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Perturbations of a topologically transitive piecewise monotonic map on the interval. (English) Zbl 0948.37026
Let \(X\) be a finite union of closed intervals and consider a piecewise monotonic map \(H:X\to \mathbb{R}\), which means that there exists a finite partition of \(X\), say \(Z\), into pairwise disjoint open interval with \(\bigcup_{z\in Z} \overline{z}= X\) such that \(H|_Z\) is bounded, strictly monotone and continuous for all \(z\in Z\). Set \(R(H)= \bigcap_{n=0}^\infty \overline{H^{-n}X}\). Note that \(R(H)\) can be considered as the set, where \(H^n\) is defined for all \(n\in \mathbb{N}\). The author deals with the influence of small perturbations on \(H\) on the dynamical system \((R(H),H)\) and presents a condition which implies the continuity of the maximal measure. Under this condition the author managed to show that certain maximal topologically transitive subsets of \(R(H)\) behave stably.

MSC:
37E05 Dynamical systems involving maps of the interval
37E15 Combinatorial dynamics (types of periodic orbits)
37A40 Nonsingular (and infinite-measure preserving) transformations
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