horseshoe map

In the horseshoe map the squeezing and stretching are uniform.

The horseshoe map is a diffeomorphism defined from a region of the plane into itself.

It is possible to describe the behavior of all initial conditions of the horseshoe map.

By folding the contracted and stretched square in different ways, other types of horseshoe maps are possible.

The notation can be extended to higher iterates of the horseshoe map.

A prominent example is the Smale horseshoe map.

Indeed, Smale (1967) showed that these points leads to horseshoe map like dynamics, which is associated with chaos.

Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in the left cap.

It converges to a point that is part of a periodic orbit of the horseshoe map.

The baker's map is topologically conjugate to the horseshoe map.

Under forward iterations of the horseshoe map, the original square gets mapped into a series of horizontal strips.

The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient.

The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit.

One definition folds over or rotates one of the sliced halves before joining it (similar to the horseshoe map) and the other does not.

For a horseshoe map:

The horseshoe map is one-to-one, which means that an inverse f exists when restricted to the image of S under f.

In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself.

Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.

The logistic map is only a second-degree polynomial; the horseshoe map is piecewise linear.

Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

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.

Many map considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature...