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Recurrent aperiodic Lorentz gases with uniform geometry are K-mixing, even though they preserve infinite measure.

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2026-07-31 13:08 UTC pith:QSQGLHAO

load-bearing objection Solid K-mixing for recurrent uniform ALGs; abstract theorem has a real but fixable gap on finite measure of enlarged rectangles. the 3 major comments →

arxiv 2607.24498 v1 pith:QSQGLHAO submitted 2026-07-27 math.DS

K-mixing for aperiodic Lorentz gases

Giovanni Canestrari , Marco Lenci This is my paper
classification math.DS MSC 37A4037A2537E0537D2537C25
keywords K-mixingaperiodic Lorentz gasinfinite ergodic theoryhyperbolic billiardsCantor rectanglesLorentz tubesdecay of correlations
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Classical proofs that hyperbolic billiards are strongly mixing rely on entropy and finite invariant measure; both tools fail for infinite-measure systems. This paper supplies a general criterion that works without those tools: any recurrent, piecewise-smooth hyperbolic map on a two-dimensional manifold that admits a covering by finite-measure Cantor rectangles decomposes into countably many invariant pieces on which a power of the map is K-mixing. The criterion is then verified for aperiodic Lorentz gases (and Lorentz tubes) whose free paths and curvatures are uniformly bounded, yielding the first K-mixing statement for truly aperiodic infinite-horizon configurations. As a direct corollary, correlations of zero-mean integrable observables against a large class of continuous test functions decay to zero. The result places aperiodic Lorentz gases on the same stochastic footing that periodic Sinai billiards have long enjoyed.

Core claim

Every recurrent aperiodic Lorentz gas (or Lorentz tube) whose free-path lengths and boundary curvatures are bounded above and below has a billiard map that is K-mixing. More abstractly, any two-dimensional map satisfying the seven structural hypotheses (H1)–(H7)—smoothness with singularities, stable/unstable manifolds, absolute continuity, distortion control, recurrence, and a countable covering by finite-measure Cantor rectangles—admits a K-decomposition: the space splits into invariant components on each of which a power of the map is K-mixing.

What carries the argument

The K-decomposition theorem (Theorem 2.1). Its engine is the tail σ-algebra generated by the stable foliation; the covering by Cantor rectangles forces every positive-measure set in the tail to fill a whole rectangle under iteration, rendering the tail atomic and thereby producing the K-property on each ergodic component.

Load-bearing premise

The phase space must be covered, up to a null set, by countably many finite-measure Cantor rectangles built from stable and unstable manifolds; without that covering the tail-atomicity argument fails.

What would settle it

Exhibit a single recurrent aperiodic Lorentz gas with uniformly bounded free paths and curvatures whose billiard map fails to be K-mixing, or show that the Cantor-rectangle covering fails for some configuration satisfying the geometric bounds.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Zero-mean L1 observables lose memory against any uniformly continuous test function, uniformly over whole Hölder or Lipschitz classes.
  • The same abstract criterion applies verbatim to other infinite-measure hyperbolic systems once a Cantor-rectangle covering is verified.
  • Global-local mixing statements for aperiodic Lorentz gases become accessible via the known implication from K-mixing.
  • Ergodic components of powers of the map are themselves K, so higher-order mixing holds on each component.

Where Pith is reading between the lines

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

  • The method may extend to other aperiodic infinite-horizon billiards once local uniformity of expansion is checked, including certain random Lorentz tubes.
  • Failure of the Growth Lemma under non-uniform free paths would produce the first natural counter-examples to K-mixing among recurrent dispersing billiards.
  • The correlation decay of Theorem 4.4 supplies a practical numerical test: sample two zero-mean densities and watch the difference of expectations of a continuous observable vanish.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proves two results. First, an abstract K-decomposition theorem (Theorem 2.1): any invertible, recurrent, piecewise-smooth hyperbolic map of a two-dimensional manifold satisfying hypotheses (H1)–(H7) (singularities, measurable stable/unstable partitions with uniform distortion and absolute continuity, a mod-0 covering by finite-measure Cantor rectangles, and conservativity) decomposes into at most countably many ergodic components, each of which is a finite cycle of sets on which a power of the map is K-mixing. The proof avoids entropy (unavailable in infinite measure) and instead shows the tail σ-algebra of the stable sub-σ-algebra is atomic, via density points on unstable manifolds, distortion, and a d-auto-intersection criterion in the spirit of Miernowski–Nogueira. Second, the framework is applied to aperiodic Lorentz gases and Lorentz tubes satisfying recurrence (ALG1) and uniform free-path/curvature bounds (ALG2): all such billiard maps are K-mixing (Theorem 4.1), with a uniform infinite-measure correlation-decay corollary for zero-mean L¹ observables (Theorem 4.4).

