Franz Hofbauer

Ergodic Properties of Invariant Measures for Piecewise Monotonic Transformations

During the last decade a lot of research has been done on onedimensional dynamics. Different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered. In most of these cases the Perron-Frobenius-operator plays an important role. In this paper we try to unify these different examples, which are discussed in detail below. First we give a discription of the results proved in this paper.

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II

The results about measures with maximal entropy, which are proved in [3], are extended to the following more general class of transformations on the unit intervalI : I=∪i=1/nJi, theJi are disjoint intervals,f/Ji is increasing or decreasing and continuous, andhtop(f)>0.

Quadratic maps without asymptotic measure

An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.

Piecewise invertible dynamical systems

SummaryThe 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 l1-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).

β-Shifts have unique maximal measure

It is proved that β-shifts have unique measure with maximal entropy by constructing an isomorphism of the β-shift with another topological dynamical system and proving it for this system.

Equilibrium states for piecewise monotonic transformations

We show that equilibrium states μ of a function φ on ([0,1], T ), where T is piecewise monotonic, have strong ergodic properties in the following three cases: (i) sup φ — inf φ h top ( T ) and φ is of bounded variation. (ii) φ satisfies a variation condition and T has a local specification property. (iii) φ = —log | T ′|, which gives an absolutely continuous μ, T is C 2 , the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

The topological entropy of the transformationx ↦ax (1−x)

A well-known conjecture about the transformationTa:x ↦ax (1−x) on [0, 1], where 2≤a≤4, says that the mapa ↦htop (Ta) is monotone. In this paper we show that this is connected with a property of the polynomialsPk (t) (4≤t≤8) given byP0 (t)=0 andPk+1(t)=(t−Pkt)2)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a∈[2,4]:htop (Tc), ifca}.

Maximal measures for simple piecewise monotonic transformations

SummaryIt is shown that the transformation x↦gbx+α (mod 1) (β>1, 0≦α<1) on [0, 1] has unique maximal measure.

Generic properties of invariant measures for continuous piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL withhtop (L)>0 of a continuous piecewise monotonic transformation on [0, 1] with the weak topology. We show that the set of periodic orbit measures is dense, that the sets of ergodic, of nonatomic, and of measures with supportL are denseGδ-sets, that the set of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGδ-set.

Topologically transitive subsets of piecewise monotonic maps, which contain no periodic points

If one splits the nonwandering set of a piecewise monotonic map into maximal subsets, which are topologically transitive, one gets two kinds of subsets. The first kind of these subsets has periodic orbits dense, the second kind contains no periodic orbits. In this paper it is shown, that there are only finitely many subsets of the second kind, each of which is minimal and has only finitely many ergodic invariant Borel probability measures.