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Continuity of the Hausdorff dimension for piecewise monotonic maps. (English) Zbl 0768.28010
If \(X\) is a finite union of closed intervals, \(T: X\to \mathbb{R}\) a piecewise monotonic map with respect to some partition \(\mathcal Z\) then \((R(T),T)\) with \(R(T)=\bigcap^ \infty_{n=0}\overline{T^{-n}(X)}\) is a dynamical system. The author considers the influence of small perturbations of \(T\) on the set \(R(T)\). He introduces some notions of convergence on the families of all \((X,T,{\mathcal Z})\) and all \((X,T,f,{\mathcal Z})\), where \(f\) is a further piecewise continuous function on \(X\) with respect to \(\mathcal Z\). The principal results are that the pressure function \((X,T,f,{\mathcal Z})\mapsto p(R(t),T,f)\) as well as the Hausdorff dimension function \((X,T,{\mathcal Z})\mapsto \dim(R(t))\) are lower semi-continuous and that jumps of these functions will be bounded in a certain sense. The proofs depend on methods of graph theory.

MSC:
28D20 Entropy and other invariants
28A78 Hausdorff and packing measures
28D10 One-parameter continuous families of measure-preserving transformations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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