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:
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 $g$.)
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.)
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.