Applications of thermodynamic formalism in complex dynamics on \(\mathbb{P}^2\) (Q5951627)

The holomorphic map \(f:\mathbb{P}^2\to\mathbb{P}^2\) on the complex 2-dimensional projective space, is investigated. It is supposed, that \(f=[P:Q:R]\), where \(P,Q,R\) are homogeneous polynomials of degree \(d\) with no common zeros other than the origin. The author studies the nonwandering set of such a mapping and its ``saddle'' part \(S_1\), i.e. the subset of points with both stable and unstable directions. Under a derivative condition, the stable manifolds of points in \(S_1\) will have a very ``thin'' intersection with \(S_1\), from the point of view of Hausdorff dimension. He also has proved that the unstable manifolds of an endomorphism depend Hölder continuously on the correspording prehistory of their base point and employed this in the end to give an estimate of the Hausdorff dimension of the global unstable set of \(S_1\).