We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.
Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.
We prove the following theorem: Let G be a compact connected graph and let f: G → G be a piecewise linear surjection which satisfies the following condition: for each nondegenerate subcontinuum A of G, there is a positive integer n such that fⁿ(A) = G. Then, for each ε > 0, there is a map which is ε-close to f such that the inverse limit is hereditarily indecomposable.
For an arbitrary topological group G any compact G-dynamical system (G,X) can be linearly G-represented as a weak*-compact subset of a dual Banach space V*. As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G,X) is weakly almost periodic (WAP). In the present paper we study the wider class of compact G-systems which can be linearly represented as a weak*-compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým...
Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean is continuous on X for any ϕ ∈ C(X), and the induced C*-valued...
We present a new technique for showing that inverse limit spaces of certain one-dimensional Markov maps are not homeomorphic. In particular, the inverse limit spaces for the three maps from the tent family having periodic kneading sequence of length five are not homeomorphic.
We describe the homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological entropy and that they are actually semi-conjugate to an algebraic homeomorphism in the case when the entropy is positive.
Let be a permutation of an abstract set . In ZFC, we find a necessary and sufficient condition it terms of cardinalities of the -orbits that allows us to topologize as a topological dynamical system on a compact Hausdorff space. This extends an early result of H. de Vries concerning compact metric dynamical systems. An analogous result is obtained for -actions without periodic points.
Hyperbolic homeomorphisms on compact manifolds are shown to have both inverse shadowing and bishadowing properties with respect to a class of δ-methods which are represented by continuous mappings from the manifold into the space of bi-infinite sequences in the manifold with the product topology. Topologically stable homeomorphisms and expanding mappings are also considered.