Smale’s spectral decomposition theorem

Let $M$ be a compact differentiable manifold and let $f:M\to M$ be an Axiom A diffeomorphism. The nonwandering set $\mathrm{\Omega }$ of $f$ can be partitioned into a finite number of compact topologically transitive blocks, called basic blocks:

$$\mathrm{\Omega }=\bigcup _{i=1}^m{\mathrm{\Lambda }}_i.$$

Moreover, each basic block is partitioned into a finite number of compact subblocks ${\mathrm{\Lambda }}_{ij}$, $j=1,\mathrm{\dots },m_i$ such that $f({\mathrm{\Lambda }}_{ij})={\mathrm{\Lambda }}_{i(j+1)}$ for and $f({\mathrm{\Lambda }}_{i{m}_i})={\mathrm{\Lambda }}_{i1}$, and ${\mathrm{\Lambda }}_{ij}$ is topologically mixing for $f^{m_i}$.