pith. sign in

arxiv: 2501.06630 · v2 · submitted 2025-01-11 · 🧮 math.DS · math.CA

Generalized dichotomies via time rescaling

Davor Dragicevic , Cesar M. Silva This is my paper

Pith reviewed 2026-05-23 05:25 UTC · model grok-4.3

classification 🧮 math.DS math.CA
keywords μ-dichotomytime rescalingdiscrete nonautonomous systemsordinary dichotomyexponential dichotomySacker-Sell spectrumlinearizationgrowth rates
0
0 comments X

The pith

Time rescaling reduces μ-dichotomies for discrete nonautonomous linear systems to ordinary and exponential dichotomies for a broad class of growth rates.

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

The paper shows that μ-dichotomies, defined with respect to sequences of norms and discrete growth rates μ, are equivalent to standard ordinary or exponential dichotomies after a suitable change of the time index. This equivalence holds for a large class of growth rates and extends an earlier result that covered only the polynomial case. The characterization lets researchers transfer known properties of ordinary and exponential dichotomies to the generalized setting. As a direct consequence the authors obtain a description of the generalized Sacker-Sell spectrum and a series of nonautonomous linearization theorems.

Core claim

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates μ, the notion of μ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. The same reduction was previously known only when μ is polynomial. The reduction also yields structural information on a generalized Sacker-Sell spectrum together with topological and smooth linearization results.

What carries the argument

A time-rescaling map that converts the growth condition imposed by μ into the growth condition of an ordinary or exponential dichotomy.

If this is right

  • Properties of ordinary and exponential dichotomies transfer directly to the corresponding μ-dichotomies.
  • The generalized Sacker-Sell spectrum admits a description in terms of the spectra obtained after rescaling.
  • Topological and smooth linearization theorems hold for the systems that satisfy a μ-dichotomy.
  • Any dichotomy notion covered by the class of admissible μ can be studied by reducing it to the ordinary or exponential case.

Where Pith is reading between the lines

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

  • The same rescaling technique might extend to continuous-time or infinite-dimensional linear systems once an appropriate discrete approximation is chosen.
  • Numerical detection of generalized dichotomies could be reduced to standard algorithms by first applying the time change.
  • Spectral theory for nonautonomous systems may gain new comparison theorems by viewing different growth rates as equivalent after rescaling.

Load-bearing premise

There exists a time rescaling that exactly matches the growth dictated by μ to the growth of an ordinary or exponential dichotomy.

What would settle it

A concrete sequence of norms and a growth rate μ in the stated class for which the μ-dichotomy condition holds but the rescaled ordinary-dichotomy condition fails (or vice versa).

read the original abstract

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $\mu$, we show that the notion of $\mu$ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.

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 / 2 minor

Summary. The manuscript shows that for discrete-time nonautonomous linear dynamics equipped with sequences of norms, and for a large class of discrete growth rates μ, the notion of μ-dichotomy can be completely characterized in terms of ordinary and exponential dichotomies via a suitable rescaling of time. This extends the known polynomial case. As an application, the structure of a generalized Sacker-Sell spectrum is studied and a series of nonautonomous topological and smooth linearization results are obtained.

Significance. If the central characterization holds, the work supplies a unified reduction of generalized dichotomies to the ordinary/exponential cases through time rescaling, extending the polynomial setting in a manner that appears independent of the result itself. The applications to the generalized Sacker-Sell spectrum and linearization results demonstrate concrete utility in nonautonomous dynamics.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a large class of discrete growth rates μ' is used without a one-sentence pointer to the precise definition or standing assumptions; adding such a pointer would improve readability.
  2. [Applications] The applications section would benefit from a brief concrete example (even a low-dimensional linear system) illustrating how the time-rescaling transfers a specific μ-dichotomy into an ordinary one.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on characterizing μ-dichotomies for discrete nonautonomous linear systems via time rescaling, the extension beyond the polynomial case, and the applications to the generalized Sacker-Sell spectrum and linearization results. The recommendation for minor revision is appreciated.

Circularity Check

0 steps flagged

No significant circularity; characterization is self-contained

full rationale

The paper establishes an equivalence between μ-dichotomies and ordinary/exponential dichotomies for a class of growth rates μ via explicit time rescaling in discrete nonautonomous linear systems. This is a direct mathematical construction that transfers dichotomy properties bidirectionally without defining the target notion in terms of itself or fitting parameters to the output. Prior polynomial results are cited as background rather than load-bearing justification for the general case. No self-definitional loops, fitted-input predictions, or uniqueness theorems imported from the authors' own prior work appear in the derivation chain. The result is externally falsifiable via the rescaling map and preserves the linear structure independently of the claimed equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; insufficient information to populate the ledger.

