Smale’s spectral decomposition theorem

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

 $\Omega=\bigcup_{i=1}^{m}\Lambda_{i}.$

Moreover, each basic block is partitioned into a finite number of compact subblocks $\Lambda_{ij}$, $j=1,\dots,m_{i}$ such that $f(\Lambda_{ij})=\Lambda_{i(j+1)}$ for $1\leq j and $f(\Lambda_{im_{i}})=\Lambda_{i1}$, and $\Lambda_{ij}$ is topologically mixing for $f^{m_{i}}$.

Title Smale’s spectral decomposition theorem SmalesSpectralDecompositionTheorem 2013-03-22 14:28:08 2013-03-22 14:28:08 Koro (127) Koro (127) 4 Koro (127) Theorem msc 37D20 spectral decomposition theorem basic block