Set Of Recurrent Points Research Articles

SMOOTH TRIANGULAR MAPS OF TYPE 2∞ WITH POSITIVE TOPOLOGICAL ENTROPY

We explicitly construct for any k in ℕ a [Formula: see text]-differentiable triangular map in the square I2 with the following properties: (a) it has periodic orbits of period 2n for any n and no other periodic orbits, (b) the topological entropy is positive, and (c) the set of recurrent points contains properly the set of uniformly recurrent points.

A Counterexample in Dynamical Systems of the Interval

In [1] it was proved that if the recurrent points of a continuous map of the unit interval form a closed set, then this map has no periodic point with period not equal to a power of 2, i.e. this map is of type ${2^\infty }$. In this paper we will construct a continuous map of the interval which is of type ${2^\infty }$ and for which the set of recurrent points is not closed. By such a counterexample it may be shown that some of the results announced in [2] are not correct.

A counterexample in dynamical systems of the interval

In [1] it was proved that if the recurrent points of a continuous map of the unit interval form a closed set, then this map has no periodic point with period not equal to a power of 2, i.e. this map is of type 2 ∞ {2^\infty } . In this paper we will construct a continuous map of the interval which is of type 2 ∞ {2^\infty } and for which the set of recurrent points is not closed. By such a counterexample it may be shown that some of the results announced in [2] are not correct.