pith-pipeline@v0.9.0 · 5617 in / 1105 out tokens · 76582 ms · 2026-05-23T05:25:15.163329+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

42 extracted references · 42 canonical work pages

  1. [1]

    Aulbach and S

    B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equa- tions, J. Differ. Equ. Appl. 7 (2001), 895–913

    work page 2001

  2. [2]

    Backes and D

    L. Backes and D. Dragičević, A characterization of (µ, ν)-dichotomies via admissibility , preprint, https://arxiv.org/pdf/2406.04126

    work page arXiv

  3. [3]

    Backes, D

    L. Backes, D. Dragičević and W. Zhang, Smooth linearization for nonautonomous dynamics under pol y- nomial behaviour, preprint, https://arxiv.org/pdf/2210.04804

    work page arXiv

  4. [4]

    Barreira, D

    L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and one-sided admissibility , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016), 235–247

    work page 2016

  5. [5]

    Barreira, D

    L. Barreira, D. Dragičević and C. Valls, Strong nonuniform spectrum for arbitrary growth rates , Commun. Contemp. Math. 19 (2017), 1650008, 25 pp

    work page 2017

  6. [6]

    Barreira, D

    L. Barreira, D. Dragičević and C. Valls, Admissibility a nd Hyperbolicity, SpringerBriefs Math., Springer, Cham, 2018

    work page 2018

  7. [7]

    Barreira and C

    L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dyn amics, Journal of Differential Equations 228 (2006), 285–310

    work page 2006

  8. [8]

    Barreira and C

    L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity , Discrete Contin. Dyn. Syst. 22 (2008), 509–528

    work page 2008

  9. [9]

    Barreira and C

    L. Barreira and C. Valls, A Grobman-Hartman theorem for general nonuniform exponent ial dichotomies , J. Funct. Anal. 257 (2009), 1976-1993

    work page 2009

  10. [10]

    Barreira and C

    L. Barreira and C. Valls, Polynomial growth rates , Nonlinear Anal. 71 (2009), 5208–5219. 28

    work page 2009

  11. [11]

    Bento, N

    A. Bento, N. Lupa, M. Megan and C. Silva, Integral conditions for nonuniform µ-dichotomy on the half- line, Discrete Contin. Dyn. Syst. B 22 (2017), 3063–3077

    work page 2017

  12. [12]

    Bento and C

    A. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies , J. Funct. Anal. 257 (2009), 122–148

    work page 2009

  13. [13]

    A. J. Bento and C. M. Silva, Nonuniform (µ, ν)-dichotomies and local dynamics of difference equations , Nonlinear Anal. 75 (2012), 78–90

    work page 2012

  14. [14]

    A. J. Bento and C. M. Silva, Generalized Nonuniform Dichotomies and Local Stable Manif olds, J. Dyn. Diff. Equat. 25 (2013), 1139–1158

    work page 2013

  15. [15]

    Boruga, M

    R. Boruga, M. Megan and D.M.M. Toth, On uniform instability with growth rates in Banach spaces , Carpathian J. Math. 38 (2022), 789–796

    work page 2022

  16. [16]

    Chicone and Yu

    C. Chicone and Yu. Latushkin, Evolution Semigroups in D ynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1 999

    work page

  17. [17]

    Chu, Robustness of nonuniform behavior for discrete dynamics , Bull

    J. Chu, Robustness of nonuniform behavior for discrete dynamics , Bull. Sci. Math. 137 (2013), 1031-1047

    work page 2013

  18. [18]

    Coppel, Dichotomies in Stability Theory, Lect

    W. Coppel, Dichotomies in Stability Theory, Lect. Note s in Math. 629, Springer, 1978

    work page 1978

  19. [19]

    L. V. Coung, T. S. Doan and S. Siegmund, A Sternberg theorem for nonautonomous differential equatio ns, J Dyn. Diff. Equat. 31 (2019), 1279–1299

    work page 2019

  20. [20]

    Dalec ′ ki ˘ ı and M

    Ju. Dalec ′ ki ˘ ı and M. Kre ˘ ın, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974

    work page 1974

  21. [21]

    Dragičević, Admissibility and nonuniform polynomial dichotomies , Math

    D. Dragičević, Admissibility and nonuniform polynomial dichotomies , Math. Nachr. 293 (2019), 226–243

    work page 2019

  22. [22]

    Dragičević, N

    D. Dragičević, N. Lup and N. Jurčević Peček, Admissibility and general dichotomies for evolution famil ies, Electron. J. Qual. Theory Differ. Equ. (2020), Paper No. 58, 1 9 pp

    work page 2020

  23. [23]

    Dragičević, W

    D. Dragičević, W. Zhang and W. Zhang, Smooth linearization of nonautonomous difference equation s with a nonuniform dichotomy , Math. Z. 292 (2019), 1175–1193

    work page 2019

  24. [24]