Significance. If correct, this is the first K-mixing result for genuinely aperiodic infinite-measure hyperbolic systems, a class about which very little is known beyond ergodicity and recurrence. The derivation is parameter-free: (H1)–(H7) and (ALG1)–(ALG2) are qualitative geometric hypotheses with no fitted constants, and the abstract theorem yields falsifiable structural conclusions (K-decomposition) for other concrete systems (Galton board, Fermi–Ulam models, cusped billiards are named). The entropy-free route to the K-property is a genuinely useful methodological contribution — it gives an alternative to Sinai's classical argument even in the finite-measure case — and the measure-theoretic appendix (Lemmas A.1–A.3) makes the paper largely self-contained. Theorem 4.4's uniformity over regularity classes is a nice strengthening of (GLM1)-type decay.

major comments (3)
  1. [§3, Lemma 3.4 (proof, p. 9)] Ergodicity of the return map F_{d,k} on the enlarged rectangle R̄_k is argued via Birkhoff averaging of a dense subset of L¹(R̄_k), justified by 'By (H4), 0 < µ(R̄_k) < ∞'. But (H4) assumes only µ(R_k) < ∞ for the plain rectangles (footnote 4 adds only measurability of R̄_k). (H1)–(H7) do not imply µ(R̄_k) < ∞: e.g., Z²-covers of Anosov diffeomorphisms satisfy (H1)–(H7) (bounded-arc subordinated partitions from a lifted Markov partition give uniform distortion, holonomy, and finite-measure rectangles), yet the saturation R̄_k is all of M mod 0. The auxiliary claim that uniformly continuous functions are dense in L¹(R̄_k) also presupposes finite measure. As written the lemma is unproved at the stated generality; it feeds Lemma 3.5 and hence atomicity of I_d.
  2. [§3, Lemma 3.12 (final estimate, p. 14)] The disintegration integral over {W̃ ∈ ξᵘ : W̃ ∩ R̄_k ≠ ∅} equals µ(R̄_k), because R̄_k is ξᵘ-saturated. The closing line 'By (H4) again, we have µ(R_k) < ∞' bounds the wrong quantity, so the finite constant C_k in the lemma's statement is not justified by the proof. Lemma 3.12 is invoked at (3.9)–(3.8) in Lemma 3.13, on which the d-auto-intersection argument and tail atomicity (Lemma 3.14) rest, so this gap propagates directly to Theorem 2.1(d). Note the obvious variant with R_k in place of R̄_k does not fix this: R_k is not ξᵘ-saturated, and µ_ξᵘ of the unstable leaves meeting R_k is not controlled by µ(R_k).
  3. [Theorem 2.1 (statement level); interaction with §4] Consequence and suggested repair: Theorem 2.1 is currently unproved at its stated level of generality. Two repairs look feasible within the paper's own tools: (i) strengthen (H4) to µ(R̄_k) < ∞ — the headline application then needs one added verification, which is available: in §4 the stable/unstable H-manifolds never leave the single-scatterer component M_i (as noted in the (H2) verification), so R̄(O) ⊆ M_i and µ(R̄(O)) ≤ µ(M_i) = 2|∂O_i| < ∞; please add this explicitly to the (H4) verification. (ii) Alternatively, rework Lemma 3.4 by inducing further onto the finite-measure R_k (inducing preserves ergodicity) and Lemma 3.12 via the local product structure of R_k. Under either repair, Theorem 4.1 should stand; the gap concerns the abstract theorem's hypotheses, not the billiard application.
minor comments (6)
  1. [§4, Lemma 4.2] This lemma (ergodicity of the induced map (F^m)_{M_i} for all m, both geometries) underlies Lemma 4.3 and hence, via Corollary 2.2, Theorem 4.1 itself. It is proved by reference to [Len03, Lemma 4.5] 'with the obvious modifications'. Since the extension from m=1 to general m and from the plane to the tube is load-bearing, please add a few lines substantiating the claim that LSMs/LUMs of (F^m)_{M_i} coincide with those of F, and what changes (if anything) for D_LT.
  2. [§3, Lemma 3.4] Notation: the return map is defined as F_{d,k} but the Birkhoff averages are written with F_{k,d}; please make consistent.
  3. [Various] Typos/wording: 'K-decoposition' (§3, before Lemma 3.3); 'homegeneity strip' and 'homegeneous component' (§4, (H2) and (H4) verifications); 'one scatter from one family' → 'scatterer' (proof of Lemma 4.3); 'the the constant' (after (4.4)); 'Our arguments are inspired by, and makes use of' (p. 3, subject–verb agreement); 'it's a basic fact' → 'it is' (Lemma 3.4); 'a.e. point. etc.' (notation paragraph, p. 4); 'zero-meanL¹' spacing in the abstract; 'the literature ... is rather skewed' (p. 2) — presumably 'sparse' is meant.
  4. [Front matter] MSC code 37E05 (dynamical systems on the interval/circle) appears inappropriate for this paper; 37D20/37A25/37A40 cover the content. Reference [Lin71] is missing volume and page numbers.
  5. [§2, (H4) and footnote 4] If (H4) is strengthened per Major Comment 3, please update footnote 4 and restate explicitly which properties of R̄_k (measurability, finiteness) are assumed; the current wording suggests the authors already anticipated the measurability issue, and the revision should make the finiteness hypothesis equally visible.
  6. [§4, proof of Theorem 4.4] The application of Lemma A.3 with B_p = S_{m*} is correct but terse; one sentence noting that (4.12) is A.3 applied at fixed p = m* (and that F_{m*} ∈ L∞(S_{m*}) with ∥F_{m*}∥∞ ≤ ∥F∥∞) would help the reader.

