On Isolated Periodic Points of Diffeomorphisms with Expanding Attractors of Codimension $1$

In this paper we consider an $\Omega$-stable $3$-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic $2$-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a $3$-torus, by Smale’s surgery. Each such attractor has bunches of degree $1$ and $2$. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with $k$ bunches and $k$ isolated periodic points.