Quotient space

of a 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 [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, 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), 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).

In general, let $X$ be a 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) $X/R$ 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) 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.

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 [Ma]. See also Proposition 14 in [Ha].

References

[Bo]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
[En]	R. Engelking, "General topology" , Heldermann (1989)
[Ha]	O. Hajek, "Prolongation in topological dynamics" , Sem. Differential Equations and Dynamical Systems II , Lect. notes in math. , 144 , Springer (1970) pp. 79–89
[Ma]	L. Markus, "Parallel dynamical systems" Topology , 8 (1969) pp. 47–57
[Mi]	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
[Ne]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)