Equidistribution of holonomy in homology classes for Anosov flows (Q1668462)

Let \(G\) be a compact connected Lie group, \(\pi: N \to M\) be a principal \(G\)-bundle over a compact manifold \(M\), \(\psi_t\) be a flow on \(N\) that commutes with the \(G\)-action on \(N\). Then the quotient by the \(G\)-action induces a flow on \(M\), say \(\phi_t\). Let \(\gamma\) be a periodic orbit for \(\phi_t\) on \(M\), \(l(\gamma)\) be the period of \(\gamma\). Associated to \(\gamma\) one has the homology class \([\gamma]\in H_1(M,\mathbb{Z})\) and its holonomy class \(\mathfrak{h}(\gamma)\), which is a conjugate class in \(G\). The flow \(\phi_t\) on \(M\) is said to have {homology-equidistributed holonomy} if for each \(\alpha\in H_1(M,\mathbb{Z})\), for each continuous class function \(F:G\to \mathbb{C}\), \[ \lim_{T\to\infty}\frac{1}{\# \mathcal{P}_T(\phi,\alpha)} \sum_{\gamma\in \mathcal{P}_T(\phi,\alpha)}F(\mathfrak{h}(\gamma)) =\int_G F(g)\, d\text{ Haar}(g). \] Here \(\text{Haar}\) is the (normalised) Haar measure on \(G\). Recall that a function \(F:G\to \mathbb{C}\) is a class function if it is constant on each conjugate class in \(G\). The main result of the paper is the following. Assume that \(\psi_t\) is topologically transitive on \(N\) and \(\phi_t\) is a homologically full Anosov flow on \(M\). Then the flow \(\phi_t\) has homology-equidistributed holonomy under some minor conditions on \(G\) (for example, when \(G\) is semisimple).