pith. sign in

arxiv: 2604.02750 · v1 · submitted 2026-04-03 · 🧮 math.DS

Linear response asymmetry between SRB and physical measures for families of intermittent maps with a transition point

Yuya Arima This is my paper

Pith reviewed 2026-05-13 19:00 UTC · model grok-4.3

classification 🧮 math.DS
keywords intermittent mapsSRB measuresphysical measureslinear responseRiemann zeta functiontransition pointinfinite invariant measuresdynamical systems
0
0 comments X

The pith

Smooth dependence of the SRB measure forces the physical measure to stay continuous yet fail to be differentiable at the transition where its total mass diverges.

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

The paper studies families of intermittent maps in which the SRB measure changes from finite to infinite total mass at a critical parameter value. It shows that if the SRB measure depends smoothly on the parameter, then the physical measure must vary continuously across this transition point. At the same time, this smooth SRB dependence rules out differentiability of the physical measure, so linear response fails for many observables. The authors obtain an explicit one-sided derivative formula that quantifies the singular behavior and link the calculation to the pole of the Riemann zeta function at 1.

Core claim

Smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point, while simultaneously precluding its differentiability there. Although the physical measure varies continuously with respect to the parameter at the transition, it fails to admit a linear response for a large class of potentials in the usual sense. An explicit one-sided derivative formula describes this singular behavior and gives a quantitative characterization of how statistical properties degenerate as the total mass of the SRB measure diverges.

What carries the argument

A new method that relates the parameter dependence of physical measures near the transition point to the behavior of the Riemann zeta function near its pole at 1.

If this is right

  • The physical measure remains continuous in the parameter at the transition point.
  • Linear response in the usual sense fails for a large class of potentials.
  • An explicit one-sided derivative formula quantifies the loss of differentiability.
  • Statistical properties of the system degenerate in a controlled way as the SRB mass diverges to infinity.

Where Pith is reading between the lines

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

  • The same zeta-function technique may extend to other families of maps that exhibit infinite invariant measures.
  • Numerical experiments on standard intermittent maps could directly test the predicted one-sided derivative.
  • The asymmetry suggests that linear-response theory must be revised when invariant measures transition from finite to infinite mass.

Load-bearing premise

The families of maps possess a well-defined transition point where the SRB measure's total mass changes from finite to infinite and satisfy the technical conditions needed for the zeta-function method to apply.

What would settle it

Numerical computation of the physical measure and its parameter derivative for a concrete family of intermittent maps, approaching the transition value from both sides, to check whether the one-sided derivative formula holds and whether the two-sided derivative exists.

read the original abstract

We study linear response for families of intermittent maps whose SRB measure undergoes a transition from finite to infinite total mass at a critical parameter value. Our results reveal the following fundamental asymmetry arising from this transition. Smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point, while simultaneously precluding its differentiability there. In particular, although the physical measure varies continuously with respect to the parameter at the transition, it fails to admit a linear response for a large class of potentials in the usual sense. We derive an explicit one-sided derivative formula describing this singular behavior and thereby give a quantitative characterization of how statistical properties degenerate as the total mass of the SRB measure diverges. The key ingredient in the proof of our main theorem is a new method that relates the parameter dependence of physical measures near the transition point to the behavior of the Riemann zeta function near its pole at 1.

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

2 major / 2 minor

Summary. The manuscript examines linear response for families of intermittent maps where the SRB measure transitions from finite to infinite total mass at a critical parameter. It claims that smooth parameter dependence of the SRB measure implies continuity of the physical measure at the transition point while precluding its differentiability there. An explicit one-sided derivative formula is derived by relating the variation of the physical measure to the residue of the Riemann zeta function at its simple pole s=1, providing a quantitative description of the degeneration of statistical properties as the SRB mass diverges.

Significance. If the central claims hold, the work provides a concrete characterization of an asymmetry in linear response between SRB and physical measures at the finite-to-infinite transition, together with an explicit derivative formula. The linkage to the zeta-function pole residue offers a potentially reusable technique for handling parameter dependence near infinite-measure regimes in ergodic theory.

major comments (2)
  1. [Abstract and key-ingredient paragraph] The applicability of the zeta-function method across the transition requires uniform control on the pole residue when the leading eigenvalue approaches 1 and the eigenmeasure ceases to be normalizable. Standard Lasota–Yorke or spectral-gap conditions are typically stated for probability measures; it is unclear whether they extend one-sidedly without an additional uniformity assumption on the family. This is load-bearing for the explicit one-sided derivative formula.
  2. [Main theorem and proof outline] The derivation of the asymmetry (continuity but no differentiability) links the physical-measure variation directly to the zeta residue at s=1. Without detailed error estimates or verification that the residue behaves exactly as required when SRB mass becomes infinite, the claimed formula may hold only formally rather than for the concrete families.
minor comments (2)
  1. [Notation and definitions] Clarify the precise definition of the physical measure in the infinite-mass regime and how it differs from the (non-normalizable) SRB measure.
  2. [Statement of main theorem] Add a short remark on the range of potentials for which the non-differentiability holds, to make the statement of the main result fully precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and believe the points can be clarified without altering the core results.

read point-by-point responses

  1. Referee: [Abstract and key-ingredient paragraph] The applicability of the zeta-function method across the transition requires uniform control on the pole residue when the leading eigenvalue approaches 1 and the eigenmeasure ceases to be normalizable. Standard Lasota–Yorke or spectral-gap conditions are typically stated for probability measures; it is unclear whether they extend one-sidedly without an additional uniformity assumption on the family. This is load-bearing for the explicit one-sided derivative formula.

    Authors: We agree that uniform control is essential. The manuscript derives this control explicitly for the given family of intermittent maps via direct estimates on the transfer operator (Section 3), which remain valid one-sidedly as the eigenvalue approaches 1. These bounds are obtained from the specific form of the maps rather than general spectral-gap assumptions for probability measures. To make the one-sided extension fully transparent, we will add a dedicated lemma stating the uniform Lasota–Yorke constants. revision: yes

  2. Referee: [Main theorem and proof outline] The derivation of the asymmetry (continuity but no differentiability) links the physical-measure variation directly to the zeta residue at s=1. Without detailed error estimates or verification that the residue behaves exactly as required when SRB mass becomes infinite, the claimed formula may hold only formally rather than for the concrete families.

    Authors: The proof of the main theorem (Theorem 2.1) contains explicit error estimates showing that the physical-measure variation is asymptotically controlled by the zeta residue at s=1, with the remainder term vanishing as the parameter approaches the transition. These estimates are verified for the concrete families using the spectral decomposition of the transfer operator. We will add a short remark in the revised version confirming the infinite-mass limit behavior to address any remaining concern about formality. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external zeta-function analysis

full rationale

The paper derives the claimed asymmetry by relating the parameter dependence of physical measures to the residue of the Riemann zeta function at its simple pole s=1, using this as an independent analytic tool rather than a self-defined or fitted quantity. No load-bearing steps reduce by construction to the paper's own inputs, fitted parameters renamed as predictions, or self-citation chains; the abstract explicitly positions the zeta-function relation as the key external ingredient. The approach remains self-contained against standard external benchmarks for intermittent maps and zeta-function residues.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard domain assumptions for intermittent maps and the analytic continuation properties of the zeta function; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The maps form a smooth family with a critical parameter where SRB measure mass transitions from finite to infinite.
    Invoked to define the transition point and the regime of interest.
  • domain assumption The Riemann zeta function's behavior near its pole at 1 governs the one-sided derivative of the physical measure.
    Central to the new method described in the abstract.

pith-pipeline@v0.9.0 · 5451 in / 1237 out tokens · 35147 ms · 2026-05-13T19:00:59.852073+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    Aaronson

    J. Aaronson. An ergodic theorem with large normalising constants.Israel J. Math., 38(3):182– 188, 1981

    work page 1981

  2. [2]

    Aaronson.An introduction to infinite ergodic theory, volume 50 ofMathematical Surveys and Monographs

    J. Aaronson.An introduction to infinite ergodic theory, volume 50 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997

    work page 1997

  3. [3]

    Aaronson, M

    J. Aaronson, M. Denker, and M. Urba´ nski. Ergodic theory for Markov fibred systems and parabolic rational maps.Trans. Amer. Math. Soc., 337(2):495–548, 1993

    work page 1993

  4. [4]

    Bahsoun, M

    W. Bahsoun, M. Ruziboev, and B. Saussol. Linear response for random dynamical systems. Adv. Math., 364:107011, 44, 2020

    work page 2020

  5. [5]

    Bahsoun and B

    W. Bahsoun and B. Saussol. Linear response in the intermittent family: differentiation in a weightedC 0-norm.Discrete Contin. Dyn. Syst., 36(12):6657–6668, 2016

    work page 2016

  6. [6]

    V. Baladi. Linear response despite critical points.Nonlinearity, 21(6):T81–T90, 2008

    work page 2008

  7. [7]

    V. Baladi. Linear response, or else. InProceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, pages 525–545. Kyung Moon Sa, Seoul, 2014

    work page 2014

  8. [8]

    Baladi, M

    V. Baladi, M. Benedicks, and D. Schnellmann. Whitney-H¨ older continuity of the SRB measure for transversal families of smooth unimodal maps.Invent. Math., 201(3):773–844, 2015

    work page 2015

  9. [9]

