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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.

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