Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. Baker[1]; Robert Rumely[2]

  • [1] Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA
  • [2] University of Georgia, Department of Mathematics, Athens, GA 30602-7403, USA

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 625-688
  • ISSN: 0373-0956

Abstract

top
Given a rational function ϕ ( T ) on ℙ 1 of degree at least 2 with coefficients in a number field k , we show that for each place v of k , there is a unique probability measure μ ϕ , v on the Berkovich space ℙ Berk , v 1 / ℂ v such that if { z n } is a sequence of points in ℙ 1 ( k ¯ ) whose ϕ -canonical heights tend to zero, then the z n ’s and their Gal ( k ¯ / k ) -conjugates are equidistributed with respect to μ ϕ , v .The proof uses a polynomial lift F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) ) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ , v ( x , y ) for each v . The measure μ ϕ , v is obtained by taking the Berkovich space Laplacian of g ϕ , v ( x , y ) . The main ingredients in the proof are an energy minimization principle for g ϕ , v ( x , y ) and a formula for the homogeneous transfinite diameter of the v -adic filled Julia set K F , v ⊂ ℂ v 2 for each place v .

How to cite

top

Baker, Matthew H., and Rumely, Robert. "Equidistribution of Small Points, Rational Dynamics, and Potential Theory." Annales de l’institut Fourier 56.3 (2006): 625-688. <http://eudml.org/doc/10160>.

@article{Baker2006,
abstract = {Given a rational function $\varphi (T)$ on $\mathbb\{P\}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure $\mu _\{\varphi ,v\}$ on the Berkovich space $\mathbb\{P\}^1_\{\rm Berk,v\} / \mathbb\{C\}_v$ such that if $\lbrace z_n \rbrace $ is a sequence of points in $\mathbb\{P\}^1(\overline\{k\})$ whose $\varphi $-canonical heights tend to zero, then the $z_n$’s and their $\{\rm Gal\}(\overline\{k\}/k)$-conjugates are equidistributed with respect to $\mu _\{\varphi ,v\}$.The proof uses a polynomial lift $F(x,y) = (F_1(x,y),F_2(x,y))$ of $\varphi $ to construct a two-variable Arakelov-Green’s function $g_\{\varphi ,v\}(x,y)$ for each $v$. The measure $\mu _\{\varphi ,v\}$ is obtained by taking the Berkovich space Laplacian of $g_\{\varphi ,v\}(x,y)$. The main ingredients in the proof are an energy minimization principle for $g_\{\varphi ,v\}(x,y)$ and a formula for the homogeneous transfinite diameter of the $v$-adic filled Julia set $K_\{F,v\} \subset \mathbb\{C\}_v^2$ for each place $v$.},
affiliation = {Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA; University of Georgia, Department of Mathematics, Athens, GA 30602-7403, USA},
author = {Baker, Matthew H., Rumely, Robert},
journal = {Annales de l’institut Fourier},
keywords = {Canonical heights; rational dynamics; equidistribution; arithmetic dynamics; potential theory; capacity theory; canonical heights},
language = {eng},
number = {3},
pages = {625-688},
publisher = {Association des Annales de l’institut Fourier},
title = {Equidistribution of Small Points, Rational Dynamics, and Potential Theory},
url = {http://eudml.org/doc/10160},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Baker, Matthew H.
AU - Rumely, Robert
TI - Equidistribution of Small Points, Rational Dynamics, and Potential Theory
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 625
EP - 688
AB - Given a rational function $\varphi (T)$ on $\mathbb{P}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure $\mu _{\varphi ,v}$ on the Berkovich space $\mathbb{P}^1_{\rm Berk,v} / \mathbb{C}_v$ such that if $\lbrace z_n \rbrace $ is a sequence of points in $\mathbb{P}^1(\overline{k})$ whose $\varphi $-canonical heights tend to zero, then the $z_n$’s and their ${\rm Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to $\mu _{\varphi ,v}$.The proof uses a polynomial lift $F(x,y) = (F_1(x,y),F_2(x,y))$ of $\varphi $ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure $\mu _{\varphi ,v}$ is obtained by taking the Berkovich space Laplacian of $g_{\varphi ,v}(x,y)$. The main ingredients in the proof are an energy minimization principle for $g_{\varphi ,v}(x,y)$ and a formula for the homogeneous transfinite diameter of the $v$-adic filled Julia set $K_{F,v} \subset \mathbb{C}_v^2$ for each place $v$.
LA - eng
KW - Canonical heights; rational dynamics; equidistribution; arithmetic dynamics; potential theory; capacity theory; canonical heights
UR - http://eudml.org/doc/10160
ER -

