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  "body": "\n## Definition\n\nLet $(X, \\mathcal{B}, \\mu)$ and $(Y, \\mathcal{C}, \\nu)$ be [[probability spaces|0NFPJVJJHX]], and let $T \\colon X \\to X$ and $S \\colon Y \\to Y$ be invertible [[measure-preserving transformations|556WBDAHNP]] with measurable measure-preserving inverses.\nLet $U_T$ and $U_S$ be their [[Koopman operators|37TRG5DBQC]] on $L^2(\\mu)$ and $L^2(\\nu)$.\n\n1. $T$ and $S$ are *spectrally isomorphic* if there is a [[unitary operator|MM0EJ2GDE4]] $W \\colon L^2(\\mu) \\to L^2(\\nu)$ with $W U_T = U_S W$.\n2. A *spectral invariant* is a property of such transformations that holds for $S$ whenever it holds for $T$ and $T$, $S$ are spectrally isomorphic.\n3. $T$ has *discrete spectrum* if $L^2(\\mu)$ has an orthonormal basis of [[eigenvectors|0BQAA2BK3Q]] of $U_T$.\n\nThe spectral theory of dynamical systems is the study of transformations up to spectral isomorphism, through the spectrum and the spectral measures of $U_T$.\n\n## Remarks\n\nIsomorphic transformations are spectrally isomorphic: an isomorphism $\\varphi$ with $\\varphi \\circ T = S \\circ \\varphi$ gives $W f = f \\circ \\varphi^{-1}$.\n\nExample of a spectral invariant: ergodicity, since $T$ is ergodic if and only if the eigenvalue $1$ of $U_T$ has multiplicity $1$.\nWeak mixing is another: it holds if and only if $1$ is a simple eigenvalue and $U_T$ has no other eigenvalue.\nAn irrational rotation $x \\mapsto x + \\alpha \\bmod 1$ has discrete spectrum, with basis $e^{2\\pi i k x}$ and eigenvalues $e^{2\\pi i k \\alpha}$, $k \\in \\Z$.\n\nNon-example: the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is not a spectral invariant.\nTwo Bernoulli shifts with different entropies, such as the $(1/2, 1/2)$ and the $(1/3, 1/3, 1/3)$ shifts, are spectrally isomorphic.\nBoth Koopman operators have countable Lebesgue spectrum on the orthogonal complement of the constants.\n",
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