Homeomorphisms of the circle were first considered by Poincare* who used them to obtain qualitative results for a class of differential equations on the torus. He classified those which have a dense orbit by showing that they are topologically equivalent to a rotation through an angle incommensurable with IT. However, Denjoy showed that there exist homeomorphisms of the circle without periodic ...

Let φ be a one-dimensional surjective cellular automaton map. We prove that if φ is a closing map, then the configurations which are both spatially and temporally periodic are dense. (If φ is not a closing map, then we do not know whether the temporally periodic configurations must be dense.) The results are special cases of results for shifts of finite type, and the proofs use symbolic dynamic...

Curves of parameter values for which an asymptotic value of

For the maps of the two-dimensional sphere into itself that preserve the latitude foliation and are differentiable at the poles, lower estimates of the number of fixed points for the maps and their iterates are obtained. Those estimates also show that the growth rate of the number of fixed points of the iterates is larger than or equal to the logarithm of the absolute value of the degree of the...