Smooth isomorphisms not having quasiconformal fractional powers (Q1088808)

The work deals with the dynamics of automorphisms \(\phi\) of unit segment I, \(\phi (x)>x\), possessing such points \(a\in I\) (relatively attractive) such that for all \(x\in [a,\phi (a))\) \[ \lim_{m\to \infty}(\phi^ m(x)-\phi^ m(a))/(\phi^{m+1}(a)-\phi^ m(a))=0. \] With their help the author obtains \(C^ 1\)-smooth automorphisms f of any n-dimensional domain D, \(n\geq 2\), arbitrarily close to identity in \(C^ 1\)-topology and which may be included in a topologic flow \[ F: {\mathbb{R}}\times D\to D\quad (F(0,x)=x,\quad F(1,x)=f(x),\quad F(t+s,x)=F(t,F(s,x))). \] However, it is impossible not only to include f in any quasiconformal flow, but there is not any quasiconformal map \(g: D\to D\) such that \(g^{\ell}=f^ k\) for some \(\ell,k\in {\mathbb{Z}}\) where \(\ell\) does not divide k.