Turning numbers for periodic orbits of disk homeomorphisms
The braid type of a pair \((f,P)\) consisting of an orientation-preserving disk homeomorphism \(f\) and its periodic orbit \(P\), is its equivalence class under the relation: \((f,P)\sim(g,Q)\) if and only if there exists a homeomorphism \(h\) such that \(Q=h(P)\) and \(g\) is isotopic to \(hfh^{-1}\) relatively to \(Q\). To any braid type, we may associate a braid, defined up to conjugacy and multiplication by the full twist braid. In order to study periodic orbits of disk homeomorphisms, the authors introduce the conjugacy invariant turning numbers of a braid. For a braid \(B\) on \(n\) strands, with associated permutation \(\tau\), the \(k\)-th turning number \(TN_k(B)\) can be defined as \(TN_k(B) = \frac{1}{2}\sum_{i=1}^n T_k(i)\), where \(T_k(i)\) is the sum of signs of the crossings between the \(i\)-th and \(\tau^k(i)\)-th strands. They can also be defined using linking numbers of the link obtained by closing the pure braid \(B^n\), and they turn out to be coefficients of the Laurent polynomials defined in [\textit{T. Fiedler}, Topology 32, No. 2, 281--294 (1993; Zbl 0787.57007)]. The paper establishes some basic properties of the turning numbers and their relation to another invariant \(\mathrm{es}(B)\), the exponent sum of \(B\), equal to the sum of signs of all crossings of \(B\). The special cases of braid extensions, positive permutation braids, which arise from interval maps, and twist braids, associated with rotations of the disk, are studied in more detail. Finally, applications to the classification of periodic orbits are presented.
scientific article; zbMATH DE number 6201879
scientific article