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Blow-up of multipliers in meromorphic families of rational maps

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Pith's one-line read For any degenerating family of rational maps, either all periodic multipliers are uniformly bounded or, for every ε>0, at least (1−ε)d^n of the period-n points have multipliers growing like |t|^{−λ}.

desk verdict Strong dichotomy result upgrading Favre–Rivera-Letelier from positive proportion to (1−ε) proportion, but the proof leans on an unpublished preprint and the cubic case needs expert checking. read the letter →

arxiv 2504.20284 v1 pith:N7Z5P7FG submitted 2025-04-28 math.DS

Charles Favre This is my paper · ORCID
classification math.DS MSC 37F1037F4537P05
keywords rationalmapsmultipliersperiodiccyclesdegeneratingfamiliesnon-ArchimedeandynamicsBerkovichspaceLyapunovexponentmoduli
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a dichotomy for one-parameter holomorphic families of rational maps that degenerate at t=0. Along such a family, either the multipliers of all periodic cycles are bounded by a constant independent of the period and the parameter, or there is a rate λ>0 such that for every ε>0 and every n, at least (1−ε)d^n of the period-n points have multiplier at least C(ε,n)|t|^{−λ} as t→0. In other words, if multipliers blow up at all, they blow up for almost every periodic orbit, and the blow-up is at least polynomial in 1/|t|. The proof passes through the non-Archimedean limit of the family over the field of Laurent series, where the dichotomy becomes a statement about the Lyapunov exponent of the limit map: zero exponent gives bounded multipliers, positive exponent forces many repelling cycles with large non-Archimedean multipliers. The paper also shows that small periods suffice to detect degeneracy: in polynomial families a period-1 or period-2 multiplier must blow up, and in cubic rational families a period-3 multiplier must blow up.

What carries the argument

The load-bearing object is the non-Archimedean limit f_na, a rational map of degree d over the complete valued field C((t)), viewed on the Berkovich projective line. Its equilibrium measure carries a Lyapunov exponent λ(f_na)=∫log|df_na| dμ, and the dichotomy is carried by the equivalence that λ(f_na)=0 iff f_na is affine Bernoulli iff f_na has no repelling rigid periodic cycle. The second ingredient is a contraction lemma in the natural extension: almost every point admits analytic inverse branches of f_na^n shrinking at rate $Le^{{−nλ(f_na)/2}}$, which lets the proof count rigid periodic points with large multiplier. A Puiseux-series comparison lemma then converts the non-Archimedean growth into the complex multiplier lower bound |t|^{−λ}, giving the precise rate of blow-up.

What would settle it

Compute the multipliers of all period-1, period-2, and period-3 cycles in an explicit degenerating cubic rational family and check whether at least one has a pole at t=0; if none does, Theorem B fails. For Theorem A, take an explicit degenerating family whose non-Archimedean limit has positive Lyapunov exponent, count the period-n solutions of f_t^n(z)=z for small |t|, and verify that the proportion with max{1,|df_t^n(z)|}^{1/n} ≥ C|t|^{−λ} tends to 1; finding a proportion bounded away from 1 for every λ>0 would refute the dichotomy.

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Extended reading notes

Core claim

The central discovery is Theorem A: a degenerating family of rational maps of degree d≥2 over the unit disk falls into one of two exclusive cases. Either there is a constant C such that |df_t^n(z)|^{1/n}≤C for all |t|≤1/2, all n, and all period-n points z; or there is λ>0 such that for every ε>0 and every n, at least (1−ε)d^n of the period-n points satisfy max{1,|df_t^n(z)|}^{1/n}≥C(ε,n)|t|^{−λ}. This upgrades an earlier result that only a positive proportion of multipliers blow up in the second case. The proof identifies the rate λ with any number below half the non-Archimedean Lyapunov exponent λ(f_na) of the limit map f_na over the field C((t)). When λ(f_na)>0, a contraction lemma for inverse branches in the natural extension yields at least (1−ε)d^n rigid periodic points of period n with non-Archimedean multiplier at least A $e^{{nλ(f_na)/2}}$; a Puiseux-series comparison transfers this into the complex lower bound |t|^{−λ}. When λ(f_na)=0, the limit is affine Bernoulli and all complex multipliers stay bounded. This mechanism makes blow-up generic rather than exceptional.