Circularity Check

0 steps flagged

No circularity: K-decomposition is derived from geometric hypotheses (H1)–(H7); ALG application imports standard external billiard technology and prior independent recurrence/ergodicity results, not the target claim.

full rationale

The paper is a pure existence/derivation theorem in infinite ergodic theory. Theorem 2.1 constructs the sub-σ-algebra S from the stable foliation and proves the three K-axioms (FS ⊇ S, ∨ F^n S = A, tail atomic) from the listed hypotheses (H1)–(H7) via Hopf argument on finite-measure Cantor rectangles, distortion, absolute continuity, and a d-auto-intersection criterion adapted from MN13. None of these steps defines the output in terms of itself or fits a parameter later called a prediction. Theorem 4.1 verifies (H1)–(H6) for ALGs by local transfer of compact Sinai-billiard technology (CM06 Growth Lemma, solid rectangles, homogeneity strips) under the uniformity assumption (ALG2), takes recurrence as (ALG1)/(H7), and obtains full K-mixing from ergodicity of all powers (Lemma 4.3) plus Corollary 2.2. Self-citations (Len03, Len06, CLS10, LT11) supply prior independent results on recurrence and ergodicity of return maps—different claims from K-mixing—and are used as ordinary external inputs. There is no self-definitional loop, no fitted-input-as-prediction, no uniqueness theorem imported to forbid alternatives, and no renaming of a known empirical pattern. (A separate correctness concern about µ of enlarged vs plain Cantor rectangles does not constitute circularity.)

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 2 invented entities

The central claims rest on seven geometric/measure-theoretic hypotheses (H1)–(H7) plus two ALG-specific assumptions. No numerical free parameters are fitted. The main invented organizational devices are the Cantor/enlarged Cantor rectangles and the tail sigma-algebra relative to the stable partition; both are standard adaptations rather than new physical entities. Recurrence and uniformity are domain assumptions imported from prior work or left as hypotheses.

