survey's selection of themes accurately captures the core and most influential aspects of Denker's contributions without major omissions or distortions in emphasis.

What would settle it

Identification of a substantial body of Denker's work on limit theorems, thermodynamic formalism, or related statistical properties that the survey omits or misrepresents would undermine its account of his contributions.

read the original abstract

This article surveys the mathematical contributions of Manfred Denker, with a focus on themes that connect ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. Denker's highly influential work includes a systematic study of the statistical properties of dynamical systems, the development of limit theorems for dependent processes, and the use of thermodynamic formalism to relate geometric and measure-theoretic properties. Particular emphasis is placed on the emergence of probabilistic behavior in deterministic systems, including central limit theorems, invariance principles or local limit theorems, under weak dependence assumptions or in infinite measure. Further topics include equilibrium states and transfer operator methods, the role of conformal measures in fractal geometry, and the asymptotic theory of statistical procedures for dependent data, such as rank statistics and U-statistics. In addition to these theoretical developments, the survey highlights contributions connecting rigorous analysis with computational and statistical methods. Taken together, these works illustrate a unifying perspective in which ergodic, probabilistic, geometric, and statistical methods interact in the study of dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. This manuscript is a survey article reviewing the mathematical contributions of Manfred Denker, with emphasis on connections between ergodic theory, probability theory, dynamical systems, fractal geometry, and statistics. It describes Denker's work on statistical properties of dynamical systems, limit theorems for dependent processes (including central limit theorems, invariance principles, and local limit theorems under weak dependence or infinite measure), thermodynamic formalism, equilibrium states, transfer operator methods, conformal measures in fractal geometry, and asymptotic theory for statistical procedures with dependent data such as rank statistics and U-statistics. The survey also notes links to computational and statistical methods.

Significance. If the survey's selection of themes accurately reflects Denker's core contributions, it offers a coherent overview of how ergodic, probabilistic, geometric, and statistical methods interact in dynamical systems. Such thematic surveys can aid in synthesizing prior work and identifying unifying perspectives, particularly given the manuscript's descriptive focus on established results rather than new derivations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for recommending acceptance of the manuscript. The report correctly identifies the survey's focus on the interconnections between ergodic theory, probability, dynamical systems, fractal geometry, and statistics in Denker's work.

Circularity Check

0 steps flagged

Descriptive survey with no derivations or quantitative claims

full rationale

This paper is a survey summarizing Manfred Denker's prior contributions across ergodic theory, probability, dynamical systems, and fractal geometry. It contains no original theorems, equations, derivations, fitted parameters, predictions, or load-bearing assumptions that could reduce to inputs by construction. The text is purely descriptive and historical, with the central claim being an overview of existing work rather than a deductive chain. No self-citations function as load-bearing justifications for new results, and no patterns of self-definition, fitted inputs called predictions, or ansatz smuggling apply. The derivation chain is empty by the nature of the document.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey paper containing no new derivations, so the ledger is empty.

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Reference graph

Works this paper leans on

93 extracted references

  1. [1]

    Aaronson, R

    J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill, and B. Weiss,Strong laws forL- and U-statistics, Trans. Amer. Math. Soc.348(1996), no. 7, 2845–2866. MR 1363941

    1996

  2. [2]

    Jon Aaronson and Manfred Denker,Lower bounds for partial sums of certain positive sta- tionary processes, Almost everywhere convergence (Columbus, OH, 1988), Academic Press, Boston, MA, 1989, pp. 1–9. MR 1035234

    1988

  3. [3]

    ,Upper bounds for ergodic sums of infinite measure preserving transformations, Trans. Amer. Math. Soc.319(1990), no. 1, 101–138. MR 1024766

    1990

  4. [4]

    Dyn.1(2001), no

    ,Local limit theorems for partial sums of stationary sequences generated by Gibbs- Markov maps, Stoch. Dyn.1(2001), no. 2, 193–237. MR 1840194

    2001

  5. [5]

    Fisher,Second order ergodic theorems for ergodic transformations of infinite measure spaces, Proc

    Jon Aaronson, Manfred Denker, and Albert M. Fisher,Second order ergodic theorems for ergodic transformations of infinite measure spaces, Proc. Amer. Math. Soc.114(1992), no. 1, 115–127. MR 1099339

    1992

  6. [6]

    Jon Aaronson, Manfred Denker, and Mariusz Urba´ nski,Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc.337(1993), no. 2, 495–548. MR 1107025

