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  "body": "\n## Statement\n\nLet $M$ be a compact smooth manifold and let $f \\colon M \\to M$ be a $C^1$ diffeomorphism that satisfies [[Axiom A|S31RTTJPDC]], with [[non-wandering set|EY2W34JZA4]] $\\Omega(f)$.\nThen:\n\n1. There are pairwise disjoint closed sets $\\Lambda_1, \\ldots, \\Lambda_k$ with $\\Omega(f) = \\Lambda_1 \\cup \\cdots \\cup \\Lambda_k$ such that $f(\\Lambda_i) = \\Lambda_i$ and $f|_{\\Lambda_i}$ is [[topologically transitive|FR49Y9GQHR]] for each $i$.\n2. Each $\\Lambda_i$ is a union of pairwise disjoint closed sets $X_{i,1}, \\ldots, X_{i,n_i}$ such that $f(X_{i,j}) = X_{i,j+1}$ for $j \u003c n_i$, $f(X_{i,n_i}) = X_{i,1}$, and $f^{n_i}|_{X_{i,j}}$ is [[topologically mixing|B763QH5521]] for each $j$.\n\n## Remarks\n\nThe sets $\\Lambda_i$ are called basic sets; the decomposition into them is unique.\nThe idea of the proof is to group periodic points whose stable and unstable manifolds meet transversally in both directions.\nThe local product structure of the hyperbolic set makes the closures of these groups open and closed in $\\Omega(f)$.\nCompactness then makes them finite in number.\nPart 1 is due to Smale, and part 2 to Bowen.\nExample: for the north-south map of the circle, the basic sets are the two hyperbolic fixed points, with $n_i = 1$.\nFor an [[Anosov diffeomorphism|RCQH8P6DPQ]] of the torus, $\\Omega(f)$ is the whole torus and there is one basic set.\nWithout hyperbolicity the conclusion fails: for the identity of the circle every point is non-wandering, and $\\Omega(f)$ is connected but $f$ is not transitive on it.\n\n## Explanations\n\n### L1 human\n\nThe long-term dynamics of an Axiom A system breaks into finitely many separate blocks.\nOn each block the motion is irreducible: some orbit comes close to every point of it.\nEach block in turn is made of a few pieces that the map visits in a cycle, and on each piece a power of the map mixes completely.\nSo the global behaviour is a finite combination of well-understood chaotic pieces.\n\n### L2 human\n\nThe theorem is the starting point of the theory of Axiom A systems.\nOrbits outside $\\Omega(f)$ go from one basic set to another.\nWhen there are no cycles among the basic sets, this gives a partial order on them, as in Conley's theory of [[chain recurrence|6MX5R482KJ]].\nEach basic set admits a [[Markov partition|4VNHVDHTNN]], so the dynamics on it is a factor of a [[subshift of finite type|624XDXX8FN]].\nThis leads to equilibrium states, [[Sinai–Ruelle–Bowen measures|256NP6JBEQ]] and the counting of periodic orbits.\n",
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