View source for Quotient space

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''of a [[Dynamical system|dynamical system]] $f^t$ given on a topological space $S$''

The quotient space of $S$ relative to the equivalence: $x\sim y$ if the points $x$ and $y$ belong to the same trajectory. In other words, the points of the quotient space are the trajectories of the dynamical system $f^t$ (in a different notation $f(t,p)$, see {{Cite|Ne}}), and the topology is the strongest in which the mapping associating each point of $S$ with its trajectory is continuous (thus,

$$\{ f^tx_k\}_{t\in\R}\xrightarrow[k\in K]{}\{ f^tx\}_{t\in\R}$$

($K$ is a directed set) if and only if there are $t_k$ such that

$$ f^{t_k}x_k\xrightarrow[k\in K]{}x;$$

if $S$ is a metric space, then $k\in\N$). The quotient spaces of many dynamical systems do not satisfy any of the separation axioms, even if $S$ does. For example, if $S$ is a [[Minimal set|minimal set]], then the closure of every non-empty set in the quotient space is the whole quotient space. If a dynamical system given on a metric space is completely unstable (see [[Complete instability|Complete instability]]), then for its quotient space to be Hausdorff it is necessary and sufficient that this dynamical system does not have saddles at infinity (cf. [[Saddle at infinity|Saddle at infinity]]).

In general, let $X$ be a [[Topological space|topological space]] and $R$ an equivalence relation on $X$ (equivalently: $X$ is the disjoint union of subsets $X_\lambda$, $\lambda$ in some index set $\Lambda$, not necessarily finite; in that case, $(x_1,x_2)\in R$ if and only if $x_1$ and $x_2$ belong to the same $X_\lambda$). The quotient space (also called decomposition space, see [[Quotient mapping|Quotient mapping]]) $X/R$ is the space whose points are the $R$-equivalence classes, endowed with the finest (i.e. strongest) topology making the quotient mapping $x\mapsto R[x]$ continuous (here $R[x]=\{ x'\in X : (x,x')\in R\}$ for $x\in X$). The object discussed above, where the equivalence classes are the trajectories of a dynamical system, is usually called the orbit space of the dynamical system. The characterization of convergence of a net (or [[Generalized sequence|generalized sequence]]) in the orbit space cannot be generalized to arbitrary quotient spaces: it is valid because for orbit spaces the quotient mapping is always an [[Open mapping|open mapping]].

That for a completely-unstable system the orbit space has a Hausdorff topology if and only if the dynamical system has no saddles at infinity is related with the results in {{Cite|Ma}}. See also Proposition 14 in {{Cite|Ha}}.

====References====
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|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)
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|valign="top"|{{Ref|En}}||valign="top"|  R. Engelking,   "General topology" , Heldermann  (1989)
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|valign="top"|{{Ref|Ha}}||valign="top"|  O. Hajek,   "Prolongation in topological dynamics" , ''Sem. Differential Equations and Dynamical Systems II'' , ''Lect. notes in math.'' , '''144''' , Springer  (1970)  pp. 79–89
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|valign="top"|{{Ref|Ma}}||valign="top"| L. Markus,   "Parallel dynamical systems"  ''Topology'' , '''8'''  (1969)  pp. 47–57
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|valign="top"|{{Ref|Mi}}||valign="top"|  V.M. Millionshchikov,   "A comment on the Nemytskii–Bebutov theorem concerning unstable dynamic system"  ''Differential Eq.'' , '''10''' :  12  (1975)  pp. 1775–1776  ''Differensial'nye Uravneniya'' , '''10''' :  12  (1975)  pp. 2292–2293
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|valign="top"|{{Ref|Ne}}||valign="top"|  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)
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