    1993

  7. [7]

    Artur Avila and Mikhail Lyubich,Hausdorff dimension and conformal measures of Feigen- baum Julia sets, J. Amer. Math. Soc.21(2008), no. 2, 305–363. MR 2373353

    2008

  8. [8]

    Berkes and H

    I. Berkes and H. Dehling,Some limit theorems in log density, Ann. Probab.21(1993), no. 3, 1640–1670. MR 1235433

    1993

  9. [9]

    Svetlana Borovkova, Robert Burton, and Herold Dehling,Limit theorems for functionals of mixing processes with applications toU-statistics and dimension estimation, Trans. Amer. Math. Soc.353(2001), no. 11, 4261–4318. MR 1851171

    2001

  10. [10]

    Hautes ´Etudes Sci

    Rufus Bowen,Hausdorff dimension of quasicircles, Inst. Hautes ´Etudes Sci. Publ. Math. (1979), no. 50, 11–25. MR 556580

    1979

  11. [11]

    Brosamler,An almost everywhere central limit theorem, Math

    Gunnar A. Brosamler,An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc.104(1988), no. 3, 561–574. MR 957261

    1988

  12. [12]

    Edgar Brunner and Manfred Denker,Rank statistics under dependent observations and appli- cations to factorial designs, J. Statist. Plann. Inference42(1994), no. 3, 353–378. MR 1309630

    1994

  13. [13]

    Robert Burton and Manfred Denker,On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc.302(1987), no. 2, 715–726. MR 891642

    1987

  14. [14]

    Richard Savage,Asymptotic normality and efficiency of certain non- parametric test statistics, Ann

    Herman Chernoff and I. Richard Savage,Asymptotic normality and efficiency of certain non- parametric test statistics, Ann. Math. Statist.29(1958), 972–994. MR 100322

    1958

  15. [15]

    Welington Cordeiro, Manfred Denker, and Xuan Zhang,On specification and measure expan- siveness, Discrete Contin. Dyn. Syst.37(2017), no. 4, 1941–1957. MR 3640582

    2017

  16. [16]

    Anirban Das, Manfred Denker, Anna Levina, and Lucia Tabacu,Estimation by stable motions and its applications, Probab. Math. Statist.43(2023), no. 1, 23–39. MR 4660830

    2023

  17. [17]

    Multivariate Anal.28(1989), no

    Herold Dehling,The functional law of the iterated logarithm for von Mises functionals and multiple Wiener integrals, J. Multivariate Anal.28(1989), no. 2, 177–189. MR 991944

    1989

  18. [18]

    Herold Dehling, Manfred Denker, and Walter Philipp,Invariance principles for von Mises andU-statistics, Z. Wahrsch. Verw. Gebiete67(1984), no. 2, 139–167. MR 758070

    1984

  19. [19]

    Theory Relat

    ,A bounded law of the iterated logarithm for Hilbert space valued martingales and its application toU-statistics, Probab. Theory Relat. Fields72(1986), no. 1, 111–131. MR 835162

    1986

  20. [20]

    Probab.14(1986), no

    ,Central limit theorems for mixing sequences of random variables under minimal conditions, Ann. Probab.14(1986), no. 4, 1359–1370. MR 866356 34 S. BOYD, H. DEHLING, M. SCHMOLL, M. STADLBAUER, AND C. WOLF

    1986

  21. [21]

    ,The almost sure invariance principle for the empirical process ofU-statistic structure, Ann. Inst. H. Poincar´ e Probab. Statist.23(1987), no. 2, 121–134. MR 891707

    1987

  22. [22]

    Woyczy´ nski,ResamplingU-statistics using p-stable laws, J

    Herold Dehling, Manfred Denker, and Wojbor A. Woyczy´ nski,ResamplingU-statistics using p-stable laws, J. Multivariate Anal.34(1990), no. 1, 1–13. MR 1062544

    1990

  23. [23]

    Denker and M

    M. Denker and M. Urba´ nski,Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. (2)43(1991), no. 1, 107–118. MR 1099090

    1991

  24. [24]

    2, 365–384

    ,On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlin- earity4(1991), no. 2, 365–384. MR 1107011

    1991

  25. [25]

    MR 310190

    Manfred Denker,Topological entropy and the construction of a unique, normalised ergodic borel measure with maximal entropy, Conference Proceedings Paimpol 1970275(1970), A735– A738. MR 310190

