Publications/Preprints

   2021--

• Salem/Pisot Numbers in the Weyl Spectrum, submitted for publication (2023), Arxiv version
• The Dynamical Degrees of Rational Surface Automorphisms, Conform. Geom. Dyn. 28 (2004), 62-87, Arxiv version
• An Automorphism group of a rational surface: Not too big not too small, Eur. J. Math 10 (2024), no.1, paper no 17, 25pp , Arxiv version
• (with Eric Klassen) Entropy of Real Rational Surface automorphisms: Actions on the Fundmental Groups, Geom. Dedicata 218 (2024), no. 2, paper no 37, 40 pp, short version , Arxiv version (long version)
• (with Jeffrey Diller) Entropy of real rational surface automorphisms, Experimental mathematics 30 (2021), no. 2 172-190 . online version

   Computer Codes

• Constructing a quadratic birational maps with given orbit data
  
  
  • SageMath : download notebooks
    
    This contains three source files and three running fiels.
    
    • NonDegenerate_Quad : Quadratic map with three distinct exceptional lines
    • Generic_Degenerate_Quad : Exceptional lines consists of one line with multiplicity 2 and one irrducible line
    • Degenerate_Degenerate_Quad : Exceptional lines are given by one line with multiplicity 3.
  
  
  
  • Mathematica : The following files will not check the availability of the orbit data.
    
    
    • degenerate map_rho means generic_degenerate and degenerate map_tau means degenerate_degenerate.
    • notebooks for mappings properly fixing a cubic with a cuspidal singularity
    • notebooks for mappings properly fixing a cubic with a conic with its tangent
    • notebooks for mappings properly fixing a cubic with three lines joining at one point

   2011--2020

• (with Eric Bedford) Julia sets for polynomial diffeomorphisms of C² are not seminanlytic, Documenta Mathematica, 24 (2019), 163-173. Arxiv version
• (with Eric Bedford) No smooth Julia sets for polynomial diffeomorphisms of C² with positive entropy, J. Geom. Anal. 27 (2017), 3085-3098 Arxiv version
• (with Eric Bedford and Jeffrey Diller) Pseudoautomorphisms with invariant curves, Complex Geometry and Dynamics, The Abel Symp. 2013, 1-27. Arxiv version
• (with Eric Bedford and Serge Cantat) Pseudo-automorphisms without dimension-reducing factors, J. Mod. Dyn. 8 (2014), no. 2, 221­250. Arxiv version ; A longer version is available here
• (with Eric Bedford) Pseudo-Automorphisms of 3-space: Periodicities and Positive Entropy in Linear Fractional Recurrences,Publ. Mat. 58 (2014), no. 1, 65-119 Arxiv version
• (with Eric Bedford) Dynamics of Rational Surface Automorphisms: Rotation Domains, Amer. J. Math. 134 (2012), no. 2, 379­405. Arxiv version
• How to Find Invariant Hypersurfaces for Birational Maps,roceedings of the International Workshop Future Directions in Difference Equations, 107-112, Universida de Vigo, Spain.
• (with Eric Bedford) Linear Fractional Recurrences: Periodicities and Integrability, Annales de la Facult\'{e} des Sciences de Toulouse, 33-56, Universit\'{e} Paul Sabatier, Toulouse, France. Arxiv version

   --2010

• (with Eric Bedford) Continuous Families of Rational Surface Automorphisms with Positive Entropy, Mathematische Annalen, 348:667-688 Arxiv version
• (with Eric Bedford) Dynamics of Rational Surface Automorphisms : Linear Fractional Recurrences, J. Geom. Anal. 19 (2009), no. 3, 553--583 preprint
• (with Nina Abarenkova et al.) Degree Complexity of a Family of Birational Maps, Math. Phys. Anal. Geom. 11 (2008), no. 1, 53--71 preprint
• (with Eric Bedford) The Number of Periodic Orbits of a Rational Difference Equation, To appear in the KiselmanFest volume Arxiv version
• (with Eric Bedford) Degree Growth of Matrix Inversion: Birational Maps of Symmetric, Cyclic Matrices, Discrete Contin. Dyn. Syst. 21 (2008), no. 4, 977--1013 Arxiv version
• (with Eric Bedford) Linear Recurrences in the Degree Sequences of Monomial Mappings, Ergodic Theory Dynam. Systems 28 (2008), no. 5, 1369--1375 Arxiv version
• (with Eric Bedford) Periodicities in Linear Fractional Recurrences: Degree growth of birational surface maps, Michigan Math. J. 54(3), 647-671 2006 Arxiv version
• (with Eric Bedford) On the Degree Growth of Birational Mappings in Higher Dimension, Journal of Geometric Analysis 14(4), 567-596 2004 Arxiv version


• (with Victor W. Goodman) Common Forward Rate Volatility, SIAM J. Finan. Math. 1, 3, 212-229.
• (with Jungmin Choi) The derivatives of Asian call option prices, Commun. Math. Sci. 6 (2008), no. 3, 557--568 preprint
• Moment Generating function of the inverse of integral of geometric Brownian Motion, Proc. Amer. Math. Soc. 132, 2753-2759 2004
• (with Victor W. Goodman) Exponential Martingales and Time Integrals of Brownian motion 2006 preprint preprint


• (with Dong-Uy Shin) The recursive dimension formula for graded Lie algebra and its applications, Communications in algebra, 27(6), 2627-2652 1999


• Integrals of Exponential Brownian motion and Derivative pricing in the log-normal bond model, Doctorate thesis 2003
• The recursive dimension formula for graded Lie algebra and its applications, Master's thesis 1997