The structure of graph maps without periodic points

In this paper we show that any graph map without periodic points has only one minimal set. We describe a class of graph maps without periodic points. Our main result is to give a structure theorem of graph maps without periodic points, which states that any graph map without periodic points must be topologically conjugate to one of the described class. In addition, we give some applications of the structure theorem.     

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.     

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).     

Some simple conditions implying topological transitivity for interval maps

Summary. We provide some sufficient conditions for topological transitivity of piecewise monotonic maps on [0,1]. Our theorems provide shorter and elementary proofs for some known recent results.     

On topological entropy of commuting tree maps

Let T$T$ be a tree with s$s$ ends and f,g$f,g$ be continuous maps from T$T$ to T$T$ with f°g=g°f$f°g=g°f$. In this note we show that if there exists a positive integer m≥2$m\ge 2$ such that gcd(m,l)=1$gcd(m,l)=1$ for any 2≤l≤s$2\le l\le s$ and f,g$f,g$ share a periodic point which is a km$km$-periodic point of f$f$ for some positive integer k$k$, then the topological entropy of f°g$f°g$ is positive.     

Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I=[c₂,c₁] denote the core and set E={(x₀,x₁,…)∈(I,f)|xi∈ω(c,f) for all i∈N}. It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of (I,f) is a proper subset of E and such that limk→∞Q(k)≠∞, where Q(k) is the kneading map.We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of (I,f) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of (I,f) is precisely E. Examples of this behavior are provided where limk→∞Q(k) does and does not equal infinity, and in the case where limk→∞Q(k)=∞, the collection of endpoints of (I,f) is always E.     

Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Transitive points via Furstenberg family

Let (X,T) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z₊ with hereditary upward property). A point x∈X is called an F-transitive one if {n∈Z₊:Tnx∈U}∈F for every non-empty open subset U of X; the system (X,T) is called F-point transitive if there exists some F-transitive point. In this paper, we aim to classify transitive systems by F-point transitivity. Among other things, it is shown that (X,T) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive).It is shown that every weakly mixing system is Fip-point transitive, while we construct an Fip-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ^?(Fwt)-transitive if and only if it is weakly disjoint from every P-system.     

On open problems concerning distributional chaos for triangular maps

We show that in the class T of the triangular maps (x,y)?(f(x),g_x(y)) of the square there is a map of type 2^∞ with non-minimal recurrent points which is not DC3. We also show that every DC1 continuous map of a compact metric space has a trajectory which cannot be (weakly) approximated by trajectories of compact periodic sets. These two results make possible to answer some open questions concerning classification of maps in T with zero topological entropy, and contribute to an old problem formulated by A.N. Sharkovsky.     

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.     

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.     

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).     

A chaotic function with a distributively scrambled set of full Lebesgue measure

Misiurewicz proved that there exists a continuous map of the interval [0, 1] onto itself for which there exists a scrambled set of full Lebesgue measure. In this paper, we form a continuous interval map which has a distributively scrambled set of full Lebesgue measure in which each point has dense orbit. This contains Misiurewicz’s result, since any distributively scrambled set must be scrambled but the converse is not generally true.     

On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

Omega-limit sets in hereditarily locally connected continua

In this paper we give a topological characterization of ω-limit sets in hereditarily locally connected continua. Moreover, we characterize also orbit-enclosing ω-limit sets in these spaces.     

Chaos via Furstenberg family couple

In this paper we define (F₁,F₂)-chaos via Furstenberg family couple F₁ and F₂. It turns out that the Li-Yorke chaos and distributional chaos can be treated as chaos in Furstenberg families sense. Some sufficient conditions such that a system is the (F₁,F₂)-chaotic (Theorems 4.2 and 4.4) are given. In addition, we construct an example as an application. It is showed that the second type of distributional chaos cannot imply the first type of distributional chaos even though the scrambled set is uncountable.     

Sharkovsky’s program for the classification of triangular maps is almost completed

For a continuous map of the interval, there are more than 50 conditions characterizing zero topological entropy. Some are applicable to the class of triangular maps (x,y)?(f(x),g_x(y)) of the square, but only a few of them are equivalent in this more general setting. In 1989, A.N. Sharkovsky posed the problem of proving or disproving all possible implications between them. During last 20 years, 32 conditions were considered, and most of the work was done. Only 45 relations out of 992 remained not clear. In this paper we give a survey of known results, provide two new examples disproving another 26 possible implications, and spell out the remaining 19 open problems; all but one concern distributional chaos.     

A note on the average shadowing property for expansive maps

Let f be a continuous map of a compact metric space. Assuming shadowing for f we relate the average shadowing property of f to transitivity and its variants. Our results extend and complete the work of Sakai [K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131 (2003) 15-31].     

Transitivities of maps of general topological spaces

In this paper we investigate orbit-transitivity, strong orbit-transitivity, ω-transitivity and open-set-transitivity of maps of general topological spaces. The relation between these transitivities is studied. We discuss various topological spaces, containing pseudo-regular spaces, partially completable spaces, and topological spaces without quasi-isolated points. Several conditions on spaces and on continuity for one transitivity to imply another transitivity are given.     

A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F(x,y)=(f(x),g(x,y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite ω-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive.