View source for Spectrum of a dynamical system

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865701.png" /> with phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865702.png" /> and invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865703.png" />''

A common name for various spectral invariants and spectral properties of the corresponding group (or semi-group) of unitary (isometric) shift operators:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865704.png" /></td> </tr></table>

in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865705.png" />. For a dynamical system in the narrow sense (a [[Measurable flow|measurable flow]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865706.png" /> or a [[Cascade|cascade]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865707.png" />), the spectral invariants of just one [[Normal operator|normal operator]] are meant: in the second case of the unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865708.png" />, and in the first, of the generating self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s0865709.png" /> that is the infinitesimal generator of the one-parameter group of unitary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657010.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657011.png" />, by Stone's theorem).

The "spectral" terminology in the theory of dynamical systems differs somewhat from the ordinary usage. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657013.png" /> of practical interest, the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657014.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657015.png" />) in the usual sense, that is, the set of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657016.png" /> for which the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657017.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657018.png" />) does not have a bounded inverse (cf. [[Spectrum of an operator|Spectrum of an operator]]), coincides with the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657019.png" /> or with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657020.png" /> (see [[#References|[1]]], [[#References|[2]]]). Therefore: a) the spectrum in the usual sense does not contain information about the properties of a given dynamical system which distinguish it from others; b) in the spectrum in the normal sense of the word, there are hardly ever any isolated points, so that it is continuous (in the ordinary sense) and this again does not contain information about specific properties of a given system. For this reason, in the theory of dynamical systems one speaks of a continuous spectrum whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657022.png" /> have no eigenfunctions other than constants, of a discrete spectrum when the eigenfunctions form a complete system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657023.png" /> and of a mixed spectrum in all other cases.

The properties of a dynamical system that are determined by its spectrum are called spectral properties. Examples are [[Ergodicity|ergodicity]] (which is equivalent with the eigenvalue 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657024.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657026.png" />, being simple) and [[Mixing|mixing]]. There is a complete metric classification of ergodic dynamical systems with a discrete spectrum: such a system is determined by its spectrum up to a [[Metric isomorphism|metric isomorphism]] [[#References|[3]]]. An analogous theory has also been developed for transformation groups more general than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657028.png" /> (see [[#References|[4]]]). In the non-commutative case formulations becomes more complicated, and, moreover, the spectrum no longer completely determines the system. If the spectrum is not discrete, then the situation is much more complex.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Ionescu Tulcea, "Random series and spectra of measure-preserving transformations" , ''Ergodic Theory (Tulane Univ. 1961)'' , Acad. Press (1963) pp. 273–292 {{MR|}} {{ZBL|0132.10703}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Goldstein, "Spectrum of measurable flows" ''Astérisque'' , '''40 (Internat. Conf. Dynam. Systems in Math. Physics)''' (1976) pp. 5–10 {{MR|0450511}} {{ZBL|0342.47006}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) {{MR|832433}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G.W. Mackey, "Ergodic transformation groups with a pure point spectrum" ''Illinois J. Math.'' , '''8''' (1964) pp. 593–600 {{MR|0172961}} {{ZBL|0255.22014}} </TD></TR></table>



====Comments====
Instead of "discrete spectrum" also the term "pure point spectrum of a dynamical systempure point spectrum" is used in the literature. For transformation groups more general than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086570/s08657030.png" /> and not necessarily commutative, also consult [[#References|[a1]]] and [[#References|[a2]]].

The next to simplest dynamical systems are those with a generalized discrete spectrum and those with a quasi-discrete spectrum. See [[#References|[a4]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Zimmer, "Extensions of ergodic group actions" ''Illinois J. Math.'' , '''20''' (1976) pp. 373–409 {{MR|0409770}} {{ZBL|0334.28015}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Zimmer, "Ergodic actions with generalized discrete spectrum" ''Illinois J. Math'' , '''20''' (1976) pp. 555–588 {{MR|0414832}} {{ZBL|0349.28011}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.M. Abramov, "Metric automorphisms with quasi-discrete spectrum" ''Transl. Amer. Math. Soc.'' , '''39''' (1964) pp. 37–56 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''26''' (1962) pp. 513–530 {{MR|0143040}} {{ZBL|0154.15704}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Parry, "Compact abelian group extensions of discrete dynamical systems" ''Z. Wahrsch. verw. Geb.'' , '''13''' (1969) pp. 95–113 {{MR|0260976}} {{ZBL|0184.26901}} </TD></TR></table>