Load-bearing premise

The dichotomy rests on the imported contraction lemma that almost every point in the natural extension of the non-Archimedean limit admits inverse branches shrinking at rate $Le^{{−nλ(f_na)/2}}$, together with the equivalence between zero Lyapunov exponent, having no repelling rigid cycle, and having all complex multipliers bounded; should either estimate fail in the stated strength, the rate |t|^{−λ} and the (1−ε)d^n count would not follow from the proof.

Editorial extensions

If this is right

  • In any degenerating family, either no multiplier grows at all or a fraction tending to 1 of period-n multipliers grows at least polynomially in 1/|t|, so intermediate behavior with only a positive proportion blowing up is impossible.
  • For polynomial families of any degree, degeneracy forces a period-1 or period-2 multiplier to have a pole at t=0; for cubic rational families, period at most 3 suffices.
  • Consequently, the multiplier spectrum map on the relevant moduli spaces is proper and birational: the boundary at infinity is detected by small-period multipliers.
  • In stable polynomial families whose non-Archimedean limit has positive Lyapunov exponent and no recurrent critical points, the blow-up rate |t|^{−λ} holds uniformly for all large n with a constant independent of n.
  • The rate of blow-up is governed by the non-Archimedean Lyapunov exponent: any λ below λ(f_na)/2 can be chosen, linking the complex multiplier growth to the expansion rate of the limit map.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dichotomy suggests a general principle for degenerating rational maps: the parameter-space boundary is governed by large multipliers, so compactness statements for moduli spaces should follow from controlling small-period multipliers.
  • One might expect a quantitative refinement: the optimal blow-up exponent should be exactly λ(f_na)/2, with the proportion (1−ε)d^n replaced by an error term depending on the mixing rate of the non-Archimedean equilibrium measure.
  • If the paper's conjecture on rigid repelling cycles of bounded period holds in every degree, then the multiplier map on moduli space would be proper in all degrees, giving a uniform finite-type description of degeneracy.
  • The affine Bernoulli exceptional case, where all multipliers stay bounded, suggests a dictionary between non-Archimedean entropy and the asymptotic growth of complex multiplier spectra that could be tested numerically on families such as z^d + t z^{−d}.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper studies one-parameter holomorphic degenerating families \{f_t\} of rational maps of degree d≥2 on the Riemann sphere. The main result, Theorem A, asserts a dichotomy: either all multipliers of periodic points satisfy a uniform bound |(f_t^n)'(z)|^{1/n}≤C for |t|≤1/2, or there exists λ>0 such that for every ε>0 and every n, at least (1−ε)d^n of the period-n points have max{1,|(f_t^n)'(z)|}^{1/n} ≥ C(ε,n)|t|^{-λ}. The proof passes to the non-Archimedean limit f_na over C((t)) and uses its Lyapunov exponent λ(f_na), together with a quantitative equidistribution of rigid repelling cycles (Theorem 1.2). The paper also proves Theorem B (existence of repelling rigid cycles of period ≤2 for polynomials and ≤3 for cubic rational maps, with fixed-point statements for quadratic and cubic-polynomial cases) and Corollary C (properness and birationality of certain multiplier maps on moduli spaces). The non-Archimedean results for polynomials (Theorem 3.1) and cubic maps (Theorem 3.2) are proved within the paper, and several known results are used and contextualized.

Significance. If correct, Theorem A is a substantial advance: it upgrades the positive-proportion blow-up in [FavRL24] to a (1−ε) proportion and gives a quantitative exponent, showing that multiplier blow-up is typical in degenerating families. The proof strategy via the non-Archimedean limit and the Lyapunov exponent is elegant and well-motivated, and the paper is clearly written. The self-contained portions — in particular the counting proof for polynomial cycles (Theorem 3.1) and the case analysis for cubic maps (Theorem 3.2) — are valuable. However, the central quantitative engine (Theorem 1.2) rests on Lemma 1.3, which is imported from the unpublished preprint [FavRL24] without proof, and the zero-Lyapunov alternative of Theorem A also relies on [FavRL24, Théorème 4.4] and Theorem 1.1 of that preprint. In addition, the proof of Theorem 1.2 contains a shift-index error in the mixing argument. These issues must be addressed before the main theorem can be fully verified.

