# 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 ${f}^{{m}_{i}}$.

Title | Smale’s spectral decomposition theorem |
---|---|

Canonical name | SmalesSpectralDecompositionTheorem |

Date of creation | 2013-03-22 14:28:08 |

Last modified on | 2013-03-22 14:28:08 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 4 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 37D20 |

Synonym | spectral decomposition theorem |

Defines | basic block |