Continuity of the hausdorff dimension for piecewise monotonic maps

In this paper a piecewise monotonic mapT:X→?, whereX is a finite union of intervals, is considered. DefineR(T)= \(\mathop \cap \limits_{n = 0}^\infty \overline {T^{ - n} X} \) . The influence of small perturbations ofT on the Hausdorff dimension HD(R(T)) ofR(T) is investigated. It is shown, that HD(R(T)) is lower semi-continuous, and an upper bound of the jumps up is given. Furthermore a similar result is shown for the topological pressure.     

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC ¹-mapX→ℝ for which {c∈X: T′c=0} is finite. Set . Fix ann ∈ ℕ. For ε>0, theC ¹-map is called an ε-perturbation ofT if is a piecewise monotonic map with at mostn intervals of monotonicity and is ε-close toT in theC ¹-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation ofT has a unique measure of maximal entropy, and the map is continuous atT in the weak star-topology.     

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.     

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ _f ⁺ isomorphic to (I, f) generalizing the \-shift and another shift Σ _M over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ _M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.     

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/n J_i, theJ _i are disjoint intervals,f/J _i is increasing or decreasing and continuous, andh _top(f)>0.     

On elliptic equations with piecewise constant coefficients

In this work we prove an existence and uniqueness theorem for solutions in ²(R^t) of second order elliptic equations with coefficients that are constant on the half-spaces R+ⁿ and R ?ⁿ     

Maximal measures for simple piecewise monotonic transformations

Summary It is shown that the transformation xgbx+ (mod 1) (>1, 0<1) on [0, 1] has unique maximal measure.     

Homogeneous maps as piecewise endomorphisms

Let G be a module over a ring R and let C = {C_a} be a cover of G by submodules CQ with certain properties. We investigate relationships among the subnear-rings, PE_R(G,C), of the centralizer near-ring M_R(G), determined by various covers C     

Conjugacy between piecewise monotonic functions and their iterative roots

It was proved that all continuous functions are topologically conjugate to their continuous iterative roots in monotonic cases. An interesting problem reads: Does the same conclusion hold in non-monotonic cases?We give a negative answer to the problem by presenting a necessary condition for the topological conjugacy,which helps us construct counter examples. We also give a sufficient condition as well as a method of constructing the topological conjugacy.     

Generalized bounded variation and applications to piecewise monotonic transformations

Summary We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Hölder-continuous or, more generally, of bounded p-variation.This work has been supported by the Deutsche Forschungsgemeinschaft     

On strongly elliptic singular integral operators with piecewise continuous coefficients

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.     

Generic properties of invariant measures for continuous piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL withh _top (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 denseG -sets, that the set of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseG -set.     

Generic properties of invariant measures for simple piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL of certain piecewise monotonic transformations 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 dense-sets, that the se of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGin/gd-set.     

Iterative roots of piecewise monotonic functions with finite nonmonotonicity height

It is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function. In this paper, we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts. By investigating the trajectory of possible continuous roots, we give a general method to find all iterative roots of those functions with finite nonmonotonicity height.