major comments (3)
  1. [§1.3, Lemma 1.3] The proof of Theorem 1.2 and hence of the blow-up alternative in Theorem A(2) is entirely dependent on Lemma 1.3, which is stated without proof and attributed to the unpublished preprint [FavRL24]. The exact contraction rate e^{-nλ(f)/2} and the full-measure union over L,τ in Lemma 1.3 are what force the exponent λ<λ(fna)/2 and the proportion (1−ε)d^n in Theorem A(2); any weakening of the rate (e.g., e^{-n(λ/2−δ)}) or a failure of the full-measure property would degrade both the exponent and the count. Because [FavRL24] is not part of the present manuscript, the central quantitative claim is not self-contained and cannot be checked from the text as it stands.
  2. [§2.1, first paragraph] The first alternative of Theorem A is obtained by invoking [FavRL24, Théorème 4.4], which itself relies on the equivalence in Theorem 1.1 (λ=0 iff affine Bernoulli iff no repelling rigid cycle). These results are also imported from the unpublished preprint [FavRL24] and are not proved or stated in sufficient detail in the present paper. The dichotomy is therefore conditional on external results; the author should either include proofs of the relevant statements or state Theorem A with an explicit dependence on [FavRL24].
  3. [§1.3, proof of Theorem 1.2] The mixing argument uses the set σ^n(\hat B(x)) ∩ \hat B(x) ∩ E_{L,τ}, but Lemma 1.3 provides an inverse branch from y_0 to y_{-n}, and the final ball B(y_{-n}, L e^{-nλ/2}) must be contained in B(x,r). This requires y_{-n}∈B(x,r), i.e., \hat y ∈ σ^{-n}(\hat B(x)). The set σ^n(\hat B(x)) only gives y_n∈B(x,r), which does not control y_{-n}. As written, the covering argument does not go through. If the intended set is σ^{-n}(\hat B(x)), the proof should be corrected; this is load-bearing because it produces the counting estimate (1−ε)d^n in Theorem 1.2.
minor comments (4)
  1. [§1.3, Theorem 1.2] The theorem states 'rigid periodic points z of period n', but the proof counts periodic points of period dividing n. The difference is O(d^{n/2}) and does not affect the asymptotic statement, but the wording should be aligned.
  2. [§2.1, final paragraph] The sentence 'Reducing A_+ if necessary, we may suppose this is true over |t|≤1/2' is potentially confusing; since the constant is allowed to depend on n and ε, it may help to spell out that one decreases C to handle the range r_n≤|t|≤1/2.
  3. [Throughout] Typos such as 'holomomorphic' (§1.4), 'We dont know' (Introduction), and 'mulitpliers' (Introduction) should be corrected.
  4. [§3.2] The proof of Theorem 3.2 is long and contains many cases; the reference to Figure 1 is helpful but the figure itself is not described in the caption, making it hard to follow. A more detailed caption or a short overview of the cases would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the blow-up dichotomy is derived from non-Archimedean dynamics, not from its own conclusion.

full rationale

The paper's central theorem A is not assumed among its hypotheses. The zero-Lyapunov half is imported from [FavRL24, Théorème 4.4], and the positive half is proved by applying Theorem 1.2, whose proof begins from Lemma 1.3 (the contraction-rate statement imported from FavRL24) and then derives, via mixing and the non-Archimedean Schwarz lemma, the lower bound |(df^n)(z)| ≥ (r/L)e^{nλ(f)/2}. Nothing in this chain is equivalent by construction to the theorem: the non-Archimedean Lyapunov exponent is defined independently of complex multipliers, the contraction lemma is a statement about inverse branches in the Berkovich tree, and the count (1−ε)d^n is obtained from full-measure mixing rather than from the statement of Theorem A. The dependency on unpublished [FavRL24] for Lemma 1.3 and Theorem 1.1 is a verification/correctness risk—if the contraction rate in Lemma 1.3 were weaker, the exponent in Theorem A(2) would degrade—but that is not circularity: the cited results are previous theorems with different content, not the present conclusion. Theorem B and Corollary C likewise use earlier results (Huguin, Gotou, Milnor, Trucco) as ingredients rather than assuming their own outputs. No step reduces to its input by definition or renames a fitted parameter as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The reader pays for the established Berkovich machinery and for several deep results quoted without proof from [FavRL24] (Theorems 1.1, the contraction lemma 1.3, and the positive-proportion blow-up theorem), as well as standard theorems by Rivera-Letelier and Faber. No free parameters are fitted to data, and no new entities are introduced. The residual characteristic zero condition is an explicit domain assumption, not a hidden one.

