Infinite, finitely generated subgroup of a group of homeomorphisms of closed unit disc

Question

Can we construct a finitely generated subgroup of infinite order of a group of homeomorphisms (which fix boundary point wise) of closed unit disc?

Answer 1

For the unit disc, in polar coordinates, $$ f_k(r, \theta) = (r, \theta + 2k\pi(1-r)) $$

Then $$ H = \{ f_k \mid k \in \Bbb Z \} $$ is generated by $f_1$, and has infinite order.

Answer 2

The disk $\Bbb D$ is topologically the same as the upper half plane $\Bbb H$. Here we have for example For every continuous $f\colon [0,\infty)\to\Bbb R$ with $f(0)=0$, we have a homeomorphism $$\phi_f\colon (x,y)\mapsto (x+f(y), y) $$ of $\Bbb H$ (with $\phi_f^{-1}=\phi_{-f}$). Also, for every continuous $g\colon \Bbb R\to \Bbb R$, we have a homeomorphism $$\psi_g\colon (x,y)\mapsto (x,e^{g(x)}y) $$ of $\Bbb H$ (with $\psi_g^{-1}=\psi_{-g}$). Now pick finitely many $\phi_f$'s and $\psi_g$'s and consider the group generated by them ...