# Piecewise invertible dynamical systems

## Authors

- Received:

DOI: 10.1007/BF00334191

- Cite this article as:
- Hofbauer, F. Probab. Th. Rel. Fields (1986) 72: 359. doi:10.1007/BF00334191

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## Summary

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 ω(*x*) of all limit points of the orbit of an *x*∈*X*. The topological entropy of *L* is the logarithm of the spectral radius of the irreducible submatrix, which is a *l*^{1}-operator. Besides these sets *L* there are two *T*-invariant sets *Y* and *P*, such that for all *x*∈*X* the set ω(*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 ω(*y*) with *y*∈*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*).