Continuity of the Hausdorff dimension for invariant subsets of interval maps.

*(English)*Zbl 0828.58014The setting of the article is expanding and piecewise monotone mappings of an interval. Suppose that a finite union of open intervals is removed from the interval. Then the limit set of points the orbits of which forever avoid these open intervals is a compact invariant set with induced dynamics. The problem addressed in the article is how the dynamics on the limit set depends on the choice of the initial open intervals. The paper shows that the Hausdorff dimension of the limit set depends continuously on the endpoints of removed intervals, and the same is true of the topological entropy for the induced dynamical system. The proof is based on first translating the problem into a symbolic dynamics problem involving an object called Markov diagrams. These are shown to depend continuously on the endpoints of the removed intervals and from there the main results are deduced.

Reviewer: G.Swiatek (Stony Brook)

##### MSC:

37E99 | Low-dimensional dynamical systems |

54C70 | Entropy in general topology |

37D99 | Dynamical systems with hyperbolic behavior |