assumptions (6)
  • standard math The Berkovich projective line P^{1,an}_L and the dynamics of rational maps over it, including the equilibrium measure, Julia set, and type II points, as developed in [BR10] and [Ben19].
    Used throughout Section 1.1 to define f_na, the equilibrium measure, the Julia set, rigid points, and local degrees.
  • standard math The Lyapunov exponent lambda(f_na) over C((t)) is non-negative, and lambda(f_na)=0 iff f_na is affine Bernoulli iff f_na has no repelling rigid periodic cycle, from [Ok15, FavRL24] (Theorem 1.1 here).
    This equivalence is the pivot of Theorem A: the lambda=0 case yields the uniform bound (Case 1) and the lambda>0 case yields blow-up (Case 2).
  • standard math Faber's structure theorem for the critical set C_f subset of P^{1,an} as a tree with endpoints at rigid critical points [Fab13, Proposition 6.9].
    Used in the proof of Theorem 3.2 to control the local degrees along segments of the Berkovich tree.
  • standard math Rivera-Letelier's theorem on the existence of a fixed type II point with local degree at least 2 for maps without potential good reduction [Ben19, Theorem 12.5].
    Used at the start of the proof of Theorem 3.2 to obtain the type II fixed point x with local degree at least 2.
  • domain assumption The ground field C((t)) and its completion L have residual characteristic 0.
    Used to ensure lambda(f_na) is non-negative, to count fixed points and critical points inside balls as in Theorem 3.1, and to guarantee tameness of the maps. The paper explicitly restricts to residual characteristic 0 in the introduction and Theorems 3.1 and 3.2.
  • standard math Artin approximation theorem [Ar68] for the convergence of formal Puiseux series solutions to analytic equations.
    Used in Section 1.5 to attach an analytic curve C_{\hat{z}} to a rigid periodic point of f_na, which is the bridge to complex multipliers in Lemma 1.7.

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Pith. "Pith review of Blow-up of multipliers in meromorphic families of rational maps." pith.science (2026). https://pith.science/paper/N7Z5P7FG

@misc{pith2026250420284,
  author       = {Pith},
  title        = {Pith review of: Blow-up of multipliers in meromorphic families of rational maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7Z5P7FG}},
  note         = {Machine review of arXiv:2504.20284}
}
read the original abstract

We study the blow-up of the multipliers of periodic cycles in one-parameter holomorphic degenerating families of rational maps of the Riemann sphere.

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Works this paper leans on

59 extracted references · 54 canonical work pages

  1. [1]

    M. Artin. On the solutions of analytic equations. Invent. Math., 5, (1968), 277--291

    work page 1968

  2. [2]

    M. Baker. A finiteness theorem for canonical heights attached to rational maps over function fields. J. Reine Angew. Math. 626 (2009) 205–233

    work page 2009

  3. [3]

    Baker and R

    M. Baker and R. Rumely. Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs, 159. American Mathematical Society, Providence, RI, 2010. xxxiv+428 pp

    work page 2010

  4. [4]

    Benedetto

    R. Benedetto. Dynamics in One Non-Archimedean Variable. Graduate Studies in Mathematics 198 (Amer. Math. Soc. Providence RI, 2019)

    work page 2019

  5. [5]

    Bergeron, K

    M. Bergeron, K. Filom, and S. Nariman. Topological aspects of the dynamical moduli space of rational maps. Adv. Math.397 (2022), Paper No. 108209, 40 pp

    work page 2022

  6. [6]

    Berkovich

    V.G. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields. Math. Surveys Monographs 33 (Amer. Math. Soc. Providence RI, 1990)

    work page 1990

  7. [7]

    Berteloot

    F. Berteloot. Bifurcation currents in holomorphic families of rational maps. Lecture Notes in Mathematics 2075 CIME Fundation subseries. Springer Verlag pp.1--93 (2013)

    work page 2013

  8. [8]

    Briend et J

    J.-Y. Briend et J. Duval. Exposants de Liapounoff et distribution des points p\'eriodiques d'un endomorphisme de CP k . Acta Math. 182 (1999), no. 2, 143--157

    work page 1999

Show all 59 references
  1. [9]

    Cornfeld, S.V

    I.P. Cornfeld, S.V. Fomin, and Y.G. Sinai. Ergodic theory. Grund. math. wissen. 245, Springer Verlag, Berlin, Heidelberg, New York, 1982

    1982

  2. [10]

    L. DeMarco. Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity. Math. Ann. 326(1) (2003), 43–73

    2003

  3. [11]

    L. DeMarco. Iteration at the boundary of the space of rational maps. Duke Math. J.130 (2005), no.1, 169–197

    2005

  4. [12]

