In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map. The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient. In the horseshoe map, the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square can be simply described. For a horseshoe map:
there are an infinite number of periodic orbits; periodic orbits of arbitrarily long period exist; the number of periodic orbits grows exponentially with the period; and close to any point of the fractal invariant set there is a point of a periodic orbit.
The horseshoe map The horseshoe map f is a diffeomorphism defined from a region S of the plane into itself. The region S is a square capped by two semi-disks. The codomain of f {\displaystyle f} (the "horseshoe") is a proper subset of its domain S {\displaystyle S} . The action of f is defined through the composition of three geometrically defined transformations. First the square is contracted along the vertical direction by a factor a < 1/2. The caps are contracted so as to remain semi-disks attached to the resulting rectangle. Contracting by a factor smaller than one half assures that there will be a gap between the branches of the horseshoe. Next the rectangle is stretched horizontally by a factor of 1/a; the caps remain unchanged. Finally the resulting strip is folded into a horseshoe-shape and placed back into S. The interesting part of the dynamics is the image of the square into itself. Once that part is defined, the map can be extended to a diffeomorphism by defining its action on the caps. The caps are made to contract and eventually map inside one of the caps (the left one in the figure). The extension of f to the caps adds a fixed point to the non-wandering set of the map. To keep the class of horseshoe maps simple, the curved region of the horseshoe should not map back into the square. The horseshoe map is one-to-one, which means that an inverse f−1 exists when restricted to the image of S under f. By folding the contracted and stretched square in different ways, other types of horseshoe maps are possible.
To ensure that the map remains one-to-one, the contracted square must not overlap itself. When the action on the square is extended to a diffeomorphism, the extension cannot always be done in the plane. For example, the map on the right needs to be extended to a diffeomorphism of the sphere by using a “cap” that wraps around the equator. The horseshoe map is an Axiom A diffeomorphism that serves as a model for the general behavior at a transverse homoclinic point, where the stable and unstable manifolds of a periodic point intersect.
Dynamics of the map The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit. As the system evolves, points in this disk remain close to the given periodic orbit, tracing out orbits that eventually intersect the disk once again. Other orbits diverge. The behavior of all the orbits in the disk can be determined by considering what happens to the disk. The intersection of the disk with the given periodic orbit comes back to itself every period of the orbit and so do points in its neighborhood. When this neighborhood returns, its shape is transformed. Among the points back inside the disk are some points that will leave the disk neighborhood and others that will continue to return. The set of points that never leaves the neighborhood of the given periodic orbit form a fractal. A symbolic name can be given to all the orbits that remain in the neighborhood. The initial neighborhood disk can be divided into a small number of regions. Knowing the sequence in which the orbit visits these regions allows the orbit to be pinpointed exactly. The visitation sequence of the orbits provide a symbolic representation of the dynamics, known as symbolic dynamics.
Orbits It is possible to describe the behavior of all initial conditions of the horseshoe map. An initial point u0 = (x, y) gets mapped into the point u1 = f(u0). Its iterate is the point u2 = f(u1) = f 2(u0), and repeated iteration generates the orbit u0, u1, u2, ... Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap. This is because the horseshoe maps the left cap into itself by an affine transformation that has exactly one fixed point. Any orbit that lands on the left cap never leaves it and converges to the fixed point in the left cap under iteration. Points in the right cap get mapped into the left cap on the next iteration, and most points in the square get mapped into the caps. Under iteration, most points will be part of orbits that converge to the fixed point in the left cap, but some points of the square never leave.
Iterating the square
Under forward iterations of the horseshoe map, the original square gets mapped into a series of horizontal strips. The points in these horizontal strips come from vertical strips in the original square. Let S0 be the original square, map it forward n times, and consider only the points that fall back into the square S0, which is a set of horizontal stripes
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