Mathematical models for time evolution can be differential equations
whose solutions represent motions developing in continuous time
$t$ or, often, maps whose $n$-th iterate
represents motions developing at discrete integer times
$n$. The point representing the state of the system at time
$t$ is denoted $S\_tx$ in the continuous time
models or, at the $n$-th observation, $S^n\backslash xi$
in the discrete time models. Here $x,\backslash xi$ will be points on
a manifold $X$ or $\backslash Xi$ respectively, called the
''phase space'', or the space of the states, of the system.
The connection between the two representations of motions is
illustrated by means of the following notion of ``timing event''.
Physical observations are always performed at discrete times: ''i.e.''
when some special, prefixed, ''timing'' event occurs, typically
when the state of the system is in a set $\backslash Xi\backslash subset\; X$ and
triggers the action of a ``measurement apparatus'', ''e.g.''
shooting a picture after noting the position of a clock arm. If
$\backslash Xi$ comprises the collection of the timing events,
''i.e.'' of the states $\backslash xi$ of the system which induce the
act of measurement, motion of the system can also be represented as a
map $\backslash xi\backslash to\; S\backslash xi$ defined on $\backslash Xi$.
For this reason mathematical models are often maps which associate
with a timing event $\backslash xi$, ''i.e.'' a point
$\backslash xi$ in the manifold $\backslash Xi$ of the measurement
inducing events, the next timing event $S\backslash xi$.
If the system motions also admit a continuous time representation on a
space of states $X\backslash supset\backslash Xi$ then there will be a simple
relation between the evolution in continuous time $x\backslash to\; S\_tx$ and the discrete representation $\backslash xi\backslash to\; S^n\backslash xi$
in discrete integer times $n$, between successive timing
events, namely $S\backslash xi\backslash equiv\; S\_\{\backslash tau(\backslash xi)\}\backslash xi$, if
$\backslash tau(\backslash xi)$ is the time elapsing between the timing event
$\backslash xi$ and the subsequent one $S\backslash xi$
The discrete time representation is particularly useful mathematically
in cases in which the continuous evolution shows singularities: the
latter can be avoided by choosing timing events which occur when the
point representing the system is not singular nor too close to a
singularity (when the physical measurements become difficult or
impossible).