2023 Spring Day Slides Outline

  1. Title

  2. Introduce the question
    • Decomposition, Ritt’s decomposition theorem and Laurent’s Theorem
    • Composition relation
    • near ring & semigroup
  3. Past result
    • $f^n = g^n$ implies they have the “same” julia set.
    • Even we knew the above properties, there is a huge gap to conclude that $<f,g>$ is a free genereated semigroup.
    • If $h_f\neq h_g$, then there exists an integer $N$ such that $<f^N, g^N>$ is free.
  4. Introduce height
    • Introduce naive height
    • Introduce dynamical canonical height
    • A test of the proof that shows the dynamical canonical height is well-defined
    • Northcott theorem
  5. Key Observation
    • Revisit the proof
    • How do we prove our past result
  6. Idea of Algorithm
    • We have right canceallation (No left canceallation)
    • Search on the tree
  7. Algorithm
    • algorithm
  8. Algorithm Complexity
    • estimate the complexity
  9. Experiment result
    • show datas
  10. Further questions
    • Investigate the gap of canonical heights
    • Show data