axioms (9)
  • domain assumption (H1) F is a diffeomorphism off countable unions of closed singularity curves S1, S-1.
    Standard piecewise-smooth hyperbolic setup; Section 2.
  • domain assumption (H2) A.e. point has LSM/LUM of uniformly bounded length satisfying the nesting and contraction properties (2.2).
    Hyperbolicity + singularity control; verified for ALGs via homogeneous manifolds and uniform expansion (4.4).
  • domain assumption (H3) Partitions into LSM/LUM are measurable; conditional measures equivalent to normalized length with uniform constant C.
    Needed for disintegration and density-point arguments; standard for billiards under uniformity.
  • domain assumption (H4) Phase space covered mod 0 by countably many finite-measure Cantor rectangles.
    Load-bearing for filling rectangles and atomicity of the tail; verified by Growth Lemma + solid rectangles under (ALG2).
  • domain assumption (H5) Uniform distortion bound D on unstable Jacobians along LUMs.
    Used in Lemma 3.9–3.10 to control densities under iteration.
  • domain assumption (H6) Absolute continuity of holonomy maps with uniform Jacobian bounds.
    Gives measure equivalence of brushes of stable manifolds (2.7); standard billiard fact.
  • domain assumption (H7) / (ALG1) The system is Poincaré recurrent.
    Essential for return maps and for Erg(Rk) belonging to the tail; assumed, not proved, for the class K.
  • domain assumption (ALG2) Uniform bounds 0 < τm ≤ τ ≤ τM and km ≤ κ ≤ kM on free path and curvature.
    Makes all local hyperbolic and distortion constants uniform so compact-case estimates transfer.
  • standard math Standard Hopf argument, Birkhoff theorem, and Rohlin disintegration apply in the σ-finite setting.
    Used throughout Section 3 and Lemmas 4.2–4.3.
invented entities (2)
  • Cantor rectangle / enlarged Cantor rectangle R independent evidence
    purpose: Provide local product structure so that density points on unstable manifolds can fill positive-measure sets under iteration, forcing tail atomicity.
    Adapted from classical Sinai-billiard Markov constructions (CM06 §7); not a new physical object.
  • Tail σ-algebra T = ∩ F^{-k}S relative to the stable partition S independent evidence
    purpose: The object whose atomicity yields the K-decomposition.
    Standard in the definition of K-mixing; constructed explicitly from ξs.

pith-pipeline@v1.2.0-grok45-kimik3 · 29448 in / 3282 out tokens · 51966 ms · 2026-07-31T13:08:06.919105+00:00 · methodology

0 comments
read the original abstract

We prove a general theorem on the decomposition into K-mixing components for recurrent, piecewise-smooth hyperbolic maps in two dimensions. In contrast with classical results, we do not assume that the map preserves a probability measure. As an application, we show that every recurrent aperiodic Lorentz gas satisfying certain uniformity assumptions is K-mixing. Finally, we discuss some consequences of K-mixing for the decay of correlations relative to zero-mean $L^1$ observables in aperiodic Lorentz gases.

Figures

Figures reproduced from arXiv: 2607.24498 by Giovanni Canestrari, Marco Lenci.

Figure 1
Figure 1. Figure 1: An aperiodic Lorentz tube. The aperiodic Lorentz gas instead extends in both directions in the plane. Remark 4.5 below). To define the billiard map F, we consider a point-like particle of unit mass and speed that is moving without friction on D and when it hits the boundary of some scatterer, the trajectory reflects according to the law that the angle of incidence equals the angle of reflection. The map F … view at source ↗
Figure 2
Figure 2. Figure 2: The tangent trajectory with velocity v∗ hits ∂O1 at q1(q, v∗). Perturbing this trajectory yields non-singular trajectories connecting O0 directly to O2 at q1(q, v), as well as connecting O0 to O1 and O1 to O2. Proof of Theorem 4.1. We first show that the billiard map associated to any ALG in K satisfies (H1)-(H6), while (H7) is precisely given by (ALG1). These properties have been established for a large c… view at source ↗
Figure 3
Figure 3. Figure 3: A solid rectangle O bounded by Wu 1 , Wu 2 , Ws 1 and Ws 2 . The point x belongs to the set R(O), as its associated H-manifolds Wu H (x) and Ws H (x) fully cross O. This concludes our work on the main theorem on the ergodic properties of ALGs. As a further application of Theorem 4.1 along the line of stochastic properties, we have the following result. For any decreasing positive sequence a = {ap}p∈N such … view at source ↗

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Reference graph

Works this paper leans on

55 extracted references · 29 canonical work pages

  1. [1]

