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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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