Distributional chaos and irregular recurrence

For a continuous map φ:X→X of a compact metric space, we study relations between distributional chaos and the existence of a point which is quasi-weakly almost periodic, but not weakly almost periodic. We provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos, DC1. Using this we prove that, even in the class of triangular maps of the square, there are no relations to DC1. This result, among others, contributes to the solution of a problem formulated by A.N. Sharkovsky in the eighties.     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

Minimal sets in compact connected subspaces

Let X be a topological space, f:X→X be a continuous map, and Y be a compact, connected and closed subset of X. In this paper we show that, if the boundary X_∂Y contains exactly one point v and f(v)∈Y, then Y contains a minimal set of f.     

A note on chaos via Furstenberg family couple

The concepts of the first type of distributional chaos in the Tan-Xiong sense (Abbrev. DC1 in the Tan-Xiong sense), the second type of strong-distributional chaos (Abbrev. strong DC2) and the third type of strong-distributional chaos (Abbrev. strong DC3) were introduced by Tan et al. [F. Tan, J. Xiong. Chaos via Furstenberg family couple, Topology Appl. (2008), doi:10.1016/j.topol.2008.08.006] for continuous maps of a metric space. However, it turns out that, for continuous maps of a compact metric space, the three mutually nonequivalent versions of distributional chaos can be discussed. Let X be a compact metric space and f:X→X a continuous map. In this paper, we show that for any integer N>0, f is strong DC2 (resp. strong DC3) if and only if f^N is strong DC2 (resp. strong DC3). We also show that the above three versions of distributional chaos are topological conjugacy invariant. In addition, as an application, we present an example.     

The Sarkovskii order for periodic continua

Suppose f is a map of a continuum X onto itself. A periodic continuum of f is a subcontinuum K of X such that fn[K]=K for some positive integer n. A proper periodic continuum of f is a periodic continuum of f that is a proper subcontinuum of X. A proper periodic continuum of f is maximal if and only if X is the only periodic continuum that properly contains it. In this paper it is shown that the maximal proper periodic continua of a map of a hereditarily decomposable chainable continuum onto itself follow the Sarkovskii order, provided the maximal proper periodic continua are disjoint. The case in which the Sarkovskii order does not hold reduces to the scenario in which the map's domain is the union of two overlapping period-two continua, each of which is maximal.     

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmüller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmüller space AT (X). The work was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science Foundation of China (Grant No. 10401036).     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

Implications of pseudo-orbit tracing property for continuous maps on compacta

We look at the dynamics of continuous self-maps of compact metric spaces possessing the pseudo-orbit tracing property (i.e., the shadowing property). Among other things we prove the following: (i) the set of minimal points is dense in the non-wandering set Ω(f), (ii) if f has either a non-minimal recurrent point or a sensitive minimal subsystem, then f has positive topological entropy, (iii) if X is infinite and f is transitive, then f is either an odometer or a syndetically sensitive non-minimal map with positive topological entropy, (iv) if f has zero topological entropy, then Ω(f) is totally disconnected and f restricted to Ω(f) is an equicontinuous homeomorphism.     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).