    1970

  26. [26]

    Z.134(1973), 231–253

    ,On strict ergodicity, Math. Z.134(1973), 231–253. MR 352402

    1973

  27. [27]

    Wahrscheinlichkeitstheorie und Verw

    ,Einige Bemerkungen zu Erzeugers¨ atzen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete29(1974), 57–64. MR 382588

    1974

  28. [28]

    Wahrschein- lichkeitstheorie und Verw

    ,Finite generators for ergodic, measure-preserving transformations, Z. Wahrschein- lichkeitstheorie und Verw. Gebiete29(1974), 45–55. MR 382587

    1974

  29. [29]

    Res., vol

    ,Statistical decision procedures and ergodic theory, Ergodic theory and related topics (Vitte, 1981), Math. Res., vol. 12, Akademie-Verlag, Berlin, 1982, pp. 35–47. MR 730767

    1981

  30. [30]

    Vieweg & Sohn, Braunschweig, 1985

    ,Asymptotic distribution theory in nonparametric statistics, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1985. MR 889896

    1985

  31. [31]

    ,Uniform integrability and the central limit theorem for strongly mixing processes, Dependence in probability and statistics (Oberwolfach, 1985), Progr. Probab. Statist., vol. 11, Birkh¨ auser Boston, Boston, MA, 1986, pp. 269–289. MR 899993

    1985

  32. [32]

    23, PWN, Warsaw, 1989, pp

    ,The central limit theorem for dynamical systems, Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 33–62. MR 1102700

    1986

  33. [33]

    Syst.24(2009), no

    Manfred Denker, Jinqiao Duan, and Michael McCourt,Pseudorandom numbers for conformal measures, Dyn. Syst.24(2009), no. 4, 439–457. MR 2572997

    2009

  34. [34]

    MR 352403

    Manfred Denker and Ernst Eberlein,Ergodic flows are strictly ergodic, Advances in Math.13 (1974), 437–473. MR 352403

    1974

  35. [35]

    Math.148 (1999), no

    Manfred Denker and Mikhail Gordin,Gibbs measures for fibred systems, Adv. Math.148 (1999), no. 2, 161–192. MR 1736956

    1999

  36. [36]

    Theory Related Fields160(2014), no

    ,Limit theorems for von Mises statistics of a measure preserving transformation, Probab. Theory Related Fields160(2014), no. 1-2, 1–45. MR 3256808

    2014

  37. [37]

    Math.48(2004), no

    Manfred Denker, Mikhail Gordin, and Anastasya Sharova,A Poisson limit theorem for toral automorphisms, Illinois J. Math.48(2004), no. 1, 1–20. MR 2048211

    2004

  38. [38]

    Manfred Denker, Christian Grillenberger, and Karl Sigmund,Ergodic theory on compact spaces, Lecture Notes in Mathematics, vol. Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 457675

    1976

  39. [39]

    Heinemann,Polynomial skew products, Ergodic theory, anal- ysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp

    Manfred Denker and Stefan-M. Heinemann,Polynomial skew products, Ergodic theory, anal- ysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 175–189, 808–

    2001

  40. [40]

    Manfred Denker and Michael Keane,Almost topological dynamical systems, Israel J. Math. 34(1979), no. 1-2, 139–160. MR 571401

    1979

  41. [41]

    Mises’ statistics for weakly de- pendent processes, Z

    Manfred Denker and Gerhard Keller,OnU-statistics and v. Mises’ statistics for weakly de- pendent processes, Z. Wahrsch. Verw. Gebiete64(1983), no. 4, 505–522. MR 717756

    1983

  42. [42]

    ,Rigorous statistical procedures for data from dynamical systems, J. Statist. Phys.44 (1986), no. 1-2, 67–93. MR 854400

    1986

  43. [43]

    Lopes, and S´ ılvia R

    Manfred Denker, Marc Kesseb¨ ohmer, Artur O. Lopes, and S´ ılvia R. C. Lopes,Parametrized families of Gibbs measures and their statistical inference, Stoch. Dyn.25(2025), no. 2, Paper No. 2550012, 45. MR 4901113

    2025

  44. [44]

    Systems28(2008), no

    Manfred Denker, Yuri Kifer, and Manuel Stadlbauer,Conservativity of random Markov fibred systems, Ergodic Theory Dynam. Systems28(2008), no. 1, 67–85. MR 2380303

