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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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