    1997 , PAGES =

    Aaronson, Jon , TITLE =. 1997 , PAGES =. doi:10.1090/surv/050 , URL =

    doi:10.1090/surv/050 1997

  2. [2]

    Blackwell, David and Freedman, David , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1964 , PAGES =. doi:10.1214/aoms/1177703284 , URL =

    arXiv 1964

  3. [3]

    Discrete Contin

    Bonanno, Claudio and Lenci, Marco , TITLE =. Discrete Contin. Dyn. Syst. , FJOURNAL =. 2021 , NUMBER =. doi:10.3934/dcds.2020309 , URL =

    work page doi:10.3934/dcds.2020309 2021

  4. [4]

    Nonlinearity , FJOURNAL =

    Bonanno, Claudio and Giulietti, Paolo and Lenci, Marco , TITLE =. Nonlinearity , FJOURNAL =. 2018 , NUMBER =. doi:10.1088/1361-6544/aadc04 , URL =

    work page doi:10.1088/1361-6544/aadc04 2018

  5. [5]

    Studia Math

    Bruin, Henk and Fougeron, Charles and Ravotti, Davide and Terhesiu, Dalia , TITLE =. Studia Math. , FJOURNAL =. 2025 , NUMBER =. doi:10.4064/sm240425-15-4 , URL =

    work page doi:10.4064/sm240425-15-4 2025

  6. [6]

    Bunimovich, L. A. and Sinai, Ya.\ G. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1980/81 , NUMBER =

    1980

  7. [7]

    Bunimovich, L. A. and Sinai, Ya.\ G. and Chernov, N. I. , TITLE =. Uspekhi Mat. Nauk , FJOURNAL =. 1990 , NUMBER =. doi:10.1070/RM1990v045n03ABEH002355 , URL =

    work page doi:10.1070/rm1990v045n03abeh002355 1990

  8. [8]

    Uniformly global observables for 1D maps with an indifferent fixed point , journal =

    Canestrari, Giovanni and Lenci, Marco , keywords =. Uniformly global observables for 1D maps with an indifferent fixed point , journal =. 2024 , issn =. doi:https://arxiv.org/abs/2405.05948 , url =

    Pith/arXiv arXiv 2024

  9. [9]

    Horocycle flows on abelian covers of surfaces of negative curvature , journal =

    Castorrini, Roberto and Ravotti, Davide , keywords =. Horocycle flows on abelian covers of surfaces of negative curvature , journal =. 2024 , issn =. doi:https://arxiv.org/abs/2405.08592 , url =

    Pith/arXiv arXiv 2024

  10. [10]

    , TITLE =

    Chernov, N. , TITLE =. J. Statist. Phys. , FJOURNAL =. 1999 , NUMBER =. doi:10.1023/A:1004581304939 , URL =

    doi:10.1023/a:1004581304939 1999

  11. [11]

    and Dolgopyat, D

    Chernov, N. and Dolgopyat, D. , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1090/S0894-0347-08-00626-7 , URL =

    work page doi:10.1090/s0894-0347-08-00626-7 2009

  12. [12]

    Chernov, N. I. and Haskell, C. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1996 , NUMBER =. doi:10.1017/S0143385700008695 , URL =

    work page doi:10.1017/s0143385700008695 1996

  13. [13]

    2006 , PAGES =

    Chernov, Nikolai and Markarian, Roberto , TITLE =. 2006 , PAGES =. doi:10.1090/surv/127 , URL =

    doi:10.1090/surv/127 2006

  14. [14]

    and Zhang, H.-K

    Chernov, N. and Zhang, H.-K. , TITLE =. Nonlinearity , FJOURNAL =. 2005 , NUMBER =. doi:10.1088/0951-7715/18/4/006 , URL =

    doi:10.1088/0951-7715/18/4/006 2005

  15. [15]

    Global-local mixing for infinite measure dynamical systems , journal =

    Coates, Douglas and Melbourne, Ian , keywords =. Global-local mixing for infinite measure dynamical systems , journal =. 2025 , issn =. doi:https://arxiv.org/abs/2505.07492 , url =

    arXiv 2025

  16. [16]

    Chaos , FJOURNAL =

    Cristadoro, Giampaolo and Lenci, Marco and Seri, Marcello , TITLE =. Chaos , FJOURNAL =. 2010 , NUMBER =. doi:10.1063/1.3405290 , URL =

    work page doi:10.1063/1.3405290 2010

  17. [17]

    and Silva, Cesar E

    Danilenko, Alexandre I. and Silva, Cesar E. , TITLE =. Ergodic theory , SERIES =. [2023] 2023 , ISBN =. doi:10.1007/978-1-0716-2388-6\_183 , URL =

    work page doi:10.1007/978-1-0716-2388-6 2023

  18. [18]

    and Liverani, Carlangelo , TITLE =

    Demers, Mark F. and Liverani, Carlangelo , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00220-023-04657-1 , URL =

    work page doi:10.1007/s00220-023-04657-1 2023

  19. [19]

    De Simoi, Jacopo and Dolgopyat, Dmitry , TITLE =. Ann. H. Lebesgue , FJOURNAL =. 2025 , PAGES =. doi:10.5802/ahl.240 , URL =

    work page doi:10.5802/ahl.240 2025

  20. [20]

    Dolgopyat, Dmitry and N\'andori, P\'eter , TITLE =. Adv. Math. , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.aim.2022.108757 , URL =

    arXiv 2022

  21. [21]

    2010 , PAGES =

    Durrett, Rick , TITLE =. 2010 , PAGES =. doi:10.1017/CBO9780511779398 , URL =

    doi:10.1017/cbo9780511779398 2010

  22. [22]

    Nonlinearity , FJOURNAL =

    Flaminio, Livio and Ravotti, Davide , TITLE =. Nonlinearity , FJOURNAL =. 2024 , NUMBER =. doi:10.1088/1361-6544/ad4aaf , URL =

    work page doi:10.1088/1361-6544/ad4aaf 2024

  23. [23]

    Friedman, N. A. and Ornstein, D. S. , TITLE =. Advances in Math. , FJOURNAL =. 1970 , PAGES =. doi:10.1016/0001-8708(70)90010-1 , URL =

    work page doi:10.1016/0001-8708(70)90010-1 1970

  24. [24]

    , TITLE =

    Gallavotti, Giovanni and Ornstein, Donald S. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1974 , PAGES =

    1974

  25. [25]

    Galli, Daniele and Lenci, Marco , TITLE =. J. Stat. Phys. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s10955-022-03039-6 , URL =

    work page doi:10.1007/s10955-022-03039-6 2023

  26. [26]

    Giulietti, Paolo and Hammerlindl, Andy and Ravotti, Davide , TITLE =. Ann. Henri Poincar\'e , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00023-021-01102-8 , URL =

    work page doi:10.1007/s00023-021-01102-8 2022

  27. [27]

    Kolmogorov, A. N. , TITLE =. Dokl. Akad. Nauk SSSR (N.S.) , FJOURNAL =. 1958 , PAGES =

    1958

  28. [28]

    Lenci, Marco , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s00220-002-0710-7 , URL =

    work page doi:10.1007/s00220-002-0710-7 2002

  29. [29]

    Ergodic Theory Dynam

    Lenci, Marco , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2003 , NUMBER =. doi:10.1017/S0143385702001529 , URL =

    work page doi:10.1017/s0143385702001529 2003

  30. [30]

    Ergodic Theory Dynam

    Lenci, Marco , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2006 , NUMBER =. doi:10.1017/S0143385706000022 , URL =

    work page doi:10.1017/s0143385706000022 2006

  31. [31]

    Lenci, Marco , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s00220-010-1043-6 , URL =

    work page doi:10.1007/s00220-010-1043-6 2010

  32. [32]

    Lenci, Marco , TITLE =. Publ. Mat. Urug. , FJOURNAL =. 2013 , PAGES =

    2013

  33. [33]

    Chaos Solitons Fractals , FJOURNAL =

    Lenci, Marco , TITLE =. Chaos Solitons Fractals , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.chaos.2015.11.024 , URL =

    work page doi:10.1016/j.chaos.2015.11.024 2016

  34. [34]

    Discrete Contin

    Lenci, Marco , TITLE =. Discrete Contin. Dyn. Syst. , FJOURNAL =. 2017 , NUMBER =. doi:10.3934/dcds.2017163 , URL =

    work page doi:10.3934/dcds.2017163 2017

  35. [35]