References

top
  1. P. Autissier, Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math. 531 (2001), 201-235 Zbl1007.11041MR1810122
  2. M. Baker, L. C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005), 61-92 Zbl1071.11040MR2164622
  3. M. Baker, R. Rumely, Harmonic analysis on metrized graphs Zbl1123.43006
  4. V. G. Berkovich, Spectral theory and analytic geometry over nonarchimedean fields, 33 (1990), AMS Mathematical Surveys and Monographs Zbl0715.14013MR1070709
  5. Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), 465-476 Zbl0918.11035MR1470340
  6. E. Bombieri, Subvarieties of linear tori and the unit equation: a survey, Analytic Number Theory 247 (1996), 1-20, Cambridge Univ. Press, Cambridge Zbl0923.11097MR1694981
  7. G. Call, S. Goldstine, Canonical heights on projective space, Journal of Number Theory 63 (1997), 211-243 Zbl0895.14006MR1443758
  8. G. Call, J. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), 163-205 Zbl0826.14015MR1255693
  9. A. Chambert-Loir, Equidistribution of small points in finite fibers Zbl1302.37065
  10. T. Chinburg, Capacity theory on varieties, Compositio Math. 80 (1991), 71-84 Zbl0761.11028MR1127060
  11. T. Chinburg, R. Rumely, The capacity pairing, J. Reine Angew. Math. 434 (1993), 1-44 Zbl0756.14013MR1195689
  12. L. DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and metrics on the sphere, Mathematische Annalen 326 (2003), 43-73 Zbl1032.37029MR1981611
  13. G. Faltings, Calculus on arithmetic surfaces, Annals of Math. 119 (1984), 387-424 Zbl0559.14005MR740897
  14. C. Favre, M. Jonsson, The valuative tree, Lecture Notes in Mathematics 1853 (2004), Springer-Verlag, Berlin and New York Zbl1064.14024MR2097722
  15. C. Favre, J. Rivera-Letelier, Equidistribution des points de petite hauteur 
  16. C. Favre, J. Rivera-Letelier, Théorème d’équidistribution de Brolin en dynamique p -adique, C. R. Math. Acad. Sci. 339 (2004), 271-276 Zbl1052.37039MR2092012
  17. A. Freire, A. Lopes, R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62 Zbl0568.58027MR736568
  18. J. H. Hubbard, P. Papadapol, Superattractive fixed points in ℂ n , Indiana Univ. Math. J. 43 (1994), 321-365 Zbl0858.32023MR1275463
  19. M. Klimek, Pluripotential Theory, 6 (1991), Oxford Science Publications Zbl0742.31001MR1150978
  20. S. Lang, Arakelov Theory, (1988), Springer–Verlag Zbl0667.14001MR969124
  21. M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems 3 (1983), 351-385 Zbl0537.58035MR741393
  22. V. Maillot, Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, 80 (2000), Mémoires de la SMF, Paris Zbl0963.14009
  23. J. Milnor, Dynamics in One Complex Variable, (2000), Vieweg Zbl0972.30014MR1721240
  24. J. Pineiro, L. Szpiro, T. Tucker, Mahler measure for dynamical systems on ℙ 1 and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory 235 (2004), 219-250, Birkhaüser Zbl1101.11020MR2166086
  25. T. Ransford, Potential Theory in the Complex Plane, 28 (1995), London Math. Soc. Zbl0828.31001MR1334766
  26. J. Rivera-Letelier, Théorie de Fatou et Julia dans la droite projective de Berkovich 
  27. J. Rivera-Letelier, Dynamique des fonctions rationelles sur les corps locaux, Astérisque 287 (2003), 147-230 Zbl1140.37336MR2040006
  28. H. L. Royden, Real Analysis, (1988), MacMillan Publishing Co., New York Zbl0704.26006MR1013117
  29. W. Rudin, Real and Complex Analysis, (1974), McGraw-Hill, New York Zbl0278.26001MR344043
  30. R. Rumely, Capacity Theory on Algebraic Curves, Lecture Notes in Mathematics 1378 (1989), Springer-Verlag, Berlin-Heidelberg-New York Zbl0679.14012MR1009368
  31. R. Rumely, An intersection theory for curves, with analytic contributions from nonarchimedean places, Canadian Mathematical Society Conference Proceedings 15 (1995), 325-357, AMS Zbl0859.11037MR1353942
  32. R. Rumely, On Bilu’s equidistribution theorem, Contemp. Math. 237 (1999), 159-166 Zbl1029.11030MR1710794
  33. R. Rumely, M. Baker, Analysis and dynamics on the Berkovich projective line Zbl1196.14002
  34. R. Rumely, C.F. Lau, Arithmetic capacities on ℙ n , Math. Zeit. 215 (1994), 533-560 Zbl0794.31008MR1269489
  35. R. Rumely, C.F. Lau, R. Varley, Existence of the Sectional Capacity, 145 (2000), American Mathematical Society, Providence, R.I. Zbl0987.14018MR1677934
  36. J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, (1994), Springer-Verlag, Berlin and New York Zbl0911.14015MR1312368
  37. L. Szpiro, E. Ullmo, S. Zhang, Équirépartition des petits points, Invent. Math. 127 (1997), 337-347 Zbl0991.11035MR1427622
  38. A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, (2005) 
  39. M. Tsuji, Potential Theory in Modern Function Theory, (1959), Maruzen, Tokyo Zbl0087.28401MR114894
  40. B. L. van der Waerden, Algebra, 1 (1970), New York Zbl0137.25403
  41. S. Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), 171-193 Zbl0795.14015MR1207481

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.