    2008

  45. [45]

    ,Thermodynamic formalism for random countable Markov shifts, Discrete Contin. Dyn. Syst.22(2008), no. 1-2, 131–164. MR 2410952

    2008

  46. [46]

    4, 435–448

    Manfred Denker and Susanne Koch,A Poisson formula for harmonic functions on the Sierpi´ nski gasket, Forum Math.12(2000), no. 4, 435–448. MR 1763899 THEMES IN THE WORK OF MANFRED DENKER 35

    2000

  47. [47]

    56, 2002, Special issue: Frontier research in theoretical statistics, 2000 (Eindhoven), pp

    ,Almost sure local limit theorems, vol. 56, 2002, Special issue: Frontier research in theoretical statistics, 2000 (Eindhoven), pp. 143–151. MR 1916315

    2002

  48. [48]

    Systems4(1984), no

    Manfred Denker and Walter Philipp,Approximation by Brownian motion for Gibbs mea- sures and flows under a function, Ergodic Theory Dynam. Systems4(1984), no. 4, 541–552. MR 779712

    1984

  49. [49]

    Systems16(1996), no

    Manfred Denker, Feliks Przytycki, and Mariusz Urba´ nski,On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems16(1996), no. 2, 255–266. MR 1389624

    1996

  50. [50]

    Syst.29 (2014), no

    Manfred Denker and Ana Rodrigues,Ergodicity of avalanche transformations, Dyn. Syst.29 (2014), no. 4, 517–536. MR 3265617

    2014

  51. [51]

    Decisions3(1985), no

    Manfred Denker and Uwe R¨ osler,Some contributions to Chernoff-Savage theorems, Statist. Decisions3(1985), no. 1-2, 49–75. MR 792458

    1985

  52. [52]

    Manfred Denker and Hiroshi Sato,Sierpi´ nski gasket as a Martin boundary. II. The intrinsic metric, Publ. Res. Inst. Math. Sci.35(1999), no. 5, 769–794. MR 1739300

    1999

  53. [53]

    ,Sierpi´ nski gasket as a Martin boundary. I. Martin kernels, Potential Anal.14(2001), no. 3, 211–232. MR 1822915

    2001

  54. [54]

    Nachr.241(2002), 32–55

    ,Reflections on harmonic analysis of the Sierpi´ nski gasket, Math. Nachr.241(2002), 32–55. MR 1912376

    2002

  55. [55]

    12, 4587–4613

    Manfred Denker, Samuel Senti, and Xuan Zhang,Lindeberg theorem for Gibbs-Markov dy- namics, Nonlinearity30(2017), no. 12, 4587–4613. MR 3734148

    2017

  56. [56]

    ,Fluctuations of ergodic sums on periodic orbits under specification, Discrete Contin. Dyn. Syst.40(2020), no. 8, 4665–4687. MR 4112026

    2020

  57. [57]

    Stratmann,The Patterson measure: classics, variations and applications, Contributions in analytic and algebraic number theory, Springer Proc

    Manfred Denker and Bernd O. Stratmann,The Patterson measure: classics, variations and applications, Contributions in analytic and algebraic number theory, Springer Proc. Math., vol. 9, Springer, New York, 2012, pp. 171–195. MR 3060460

    2012

  58. [58]

    Manfred Denker and Lucia Tabacu,Logarithmic quantile estimation for rank statistics, J. Stat. Theory Pract.9(2015), no. 1, 146–170. MR 3275537

    2015

  59. [59]

    6, 561–579

    Manfred Denker and Mariusz Urba´ nski,Absolutely continuous invariant measures for expan- sive rational maps with rationally indifferent periodic points, Forum Math.3(1991), no. 6, 561–579. MR 1129999

    1991

  60. [60]

    ,On the existence of conformal measures, Trans. Amer. Math. Soc.328(1991), no. 2, 563–587. MR 1014246

    1991

  61. [61]

    Manfred Denker and Wojbor A. Woyczy´ nski,Introductory statistics and random phenomena, Statistics for Industry and Technology, Birkh¨ auser Boston, Inc., Boston, MA, 1998, Uncer- tainty, complexity and chaotic behavior in engineering and science, Appendix E by Bernard Ycart. MR 1643201

    1998

  62. [62]

    Math., vol

    Manfred Denker and Michiko Yuri,Conformal families of measures for general iterated func- tion systems, Recent trends in ergodic theory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 93–108. MR 3330340

