Stability of the topological pressure for piecewise monotonic maps under C1-perturbations

Assume that X is a finite union of closed intervals and consider a C1-map T : X ? R for which {c ? X : T'c = 0} is finite. Set R(T) = nj T-jX. Fix an n ? N. For e > 0, the C1-map T~ : X ? R is called an e-perturbation of T if T~ is a piecewise monotonic map with at most n intervals of monotonicity and T~ is e-close to T in the C1-topology. The influence of small perturbations of T on the dynamical system (R(T), T) is investigated. Under a certain condition on the continuous function f : X ? R, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous function f : X ? R. If (R(T), T) has positive topological entropy and a unique measure œ of maximal entropy, then every sufficiently small perturbation T~ of T has a unique measure œ~ of maximal entropy, and the map T~ ? œ~ is continuous at T in the weak star-topology.