On the Ruelle rotation for torus diffeomorphisms (Q1574454)

This note deals with the Ruelle rotation number, which is defined for a \(C^1\) diffeomorphism \(h\) of the 2-torus \(T^2\) that is isotopic to the identity. Note that the Ruelle rotation number depends on the trivialization of the tangent bundle \(TT^2\) of \(T^2\), as well as on the isotopy from id to \(h\). The author shows that two points of \(S^1\) taken starting with two different trivializations of \(TT^2\) differ by an element of the group \(\operatorname {Im}R_\nu(h)\), where \[ R_\nu(h): H^1(T^2,\mathbb{Z})\to S^1 \] is the \(\nu\)-rotation number. Let \(\nu\) be a Borel probability measure on \(T^2\) and \(\text{Diff}_0^1 (T^2,\nu)\) be the path component of the identity in the \(C^1\) topology of the group of \(C^1\) diffeomorphisms of \(T^2\) which preserve \(\nu\). In the space of continuous paths in \(\text{Diff}_0^1 (T^2,\nu)\) with initial point Id endowed with the compact-open topology the relation of homotopy with fixed endpoints is an equivalence relation. Let \(\widetilde{\text{Diff}}_0^1 (T^2,\nu)\) be the corresponding quotient space. The author studies properties of \(\widetilde{\text{Diff}}_0^1 (T^2,\nu)\).