Symbolic Dynamics

 

We can identify points in the invariant set according to the following scheme. After one forward iteration and one backward iteration, the invariant set is located within the four shaded rectangular regions shown in Figure 4.20(c). After two forward iterations and two backward iterations, the invariant set is a subset of the 16 shaded regions. The shaded regions are the intersection of the horizontal and vertical strips. To each shaded region we associate a bi-infinite  symbol sequence,

constructed from the label of the vertical and horizontal strips forming a point in the invariant set. The right-hand side of the symbolic name, , is the label from the horizontal strip . The left-hand side of the symbolic name, , is the label from the vertical strip written backwards, . For instance, the shaded region labeled L in Figure 4.20(c) has a symbolic name ``10.01.'' The ``.01'' to the right of the dot indicates horizontal strip . The ``10.'' (``01'' backwards) to the left of the dot indicates that the shaded region comes from the vertical strip .

We hone in closer and closer to the invariant set by iterating the horseshoe map both forward and backward. Moreover, the above labeling scheme generates a symbolic name, or symbolic coordinate, for each point of the invariant set. This symbolic name contains information about the dynamics of the invariant point.

To see how this works more formally, let us call the symbol space  of all bi-infinite sequences of 0's and 1's. A metric on between the two sequences

is defined by

 

Next we define a shift map  on by

 

The shift map is continuous and it has two fixed points consisting of a string of all 0's or all 1's. A period n orbit of is written as , where the overbar indicates that the symbolic sequence repeats forever. A few of the periodic orbits and their ``shift equivalent''  representations are listed below,

and so on. In addition to periodic orbits of arbitrarily high period, the shift map also possesses an uncountable infinity of nonperiodic orbits as well as a dense orbit. See Devaney [13] or Wiggins [14] for the details.

In section 2.11 we showed that the shift map on the space of one-sided symbol sequences is topologically semiconjugate to the quadratic map. A similar results holds for the shift map on the space of bi-infinite sequences and the horseshoe map; namely, there exists a homeomorphism connecting the dynamics of f on and on such that :

The correspondence between the shift map on and the horseshoe map on the invariant set is pretty easy to see (again, for the mathematical details see Wiggins [14]). The invariant set consists of an infinite intersection of horizontal and vertical strips. These intersection points are labeled by their symbolic itinerary, and the horseshoe map carries one point of the invariant set to another precisely by a shift map. Consider, for instance, the period two orbit

The shift map sends to and back again. This corresponds to an orbit of the horseshoe map that bounces back and forth between the points labeled 01.01 and 10.10 in (see Fig.\ 4.21).

  
Figure 4.21: Equivalence between the dynamics of the horseshoe map on the unit square and the shift map on the bi-infinite symbol space.

The topological conjugacy between and f allows us to immediately conclude that, like the shift map, the horseshoe map has

1. a countable infinity of periodic orbits (and all the periodic orbits are hyperbolic saddles);
2. an uncountable infinity of nonperiodic orbits;
3. a dense orbit.

The shift map (and hence the horseshoe map) exhibits sensitive dependence on initial conditions (see section 4.10). As stated above, it possesses a dense orbit. These two properties are generally taken as defining a chaotic set.