Figure 4.6: Deformation of an infinitesimal region under a map.
In Figure 4.6 we show how a small rectangular region R of the plane
is transformed to f(R) under one iteration of the map
where and 
. The
Jacobian of f, written 
, is the
determinant of the derivative matrix 
of f:
In the example just considered, 
,
the
Jacobian is
The Jacobian of a map at 
determines whether the area
about expands or contracts. If the absolute value of the
Jacobian is less than one, then the map is contracting; if the absolute value
of the Jacobian is greater than one, then the map is expanding.
A simple example of
a contracting map is provided by the Hénon map for the parameter
range 
. In this case,
and a quick calculation shows
The Jacobian is constant for the Hénon map; it does not depend on
the initial position 
. When iterating the Hénon map, the area
is multiplied each time by 
, and after k iterations the
size of an initial area 
is
In particular, if , then the area is contracting.