    2015

  63. [63]

    Albert Fisher,Convex-invariant means and a pathwise central limit theorem, Adv. in Math. 63(1987), no. 3, 213–246. MR 877784

    1987

  64. [64]

    3, 191–198

    Lukas Geyer,Porosity of parabolic Julia sets, Complex Variables Theory Appl.39(1999), no. 3, 191–198. MR 1717570

    1999

  65. [65]

    Halmos,The theory of unbiased estimation, Ann

    Paul R. Halmos,The theory of unbiased estimation, Ann. Math. Statistics17(1946), 34–43. MR 15746

    1946

  66. [66]

    Systems 13(1993), no

    Masaki Hirata,Poisson law for Axiom A diffeomorphisms, Ergodic Theory Dynam. Systems 13(1993), no. 3, 533–556. MR 1245828

    1993

  67. [67]

    Wassily Hoeffding,A class of statistics with asymptotically normal distribution, Ann. Math. Statistics19(1948), 293–325. MR 26294

    1948

  68. [68]

    Systems2(1982), no

    Franz Hofbauer and Gerhard Keller,Equilibrium states for piecewise monotonic transforma- tions, Ergodic Theory Dynam. Systems2(1982), no. 1, 23–43. MR 684242

    1982

  69. [69]

    Z.180(1982), no

    ,Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z.180(1982), no. 1, 119–140. MR 656227

    1982

  70. [70]

    I. A. Ibragimov,A remark on the central limit theorem for dependent random variables, Teor. Verojatnost. i Primenen.20(1975), 134–140. MR 362448

    1975

  71. [71]

    I. A. Ibragimov and Yu. ˜V. Linnik,Independent and stationary sequences of random vari- ables, Wolters-Noordhoff Publishing, Groningen, 1971, With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. MR 322926 36 S. BOYD, H. DEHLING, M. SCHMOLL, M. STADLBAUER, AND C. WOLF

    1971

  72. [72]

    Michael Keane and Meir Smorodinsky,Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2)109(1979), no. 2, 397–406. MR 528969

    1979

  73. [73]

    Lacey,On weak convergence in dynamical systems to self-similar processes with spectral representation, Trans

    Michael T. Lacey,On weak convergence in dynamical systems to self-similar processes with spectral representation, Trans. Amer. Math. Soc.328(1991), no. 2, 767–778. MR 1066446

    1991

  74. [74]

    E. L. Lehmann,Nonparametrics: statistical methods based on ranks, Holden-Day Series in Probability and Statistics, Holden-Day, Inc., San Francisco, CA; McGraw-Hill International Book Co., New York-D¨ usseldorf, 1975, With the special assistance of H. J. M. d’Abrera. MR 395032

    1975

  75. [75]

    Emmanuel Lesigne,Almost sure central limit theorem for strictly stationary processes, Proc. Amer. Math. Soc.128(2000), no. 6, 1751–1759. MR 1641132

    2000

  76. [76]

    thesis, Georg-August Universit¨ at G¨ ottingen, G¨ ottingen, July 2008, Optional note

    Ana Levina,A mathematical approach to self-organized criticality in neural networks, Ph.D. thesis, Georg-August Universit¨ at G¨ ottingen, G¨ ottingen, July 2008, Optional note

    2008

  77. [77]

    Math.98(1997), 29–60

    Pierre Liardet and Dalibor Voln´ y,Sums of continuous and differentiable functions in dynam- ical systems, Israel J. Math.98(1997), 29–60. MR 1459847

    1997

  78. [78]

    Sarig,Symbolic dynamics for three-dimensional flows with positive topological entropy, J

    Yuri Lima and Omri M. Sarig,Symbolic dynamics for three-dimensional flows with positive topological entropy, J. Eur. Math. Soc. (JEMS)21(2019), no. 1, 199–256. MR 3880208

    2019

  79. [79]

    McMullen,Thermodynamics, dimension and the Weil-Petersson metric, Invent

    Curtis T. McMullen,Thermodynamics, dimension and the Weil-Petersson metric, Invent. Math.173(2008), no. 2, 365–425 (English)

    2008

  80. [80]

    Ann.393 (2025), no

    Matthias Meiwes,Topological entropy and orbit growth in link complements, Math. Ann.393 (2025), no. 1, 617–665. MR 4966568

    2025

Showing first 80 references.