    L. DeMarco. The moduli space of quadratic rational maps. J. Amer. Math. Soc.20 (2007), no.2, 321–355

    2007

  5. [13]

    L. DeMarco. Bifurcations, intersections, and heights. Algebra Number Theory 10 (2016), 1031–1056

    2016

  6. [14]

    DeMarco, and X

    L. DeMarco, and X. Faber. Degenerations of complex dynamical systems. Forum Math. Sigma 2 (2014), e6, 36 pp

    2014

  7. [15]

    DeMarco, and X

    L. DeMarco, and X. Faber. Degenerations of complex dynamical systems II: Analytic and algebraic stability. Avec un appendice de Jan Kiwi. Math. Ann., 365 (2016), 1669-1699

    2016

  8. [16]

    DeMarco, and C

    L. DeMarco, and C. T. McMullen. Trees and the dynamics of polynomials. Ann. Sci. \'Ec. Norm. Sup\'er. (4) 41(3) (2008), 337--382

    2008

  9. [17]

    Devaney, D

    R. Devaney, D. Look, and D. Uminsky. The escape trichotomy for singularly perturbed rational maps. Indiana Univ. Math. J. 54, No. 6, 1621-1634 (2005)

    2005

  10. [18]

    X. Faber. Topology and geometry of the Berkovich ramification locus for rational functions I. Manuscripta Math. 142, 439--474 (2013)

    2013

  11. [19]

    X. Faber. Topology and geometry of the Berkovich ramification locus for rational functions II. Math. Ann. 356, 819--844 (2013)

    2013

  12. [20]

    C. Favre. Degenerations of endomorphisms of the complex projective space in the hybrid space. Journal de l’IMJ. 19(4), 1141--1183 (2018)

    2018

  13. [21]

    Favre et T

    C. Favre et T. Gauthier. The arithmetic of polynomial dynamical pairs.. Annals of Mathematics Studies 214. Princeton, NJ: Princeton Univer- sity Press (ISBN 978-0-691-23546-2/hbk; 978-0-691-23547-9/pbk). 252 p. (2022)

    2022

  14. [22]

    Favre et C

    C. Favre et C. Gong. Non-Archimedean techniques and dynamical degenerations. To appear in Peking Math. J. (2025). arXiv:2406.15892

    2025 arXiv

  15. [23]

    Favre et J

    C. Favre et J. Rivera-Letelier. Th \'e orie ergodique des fractions rationnelles sur un corps ultra-m\'etrique. Proc. London Math Soc. (3) 100 (2010), no. 1, 116--154

    2010

  16. [24]

    Favre et J

    C. Favre et J. Rivera-Letelier. Expansion et entropie en dynamique non-archimédienne. Preprint

  17. [25]

    Fujimura

    M. Fujimura. Projective moduli space for the polynomials. Dynamics of Continuous, Discrete and Impulsive Systems Series A 13 (2006), no. 6, 787

    2006

  18. [26]

    Fujimura, and K

    M. Fujimura, and K. Nishisawa. Projective Moduli Space of the Polynomials: Cubic Case. Josai mathematical monographs Vol. 2 (2000) pp. I - 10

    2000

  19. [27]

    Fujimura, and M

    M. Fujimura, and M. Taniguchi. A compactification of the moduli space of polynomials. Proc. Amer. Math. Soc.136 (2008), no.10, 3601–3609

    2008

  20. [28]

    Gauthier, Y

    T. Gauthier, Y. Okuyama et G. Vigny. Hyperbolic components of rational maps: Quantitative equidistribution and counting. Commentarii Math. Helv., Volume 94, Issue 2, (2019), pp. 347--398

    2019

  21. [29]

    Gauthier, Y

    T. Gauthier, Y. Okuyama et G. Vigny. Approximation of non-archimedean Lyapunov exponents and applications over global fields. Trans. Amer. Math. Soc. 373 (2020), no. 12, 8963–-9011

    2020

  22. [30]

    C. Gong. Multipliers of rational maps and rescaling limits. Preprint

  23. [31]

    R. Gotou. Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps. Preprint (2023). arXiv:2309.15404

    2023 arXiv

  24. [32]

    J.-H. Hubbard. Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory. Ithaca, NY: Matrix Editions (ISBN 0-9715766-2-9/hbk). xx, 459 p. (2006)

    2006

  25. [33]

    V. Huguin. Multipliers at small cycles for degenerating sequences of polynomials. Preprint (2023)

    2023

  26. [34]