    Dragičević, W

    D. Dragičević, W. Zhang and W. Zhang, Smooth linearization of nonautonomous differential equati ons with a nonuniform dichotomy , Proc. London. Math. Soc. 121 (2020), 32–50

    work page 2020

  25. [25]

    Dragičević, A

    D. Dragičević, A. L. Sasu and B. Sasu, On Polynomial Dichotomies of Discrete Nonautonomous Syste ms on the Half-Line , Carpathian J. Math. 38 (2022), 663–680

    work page 2022

  26. [26]

    Dragičević, A

    D. Dragičević, A. L. Sasu and B. Sasu, Admissibility and polynomial dichotomy of discrete nonaut onomous systems, Carpathian J. Math. 38 (2022), 737–762

    work page 2022

  27. [27]

    Dragičević, C

    D. Dragičević, C. Silva and H. Vilarinho, Admissibility and generalized nonuniform dichotomies for nonau- tonomous random dynamical systems , preprint, https://arxiv.org/abs/2410.1369

    work page arXiv

  28. [28]

    Henry, Geometric Theory of Semilinear Parabolic Equ ations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

    D. Henry, Geometric Theory of Semilinear Parabolic Equ ations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

    work page 1981

  29. [29]

    Jara and C

    N. Jara and C. A. Gallegos, Spectrum invariance dilemma for nonuniformly kinematical ly similar systems , Math. Ann. (2024). https://doi.org/10.1007/s00208-024- 02969-8

    work page doi:10.1007/s00208-024- 2024

  30. [30]

    Jiang, C

    Y. Jiang, C. Zhang and Z. Feng, A Perron-type theorem for nonautonomous differential equat ions with different growth rates , Discrete Contin. Dyn. Syst. S 10 (2017), 995–1008

    work page 2017

  31. [31]

    Lupa and L

    N. Lupa and L. Popescu, Generalized exponential behavior on the half-line via evol ution semigroups , Carpathian J. Math. 38 (2022), 691–705

    work page 2022

  32. [32]

    Lupa and L

    N. Lupa and L. Popescu, Generalized evolution semigroups and general dichotomies , Results Math. 78 (2023), Paper No. 112, 26 pp

    work page 2023

  33. [33]

    Kloeden and M

    P.E. Kloeden and M. Rasmussen, Nonautonomous Dynamica l Systems. Math. Surveys and Monogr. vol. 176, Amer. Math. Soc., Providence, RI 2011

    work page 2011

  34. [34]

    Massera and J

    J. Massera and J. Schäffer, Linear Differential Equation s and Function Spaces, Pure and Applied Mathe- matics 21, Academic Press, New York-London, 1966

    work page 1966

  35. [35]

    J. S. Muldowney, Dichotomies and asymptotic behaviour for linear differenti al systems , Trans. Amer. Math. Soc. 283 (1984), 465–484

    work page 1984

  36. [36]

    Naulin, M

    R. Naulin, M. Pinto, Roughness of (h, k)-dichotomies, J. Differential Equations 118 (1995), 20–35

    work page 1995

  37. [37]

    K. J. Palmer, A generalization of Hartman ’s linearization theorem , J. Math. Anal. Appl. 41 (1973), 753– 758

    work page 1973

  38. [38]

    Perron, Die stabilitätsfrage bei differentialgleichungen , Math

    O. Perron, Die stabilitätsfrage bei differentialgleichungen , Math. Z. 32 (1930), 703–728

    work page 1930

  39. [39]

    Pötzsche, Geometric Theory of Discrete Nonautonomo us Dynamical Systems

    C. Pötzsche, Geometric Theory of Discrete Nonautonomo us Dynamical Systems. Lecture Notes in Math- ematics vol. 2002, Springer, 2010

    work page 2002

  40. [40]

    G. R. Sell and Y. You, Dynamics of Evolutionary Equation s, Appl. Math. Sci. 143 Springer-Verlag, New York, 2002

    work page 2002

  41. [41]

    Silva, Admissibility and generalized nonuniform dichotomies for discrete dynamics, Comm

    C. Silva, Admissibility and generalized nonuniform dichotomies for discrete dynamics, Comm. Pure Appl. Anal. 20 (2021), 3419–3443

    work page 2021

  42. [42]

    Silva, Nonuniform µ-dichotomy spectrum and kinematic similarity , J

    C. Silva, Nonuniform µ-dichotomy spectrum and kinematic similarity , J. Differential Equations 375 (2023), 618–652. 29 F aculty of Mathematics, University of Rijeka, Croatia Email address, Davor Dragičević: ddragicevic@math.uniri.hr Centro de Matemática e Aplicações, Universidade da Beira Inte rior, 6201-001 Covilhã, Portugal Email address, César M. Silva:...

    work page 2023