## Piecewise invertible dynamical systems.(English)Zbl 0578.60069

The aim of the paper is the investigation of piecewise monotonic maps T of an interval X. The main tool is an isomorphism of (X,T) with a topological Markov chain with countable state space which is described by a 0-1-transition matrix M. The behavior of the orbits of points in X under T is very similar to the behavior of the paths of the Markov chain. Every irreducible submatrix of M gives rise to a T-invariant subset L of X such that L is the set $$\omega$$ (x) of all limit points of the orbit of an $$x\in X$$. The topological entropy of L is the logarithm of the spectral radius of the irreducible submatrix, which is an $$l^ 1$$- operator. Besides these sets L there are two T-invariant sets Y and P, such that for all $$x\in X$$ the set $$\omega$$ (x) is either contained in one of the sets L or in Y or in P. The set P is a union of periodic orbits and Y is contained in a finite union of sets $$\omega$$ (y) with $$y\in X$$ and has topological entropy zero.
This isomorphism of (X,T) with a topological Markov chain is also an important tool for the investigation of T-invariant measures on X. Results in this direction, which are published elsewhere, are described at the end of the paper. Furthermore, a part of the proofs in the paper is purely topological without using the order relation of the interval X, so that some results hold for more general dynamical systems (X,T).

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 28D05 Measure-preserving transformations
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### References:

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