A compactification of the space of expanding maps on the circle
Let \(\mathrm{Exp}_d(S^1)\) denote the space of topological covering maps \(f:S^1\rightarrow S^1\) of degree \(d>1\) that preserve Lebesgue measure. In this paper a compactification \(\widehat{\mathrm{Exp}_d}(S^1)\) for the space of expanding maps is introduced. It is well known that every \(f \in \mathrm{Exp}_d(S^1)\) is topologically conjugate to the model mapping \(p_d(t)=d\cdot t \mathrm{mod}~1\). There are \((d-1)\) choices for conjugacy \(\phi\); the choice of one is a marking of \(f\). Using the marking, one can transport Lebesgue measure to an invariant measure \(\nu_{(f,\phi)}=\phi_{\ast}(\lambda)\) for the model mapping \(p_d.\) The author prove that the map \[ \nu : T\widehat{\mathrm{Exp}_d}(S^1)\rightarrow M_d(S^1) \] gives a bijective homeomorphism between the compactified Teichmüller space of expanding maps and the space of invariant probability measures for \(t\mapsto d\cdot t \mathrm{mod}~1.\) The analogous result holds for the space \(\overline{\mathcal{B}}_d\) of expanding Blaschke products on \(S^1.\) Several interesting properties of \(\overline{\mathcal{B}}_d\) are obtained.
scientific article; zbMATH DE number 5603850