On some modifications of Arnold's cat map

An effective method is proposed for constructing specific examples of Anosov diffeomorphisms on the torus $\mathbb{T}^2$, that are different from linear hyperbolic automorphisms. We introduce a special class of diffeomorphisms that are compositions of the well-known linear Arnold’s cat map and some diffeomorphisms homotopic to the identity. Constructively verified sufficient hyperbolicity conditions are established for this class of mappings.