Discontinuities of the pressure for piecewise monotonic interval maps

Abstract

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.