Why is it so hard to find any composition relation between two rational functions?

Wayne Peng

joint with T. Tucker, and S.-F. Tsai

60 TMS Annual Meeting

Growth rate

Let $\mathcal{R}$ be the set of composition relations. The existence of any relation between generators results in a reduction in the count of distinct elements within $\Sigma^*_n$, leading to the inequality $\#\Sigma^*_n/\mathcal{R}\leq (\#\Sigma)^n$.

Let $\Sigma=\{f_1,\ f_2,\ \ldots,\ f_m\}$, and assume $f_if_j=f_jf_i$ for all $i$ and $j$.

  • For $n=2$, $\#\Sigma_n^*/\mathcal{R}=n+1$.
  • For $n>2$, $\#\Sigma_n^*/\mathcal{R}=H^m_n$.

  • Monomial $x^n$: The Julia set is the unit circle centered at the origin (0,0).
  • Chebyshev polynomial: The Julia set is the closed interval $[-2,2]$
  • Lattès map: The Julia set is the entire space.

The only remaining case is when we have a semigroup generated by monomials like $$ \langle \zeta_1x^{n_1},\zeta_2x^{n_2},\ldots,\zeta_mx^{n_m}\rangle. $$

A finitely generated semigroup $S=\Sigma^*_{<\infty}/\mathcal{R}$ is a semigroup with the following properties:

  1. Let equivalent classes $[w_1]$ and $[w_2]$ belong to $G$. If $[w_1]=[w_2]$, then $|w_1|_s = |w_2|_s$ (counting the number of characters to be $s$) for all $s\in S$ (If two composition are the same, then they use the same number of $f$ and $g$.)
  2. There exists a positive integer $N$ such that $\#\mathcal{U}_I/\mathcal{R}\leq N$ for all indices $I$ (Number of possible composition result is bounded.)
  3. The semigroup $G$ has one-side cancellation property.

:) - Tsai - Tucker

If $S$ is a finitely generated semigroup that meets the three specified conditions, then the cardinality of $\mathcal{R'}$ is finite and does not depend on our choice of relations. Furthermore, the process of finding relations can be finished in a finite number of steps.

Thank you

Takeaways:

  • We introduced the Ping-Pong lemma and how we can use it to reprove tits alternative for endomorphisms of the projective line.
  • We confirmed a surmise about the composition: there is no relation between rational functions generically.
  • We define finitely represented semigroup, and a set of basic relations can be considered like a basis for a vector space.
  • We attempt to define the nilpotent-by-finite for non-cancellative semigroup.