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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 \(X\)”.

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