Louis Block

An improved algorithm for computing topological entropy

A new algorithm is presented for computing the topological entropy of a unimodal map of the interval. The accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed.

Stratification of continuous maps of an interval

We define the notion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points. We define also strongly simple orbits and show, in particular, that they represent periodic orbits with minimum entropy. Further results are obtained for unimodal maps with negative Schwarzian, which sharpen recent results of Block and Hart.

Periods of periodic points of maps of the circle which have a fixed point

For a continuous map / of the circle to itself, let P(f) denote the set of positive integers n such that / has a periodic point of (least) period n. Results are obtained which specify those sets, which occur as P(f), for some continuous map/ of the circle to itself having a fixed point. These results extend a theorem of Sarkovskii on maps of the interval to maps of the circle which have a fixed point.

Computing the topological entropy of maps of the interval with three monotone pieces

An algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces. The accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed. Some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces.

The chain recurrent set, attractors, and explosions

Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x . In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Strange adding machines

We show that given an adding machine of type

ω-limit sets for maps of the interval

\alpha

The chain recurrent set for maps of the interval

, for a dense set of parameters

Homoclinic and non-wandering points for maps of the circle

s

The bifurcation of periodic orbits of one-dimensional maps

in the interval