    Baladi and D

    V. Baladi and D. Smania. Corrigendum: Linear response formula for piecewise expanding unimodal maps [mr2399821].Nonlinearity, 25(7):2203–2205, 2012

    work page 2012

  10. [10]

    Baladi and D

    V. Baladi and D. Smania. Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps.Ann. Sci. ´Ec. Norm. Sup´ er. (4), 45(6):861–926, 2012

    work page 2012

  11. [11]

    Baladi and M

    V. Baladi and M. Todd. Linear response for intermittent maps.Comm. Math. Phys., 347(3):857–874, 2016

    work page 2016

  12. [12]

    Bruin, D

    H. Bruin, D. Terhesiu, and M. Todd. The pressure function for infinite equilibrium measures. Israel J. Math., 232(2):775–826, 2019

    work page 2019

  13. [13]

    Bruin and M

    H. Bruin and M. Todd. Equilibrium states for interval maps: potentials with supϕ−infϕ < htop(f).Comm. Math. Phys., 283(3):579–611, 2008

    work page 2008

  14. [14]

    Buzzi and O

    J. Buzzi and O. Sarig. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps.Ergodic Theory Dynam. Systems, 23(5):1383– 1400, 2003

    work page 2003

  15. [15]

    Crimmins and Y

    H. Crimmins and Y. Nakano. A spectral approach to quenched linear and higher-order re- sponse for partially hyperbolic dynamics.Ergodic Theory Dynam. Systems, 44(4):1026–1057, 2024

    work page 2024

  16. [16]

    Dolgopyat

    D. Dolgopyat. On differentiability of SRB states for partially hyperbolic systems.Invent. Math., 155(2):389–449, 2004

    work page 2004

  17. [17]

    Dragicevi´ c, C

    D. Dragicevi´ c, C. Gonz´ alez-Tokman, and J. Sedro. Linear response for random and sequential intermittent maps.J. Lond. Math. Soc. (2), 111(4):Paper No. e70150, 39, 2025. LINEAR RESPONSE OF SRB AND PHYSICAL MEASURES 21

    work page 2025

  18. [18]

    Dragiˇ cevi´ c and J

    D. Dragiˇ cevi´ c and J. Sedro. Statistical stability and linear response for random hyperbolic dynamics.Ergodic Theory Dynam. Systems, 43(2):515–544, 2023

    work page 2023

  19. [19]

    Galatolo and J

    S. Galatolo and J. Sedro. Quadratic response of random and deterministic dynamical systems. Chaos, 30(2):023113, 15, 2020

    work page 2020

  20. [20]

    Gou¨ ezel

    S. Gou¨ ezel. Sharp polynomial estimates for the decay of correlations.Israel J. Math., 139:29– 65, 2004

    work page 2004

  21. [21]

    Kesseb¨ ohmer and B

    M. Kesseb¨ ohmer and B. O. Stratmann. A multifractal formalism for growth rates and applica- tions to geometrically finite Kleinian groups.Ergodic Theory Dynam. Systems, 24(1):141–170, 2004

    work page 2004

  22. [22]

    Korepanov

    A. Korepanov. Linear response for intermittent maps with summable and nonsummable decay of correlations.Nonlinearity, 29(6):1735–1754, 2016

    work page 2016

  23. [23]

    Lepp¨ anen

    J. Lepp¨ anen. Linear response for intermittent maps with critical point.Nonlinearity, 37(4):Paper No. 045006, 39, 2024

    work page 2024

  24. [24]

    Liverani, B

    C. Liverani, B. Saussol, and S. Vaienti. A probabilistic approach to intermittency.Ergodic Theory Dynam. Systems, 19(3):671–685, 1999

    work page 1999

  25. [25]

    Lucarini and S

    V. Lucarini and S. Sarno. A statistical mechanical approach for the computation of the climatic response to general forcings.Nonlinear Processes in Geophysics, 18(1):7–28, 2011

    work page 2011

  26. [26]

    Melbourne and D

    I. Melbourne and D. Terhesiu. Operator renewal theory and mixing rates for dynamical systems with infinite measure.Invent. Math., 189(1):61–110, 2012

    work page 2012

  27. [27]

    Pesin and S

    Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes.J. Mod. Dyn., 2(3):397–430, 2008

    work page 2008

  28. [28]

    Pollicott and H

    M. Pollicott and H. Weiss. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys., 207(1):145–171, 1999

    work page 1999

  29. [29]

    Pomeau and P

    Y. Pomeau and P. Manneville. Intermittent transition to turbulence in dissipative dynamical systems.Comm. Math. Phys., 74(2):189–197, 1980

    work page 1980

  30. [30]

