Dynamics of affine interval transformations of Higman-Thompson groups \(V_{r, m}\) (Q486806)

The authors study the dynamics of affine interval exchange transformations, whose slopes are integer powers of the same integer \(m\), and whose cuts and their images are rational. The paper proves that such a map has very simple dynamics: all its orbits are proper and it has at least one periodic orbit or periodic cycle. As a corollary; a distortion element of the Higman-Thompson group \(V_{r,m}\) is of finite order. Following [\textit{I. Liousse}, Ann. Inst. Fourier 55, No. 2, 431--482 (2005; Zbl 1079.37033)], the spirit of the proofs is dynamical. The present results are related to [\textit{C. Bleak} et al., Groups Geom. Dyn. 7, No. 4, 821--865 (2013; Zbl 1298.20052)], where combinatorial technics were employed. The origin of the Higman-Thompson group \(V_{r,m}\) goes back to \textit{G. Higman} [Finitely presented infinite simple groups. Canberra: Australian National University (1974)].