Value distribution of derivatives in polynomial dynamics
For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the ha...
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| Veröffentlicht in: | Ergodic theory and dynamical systems Jg. 41; H. 12; S. 3780 - 3806 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge, UK
Cambridge University Press
01.12.2021
Cambridge University Press (CUP) |
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| ISSN: | 0143-3857, 1469-4417 |
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| Abstract | For every
$m\in \mathbb {N}$
, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in
$\mathbb {C}\setminus \{0\}$
under the
$m$
th order derivatives of the iterates of a polynomials
$f\in \mathbb {C}[z]$
of degree
$d>1$
towards the harmonic measure of the filled-in Julia set of f with pole at
$\infty $
. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on
$\mathbb {P}^1(\overline {k})$
having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of
$\mathbb {C}^2$
has a given eigenvalue. |
|---|---|
| AbstractList | Abstract For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue. For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue. For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue. For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue. |
| Author | OKUYAMA, YÛSUKE VIGNY, GABRIEL |
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| Keywords | higher derivative 37P30 32H50 value distribution 37F10 Hénon map non-archimedean dynamics complex dynamics |
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| References | Jonsson (S014338572000125X_r19) 2015 S014338572000125X_r2 S014338572000125X_r1 S014338572000125X_r4 S014338572000125X_r3 S014338572000125X_r5 S014338572000125X_r14 S014338572000125X_r8 S014338572000125X_r15 S014338572000125X_r12 S014338572000125X_r13 Berkovich (S014338572000125X_r6) 1990 S014338572000125X_r16 Dinh (S014338572000125X_r11) 2014; 8 Gauthier (S014338572000125X_r17) 2017; 8 S014338572000125X_r10 S014338572000125X_r30 S014338572000125X_r25 Milnor (S014338572000125X_r20) 2006 S014338572000125X_r23 Rivera-Letelier (S014338572000125X_r26) 2003 S014338572000125X_r24 S014338572000125X_r29 Okuyama (S014338572000125X_r22) 2017 S014338572000125X_r27 S014338572000125X_r28 Chambert-Loir (S014338572000125X_r9) 2006; 595 Berteloot (S014338572000125X_r7) 2001 Hörmander (S014338572000125X_r18) 1983 S014338572000125X_r21 |
| References_xml | – ident: S014338572000125X_r10 doi: 10.1007/978-3-642-13171-4_4 – volume-title: Dynamics in One Complex Variable year: 2006 ident: S014338572000125X_r20 – volume: 8 start-page: 247 year: 2017 ident: S014338572000125X_r17 article-title: Distribution of points with prescribed derivative in polynomial dynamics publication-title: Riv. Math. Univ. Parma (N.S.) – ident: S014338572000125X_r5 doi: 10.1090/gsm/198 – ident: S014338572000125X_r14 doi: 10.1112/plms/pdp022 – ident: S014338572000125X_r30 doi: 10.1090/S1088-4173-2011-00229-3 – ident: S014338572000125X_r4 doi: 10.1007/BF02921791 – ident: S014338572000125X_r23 doi: 10.5186/aasfm.2017.4233 – ident: S014338572000125X_r3 doi: 10.1090/surv/159 – ident: S014338572000125X_r28 – ident: S014338572000125X_r24 – ident: S014338572000125X_r25 doi: 10.1017/CBO9780511623776 – ident: S014338572000125X_r8 doi: 10.1007/BF02591353 – volume-title: Spectral Theory and Analytic Geometry over Non-Archimedean Fields year: 1990 ident: S014338572000125X_r6 – ident: S014338572000125X_r13 doi: 10.1007/s00208-006-0751-x – volume: 595 start-page: 215 year: 2006 ident: S014338572000125X_r9 article-title: Mesures et équidistribution sur les espaces de Berkovich publication-title: J. Reine Angew. Math. – ident: S014338572000125X_r2 doi: 10.5802/aif.2196 – ident: S014338572000125X_r12 doi: 10.1007/b100262 – volume-title: The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis year: 1983 ident: S014338572000125X_r18 – ident: S014338572000125X_r1 doi: 10.1515/crll.2005.2005.585.61 – ident: S014338572000125X_r29 doi: 10.1112/plms/pds051 – ident: S014338572000125X_r15 doi: 10.1007/978-94-011-0934-5_4 – start-page: 205 volume-title: Berkovich Spaces and Applications year: 2015 ident: S014338572000125X_r19 doi: 10.1007/978-3-319-11029-5_6 – start-page: 147 volume-title: Geometric Methods in Dynamics. II year: 2003 ident: S014338572000125X_r26 – start-page: 55 volume-title: Algebraic Number Theory and Related Topics 2014 year: 2017 ident: S014338572000125X_r22 – ident: S014338572000125X_r27 doi: 10.1512/iumj.1997.46.1441 – ident: S014338572000125X_r21 doi: 10.2140/pjm.2016.280.141 – ident: S014338572000125X_r16 doi: 10.1007/978-1-4612-0041-3 – volume-title: Rudiments de Dynamique Holomorphe year: 2001 ident: S014338572000125X_r7 – volume: 8 start-page: 499 year: 2014 ident: S014338572000125X_r11 article-title: Rigidity of Julia sets for Hénon type maps publication-title: J. Mod. Dyn. |
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| Snippet | For every
$m\in \mathbb {N}$
, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in
$\mathbb... For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb... Abstract For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in... |
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| SubjectTerms | Automorphisms Eigenvalues Mathematical analysis Mathematics Number theory Original Article Polynomials Potential theory |
| Title | Value distribution of derivatives in polynomial dynamics |
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