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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