# Chaotic hypothesis/timing event

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, \(S^n\xi\)
in the discrete time models. Here \(x,\xi\) will be points on
a manifold \(X\) or \(\Xi\) 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 \(\Xi\subset X\) and
triggers the action of a ``measurement apparatus*, *e.g.
shooting a picture after noting the position of a clock arm. If
\(\Xi\) comprises the collection of the timing events,
*i.e.* of the states \(\xi\) of the system which induce the
act of measurement, motion of the system can also be represented as a
map \(\xi\to S\xi\) defined on \(\Xi\).

For this reason mathematical models are often maps which associate
with a timing event \(\xi\), *i.e.* a point
\(\xi\) in the manifold \(\Xi\) of the measurement
inducing events, the next timing event \(S\xi\).

If the system motions also admit a continuous time representation on a space of states \(X\supset\Xi\) then there will be a simple relation between the evolution in continuous time \(x\to S_tx\) and the discrete representation \(\xi\to S^n\xi\) in discrete integer times \(n\), between successive timing events, namely \(S\xi\equiv S_{\tau(\xi)}\xi\), if \(\tau(\xi)\) is the time elapsing between the timing event \(\xi\) and the subsequent one \(S\xi\)

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).