대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제60권3호
  • /
  • Pages.659-675
  • /
  • 2023
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

THE CRITICAL PODS OF PLANAR QUADRATIC POLYNOMIAL MAPS OF TOPOLOGICAL DEGREE 2

  • Misong Chang (Department of Mathematics Soongsil University) ;
  • Sunyang Ko (Department of Mathematics Soongsil University) ;
  • Chong Gyu Lee (Department of Mathematics Soongsil University) ;
  • Sang-Min Lee (Department of Mathematics Soongsil University)
  • 투고 : 2022.04.22
  • 심사 : 2022.10.28
  • 발행 : 2023.05.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let K be an algebraically closed field of characteristic 0 and let f be a non-fibered planar quadratic polynomial map of topological degree 2 defined over K. We assume further that the meromorphic extension of f on the projective plane has the unique indeterminacy point. We define the critical pod of f where f sends a critical point to another critical point. By observing the behavior of f at the critical pod, we can determine a good conjugate of f which shows its statue in GIT sense.

키워드

과제정보

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2016R1D1A1B01009208 and NRF-2021R1A6A1A10044154).

참고문헌

  1. E. Bedford, M. Lyubich, and J. Smillie, Polynomial diffeomorphisms of ℂ2. IV. The measure of maximal entropy and laminar currents, Invent. Math. 112 (1993), no. 1, 77-125. https://doi.org/10.1007/BF01232426
  2. E. Bedford, M. Lyubich, and J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of ℂ2, Invent. Math. 114 (1993), no. 2, 277-288. https://doi.org/10.1007/BF01232671
  3. E. Bedford and J. Smillie, Polynomial diffeomorphisms of ℂ2: currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), no. 1, 69-99. https://doi.org/10.1007/BF01239509
  4. E. Bedford and J. Smillie, Polynomial diffeomorphisms of ℂ2. II. Stable manifolds and recurrence, J. Amer. Math. Soc. 4 (1991), no. 4, 657-679. https://doi.org/10.2307/2939284
  5. E. Bedford and J. Smillie, Polynomial diffeomorphisms of ℂ2. III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann. 294 (1992), no. 3, 395-420. https://doi.org/10.1007/BF01934331
  6. E. Bedford and J. Smillie, External rays in the dynamics of polynomial automorphisms of ℂ2, in Complex geometric analysis in Pohang (1997), 41-79, Contemp. Math., 222, Amer. Math. Soc., Providence, RI, 1999. https://doi.org/10.1090/conm/222/03175
  7. J. Diller, R. Dujardin, and V. Guedj, Dynamics of meromorphic maps with small topological degree I: from cohomology to currents, Indiana Univ. Math. J. 59 (2010), no. 2, 521-561. https://doi.org/10.1512/iumj.2010.59.4023
  8. J. Diller, R. Dujardin, and V. Guedj, Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory, Ann. Sci. Ec. Norm. Super. (4) 43 (2010), no. 2, 235-278. https://doi.org/10.24033/asens.2120
  9. J. Diller, R. Dujardin, and V. Guedj, Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure, Comment. Math. Helv. 86 (2011), no. 2, 277-316. https://doi.org/10.4171/CMH/224
  10. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326. https://doi.org/10.2307/1970547
  11. H. Kwon, C. G. Lee, and S.-M. Lee, Regular polynomial automorphisms in the space of planar quadratic rational maps, J. Fixed Point Theory Appl. 22 (2020), no. 4, Paper No. 84, 12 pp. https://doi.org/10.1007/s11784-020-00819-z
  12. C. G. Lee, The equidistribution of small points for strongly regular pairs of polynomial maps, Math. Z. 275 (2013), no. 3-4, 1047-1072. https://doi.org/10.1007/s00209-013-1169-2
  13. C. G. Lee and J. H. Silverman, GIT stability of Henon maps, Proc. Amer. Math. Soc. 148 (2020), no. 10, 4263-4272. https://doi.org/10.1090/proc/15055
  14. D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994.
  15. J. H. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007. https://doi.org/10.1007/978-0-387-69904-2
  16. J. Xie, The dynamical Mordell-Lang conjecture for polynomial endomorphisms of the affine plane, Asterisque 394 (2017), vi+110 pp.