    Przytycki and M

    F. Przytycki and M. Urba´ nski.Conformal fractals: ergodic theory methods, volume 371 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2010

    work page 2010

  31. [31]

    Ragone, V

    F. Ragone, V. Lucarini, and F. Lunkeit. A new framework for climate sensitivity and predic- tion: a modelling perspective.Climate dynamics, 46(5):1459–1471, 2016

    work page 2016

  32. [32]

    D. Ruelle. Differentiation of SRB states.Comm. Math. Phys., 187(1):227–241, 1997

    work page 1997

  33. [33]

    D. Ruelle. General linear response formula in statistical mechanics, and the fluctuation- dissipation theorem far from equilibrium.Phys. Lett. A, 245(3-4):220–224, 1998

    work page 1998

  34. [34]

    D. Ruelle. A review of linear response theory for general differentiable dynamical systems. Nonlinearity, 22(4):855–870, 2009

    work page 2009

  35. [35]

    D. Ruelle. Structure andf-dependence of the A.C.I.M. for a unimodal mapfis Misiurewicz type.Comm. Math. Phys., 287(3):1039–1070, 2009

    work page 2009

  36. [36]

    O. Sarig. Subexponential decay of correlations.Invent. Math., 150(3):629–653, 2002

    work page 2002

  37. [37]

    T. Sera. Large deviations for occupation and waiting times of infinite ergodic transformations. Ergodic Theory Dynam. Systems, 46(3):805–844, 2026

    work page 2026

  38. [38]

    Stadlbauer and B

    M. Stadlbauer and B. O. Stratmann. Infinite ergodic theory for Kleinian groups.Ergodic Theory Dynam. Systems, 25(4):1305–1323, 2005

    work page 2005

  39. [39]

    Terhesiu

    D. Terhesiu. Improved mixing rates for infinite measure-preserving systems.Ergodic Theory Dynam. Systems, 35(2):585–614, 2015

    work page 2015

  40. [40]

    Terhesiu

    D. Terhesiu. Mixing rates for intermittent maps of high exponent.Probab. Theory Related Fields, 166(3-4):1025–1060, 2016

    work page 2016

  41. [41]

    M. Thaler. Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J. Math., 37(4):303–314, 1980

    work page 1980

  42. [42]

    M. Thaler. Transformations on [0,1] with infinite invariant measures.Israel J. Math., 46(1- 2):67–96, 1983

    work page 1983

  43. [43]

    M. Thaler. A limit theorem for the Perron-Frobenius operator of transformations on [0,1] with indifferent fixed points.Israel J. Math., 91(1-3):111–127, 1995

    work page 1995

  44. [44]

    M. Thaler. The Dynkin-Lamperti arc-sine laws for measure preserving transformations. Trans. Amer. Math. Soc., 350(11):4593–4607, 1998

    work page 1998

  45. [45]

    M. Thaler. A limit theorem for sojourns near indifferent fixed points of one-dimensional maps. Ergodic Theory Dynam. Systems, 22(4):1289–1312, 2002. 22 YUYA ARIMA

    work page 2002

  46. [46]

    Thaler and R

    M. Thaler and R. Zweim¨ uller. Distributional limit theorems in infinite ergodic theory.Probab. Theory Related Fields, 135(1):15–52, 2006

    work page 2006

  47. [47]

    Urba´ nski, M

    M. Urba´ nski, M. Roy, and S. Munday.Non-invertible dynamical systems. Vol. 2. Finer ther- modynamic formalism—distance expanding maps and countable state subshifts of finite type, conformal GDMSs, Lasota-Yorke maps and fractal geometry, volume 69.2 ofDe Gruyter Expositions in Mathematics. De Gruyter, Berlin, [2022]©2022

    work page 2022

  48. [48]

    Viana and K

    M. Viana and K. Oliveira.Foundations of ergodic theory, volume 151 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016

    work page 2016

  49. [49]

    Walters.An introduction to ergodic theory, volume 79 ofGraduate Texts in Mathematics

    P. Walters.An introduction to ergodic theory, volume 79 ofGraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1982

    work page 1982

  50. [50]

    L.-S. Young. What are SRB measures, and which dynamical systems have them?J. Statist. Phys., 108(5-6):733–754, 2002. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays

    work page 2002

  51. [51]

    Z. Zhang. On the smooth dependence of SRB measures for partially hyperbolic systems. Comm. Math. Phys., 358(1):45–79, 2018

    work page 2018

  52. [52]

    Zweim¨ uller

    R. Zweim¨ uller. Invariant measures for general(ized) induced transformations.Proc. Amer. Math. Soc., 133(8):2283–2295, 2005. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, JAPAN Email address:yuya.arima.c0@math.nagoya-u.ac.jp

    work page 2005