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Discontinuities of the pressure for piecewise monotonic interval maps. (English) Zbl 0972.37024
From the abstract: “For a piecewise monotonic map $$T: X \to \mathbb R$$, 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 $$f_0 : P \to \mathbb R$$ 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 functions $$f : X \to \mathbb R$$ with $$f$$ restricted to $$P$$ equal to $$f_0$$, equals the maximum of $$h_{\text{top}} (R(T),T)$$ 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 $$f: X \to \mathbb R$$. In the case of a continuous piecewise monotonic map $$T: X \to \mathbb R$$ 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 $$X$$”.

##### MSC:
 37E05 Dynamical systems involving maps of the interval 37C75 Stability theory for smooth dynamical systems
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