Abstract
We look at density of periodic points and Devaney Chaos. We prove that if f is
Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f 2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)
Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f 2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)
| Original language | English |
|---|---|
| Pages (from-to) | 773-780 |
| Number of pages | 8 |
| Journal | The American Mathematical Monthly |
| Volume | 122 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - Oct 2015 |