    V. Huguin. Moduli spaces of polynomial maps and multipliers at small cycles. Preprint (2024)

    2024

  27. [35]

    Hutz, and M

    B. Hutz, and M. Tepper. Multiplier spectra and the moduli space of degree 3 morphisms on P ^1 . Journal of Algebra, Number Theory and Applications 29 (2013), no. 2, 189–206

    2013

  28. [36]

    R. Irokawa. Ramification loci of non-Archimedean cubic rational functions. Preprint (2021). arXiv:2107.06358

    2021 arXiv

  29. [37]

    Ji, and J

    Z. Ji, and J. Xie. Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics Forum of Mathematics, Pi , Volume 11 , 2023 , e11

    2023

  30. [38]

    Ji, and J

    Z. Ji, and J. Xie. The multiplier spectrum morphism is generically injective. Preprint (2023). arXiv:2309.15382

    2023

  31. [39]

    J. Kiwi. Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier (Grenoble), 56(5): 1337--1404, (2006)

    2006

  32. [40]

    J. Kiwi. Puiseux series dynamics of quadratic rational maps. Israel J. Math. 201 (2014), no. 2, 631--700

    2014

  33. [41]

    J. Kiwi. Rescaling limits of complex rational maps. Duke Math. J. 164 (7) 1437 - 1470, 15. 2015

    2015

  34. [42]

    M. Ju. Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynamical Systems 3 (1983)

    1983

  35. [43]

    A. Levy. The space of morphisms on projective space. Acta Arith.146 (2011), no.1, 13–31

    2011

  36. [44]

    Y. Luo. Limits of rational maps, -trees and barycentric extension. Adv. Math. 393 (2021), Paper No. 108075, 46 pp

    2021

  37. [45]

    Y. Luo. Trees, length spectra for rational maps via barycentric extensions and Berkovich spaces. Duke Math. J. 171(14): 2943-3001 (2022)

    2022

  38. [46]

    McMullen

    C.T. McMullen. Automorphisms of rational maps . In Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ. 10, Springer, New York, 1988

    1988

  39. [47]

    J. Milnor. Geometry and dynamics of quadratic rational maps. With an appendix by the author and Lei Tan. Experiment. Math. 2 (1993), no. 1, 37 -- 83

    1993

  40. [48]

    J. Milnor. On Latt\`es maps. Dynamics on the Riemann sphere, 9--43, Eur. Math. Soc., Z\"urich, 2006

    2006

  41. [49]

    H. Nie. Non-Archimedean Lyapunov exponents of polynomials. Preprint (2022). arXiv:2202.13550

    2022 arXiv

  42. [50]

    Y. Okuyama. Quantitative approximations of the Lyapunov exponent of a rational function over valued fields. Mathematische Zeitschrift, (2015), Volume 280, Issue 3--4, pp 691--706

    2015

  43. [51]

    Petsche, L

    C. Petsche, L. Szpiro, and M. Tepper. Isotriviality is equivalent to potential good reduction for endomorphisms of P ^N over function fields. J. Algebra 322 (2009), no.9, 3345–3365

    2009

  44. [52]

    J. Poineau. Dynamique analytique sur Z I: Mesures d'équilibre sur une droite projective relative . Preprint (2022). arXiv:2201.08480

    2022 arXiv

  45. [53]

    Qiu, Weiyuan, X

    W. Qiu, Weiyuan, X. Wang, and Y. Yin. Dynamics of McMullen maps. Adv. Math. 229, No. 4, 2525-2577 (2012)

    2012

  46. [54]

    Rivera-Letelier

    J. Rivera-Letelier. Points p\'eriodiques des fonctions rationnelles dans l'espace hyperbolique p -adique. Comment. Math. Helv., 80(3):593--629, 2005

    2005

  47. [55]

    J. Schmitt. A compactification of the moduli space of self-maps of CP ^1 via stable maps. Conform. Geom. Dyn.21 (2017), 273–318

    2017

  48. [56]

    Silverman

    J. Silverman. The space of rational maps on P ^1 . Duke Math. J.94 (1998), no.1, 41–77

    1998

  49. [57]

    Sugiyama

    T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math.322 (2017), 132–185

    2017

  50. [58]

    E. Trucco. Wandering Fatou components and algebraic Julia sets. Bull. Soc. Math. France 142 (2014), no. 3, 411--464

    2014

  51. [59]

    L. W. West. The moduli space of cubic rational maps. Preprint (2014). arXiv:1408.3247

    2014 arXiv

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