    Lenci, Marco and Troubetzkoy, Serge , TITLE =. Phys. D , FJOURNAL =. 2011 , NUMBER =. doi:10.1016/j.physd.2011.06.020 , URL =

    work page doi:10.1016/j.physd.2011.06.020 2011

  36. [36]

    Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , YEAR =

    Lin, Michael , title =. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , YEAR =

  37. [37]

    Ergodic Theory Dynam

    Liverani, Carlangelo and Saussol, Beno\^it and Vaienti, Sandro , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 1999 , NUMBER =. doi:10.1017/S0143385799133856 , URL =

    doi:10.1017/s0143385799133856 1999

  38. [38]

    , TITLE =

    Liverani, Carlangelo and Wojtkowski, Maciej P. , TITLE =. Dynamics reported , SERIES =. 1995 , ISBN =

    1995

  39. [39]

    Koninklijke Akademie van Wetenschappen te Amsterdam, Section of Sciences , FJOURNAL =

    Lorentz, Hendrik Antoon , TITLE =. Koninklijke Akademie van Wetenschappen te Amsterdam, Section of Sciences , FJOURNAL =. 1905 , NUMBER =. doi:, URL =

    1905

  40. [40]

    Ergodic Theory Dynam

    Markarian, Roberto , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2004 , NUMBER =. doi:10.1017/S0143385703000270 , URL =

    doi:10.1017/s0143385703000270 2004

  41. [41]

    Advances in Math

    Ornstein, Donald , TITLE =. Advances in Math. , FJOURNAL =. 1970 , PAGES =. doi:10.1016/0001-8708(70)90029-0 , URL =

    work page doi:10.1016/0001-8708(70)90029-0 1970

  42. [42]

    Pomeau, Yves and Manneville, Paul , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1980 , NUMBER =

    1980

  43. [43]

    Seri, Marcello and Lenci, Marco and degli Esposti, Mirko and Cristadoro, Giampaolo , TITLE =. J. Stat. Phys. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s10955-011-0244-5 , URL =

    work page doi:10.1007/s10955-011-0244-5 2011

  44. [44]

    Ergodic Theory Dynam

    Miernowski, Tomasz and Nogueira, Arnaldo , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2013 , NUMBER =. doi:10.1017/S014338571100085X , URL =

    work page doi:10.1017/s014338571100085x 2013

  45. [45]

    Parry, William , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1965 , PAGES =. doi:10.2307/2035593 , URL =

    work page doi:10.2307/2035593 1965

  46. [46]

    Pène, Françoise and Terhesiu, Dalia , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/tran/9323 , URL =

    work page doi:10.1090/tran/9323 2025

  47. [47]

    Pène, Françoise , TITLE =. J. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1063/5.0244180 , URL =

    work page doi:10.1063/5.0244180 2025

  48. [48]

    , TITLE =

    Pesin, Ja.\ B. , TITLE =. Uspehi Mat. Nauk , FJOURNAL =. 1977 , NUMBER =

    1977

  49. [49]

    , TITLE =

    Pesin, Ja.\ B. , TITLE =. Mat. Zametki , FJOURNAL =. 1977 , NUMBER =

    1977

  50. [50]

    Rohlin, V. A. , TITLE =. Amer. Math. Soc. Translation , FJOURNAL =. 1952 , NUMBER =

    1952

  51. [51]

    Rohlin, V. A. , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1961 , PAGES =

    1961

  52. [52]

    , TITLE =

    Sinai, Ja.\ G. , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1966 , PAGES =

    1966

  53. [53]

    , TITLE =

    Sinai, Ja.\ G. , TITLE =. Uspehi Mat. Nauk , FJOURNAL =. 1970 , NUMBER =

    1970

  54. [54]

    Trevi\ no, Rodrigo and Zelerowicz, Agnieszka , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00220-022-04493-9 , URL =

    work page doi:10.1007/s00220-022-04493-9 2022

  55. [55]

    Troubetzkoy, Serge , TITLE =. J. Stat. Phys. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s10955-010-0026-5 , URL =

    work page doi:10.1007/s10955-010-0026-5 2010