Let \(S\) be a closed Riemannian surface without focal points and \(g_t\) be the corresponding geodesic flow on its unit tangent bundle. Denote by \(E^u_v\) the unstable space of the unit vector \(v\). Recall that the geometric potential is defined as
\[ \varphi^{\mathrm{geo}}(v)=-\lim_{t\rightarrow 0}\frac{1}{t} \log \left\lVert dg_t |_{E_v^u} \right\rVert. \] Let \(\mathcal P(t)\) stand for the pressure of \(t\varphi^{\mathrm{geo}}\) and \(\mathcal E\) for the Legendre transform of \(\mathcal P\). The Lyapunov level set \(\mathcal L(\beta)\) is defined as the set of those \(v\) such that \(\chi(v)=\beta\), being \(\chi(v)\) be the Lyapunov exponent of \(v\). Let \(h(\mathcal L(\beta))\) denote the topological entropy of \(\mathcal L(\beta)\) and \(\dim_H(\mathcal L(\beta))\) the Hausdorff dimension of \(\mathcal L(\beta)\). Set \(\alpha_1=\lim_{t \rightarrow-\infty} D^+ \mathcal P(t)\), where \(D^+\) denotes the right derivative.
In the main result of the article the authors show that if \(S\) is a rank-1 Riemannian surface with no focal points then the Lyapunov level set \(\mathcal L(-\alpha)\) is non empty if and only if \(\alpha\in [\alpha_1, 0]\). It is also shown that if \(\alpha\in (\alpha_1, 0)\) then \(h(\mathcal L(-\alpha))=\mathcal E(\alpha)\) and \(\dim_H(\mathcal L(-\alpha))\geq 1-2\mathcal E(\alpha)/\alpha\) .
These results extend previous work by K. Burns and K. Gelfert [Discrete Contin. Dyn. Syst. 34, No. 5, 1841–1872 (2014; Zbl 1325.37018)] for non-positively curved surfaces .