Dendrite Map Research Articles

A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled

A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled

Origin of wiring specificity in an olfactory map revealed by neuron type-specific, time-lapse imaging of dendrite targeting.

How does wiring specificity of neural maps emerge during development? Formation of the adult Drosophila olfactory glomerular map begins with the patterning of projection neuron (PN) dendrites at the early pupal stage. To better understand the origin of wiring specificity of this map, we created genetic tools to systematically characterize dendrite patterning across development at PN type-specific resolution. We find that PNs use lineage and birth order combinatorially to build the initial dendritic map. Specifically, birth order directs dendrite targeting in rotating and binary manners for PNs of the anterodorsal and lateral lineages, respectively. Two-photon- and adaptive optical lattice light-sheet microscope-based time-lapse imaging reveals that PN dendrites initiate active targeting with direction-dependent branch stabilization on the timescale of seconds. Moreover, PNs that are used in both the larval and adult olfactory circuits prune their larval-specific dendrites and re-extend new dendrites simultaneously to facilitate timely olfactory map organization. Our work highlights the power and necessity of type-specific neuronal access and time-lapse imaging in identifying wiring mechanisms that underlie complex patterns of functional neural maps.

On conjugacy of natural extensions of one-dimensional maps

AbstractWe prove that for any non-degenerate dendrite D, there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$ such that the natural extensions (as known as shift homeomorphisms) $\sigma _F$ and $\sigma _f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map f does not depend on the dendrite D and can be selected so that the inverse limit $\underleftarrow {\lim } (D,F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and Hénon maps to natural extensions of dendrite maps.

Recurrence and Chaos of Local Dendrite Maps

Let X be a local dendrite, and let f be a continuous self‐mapping of X. Let E(X) represent the subset of endpoints of X. Let AP(f) denote the subset of almost periodic points of f, R(f) be the subset of recurrent points of f, and P(f) be the subset of periodic points of f. In this work, it is shown that if and only if E(X) is countable. Also, we show that if E(X) is countable, then R(f) = X (respectively, ) if and only if either , and f is a homeomorphism topologically conjugate to an irrational rotation, or P(f) = X (respectively, ). In this setting, we derive that if E(X) is countable, then, on local dendrites , transitivity = chaos.

One-dimensional dynamical systems

The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.

Properties of Dynamical Systems on Dendrites and Graphs

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

Equicontinuity and Li–Yorke pairs of dendrite maps

ABSTRACT In this paper, relationships between equicontinuity of a dendrite map f on , absence of Li–Yorke pairs, collection of minimal sets and regularly recurrent points are investigated.

The depths of the centres and the attracting centres of a class of dendrite maps

Let D1 be the set of dendrites with the number of the accumulation points of branch points being finite, D∈D1 and f be a continuous map from D to D. Denote by R(f), Ω(f) and ω(x,f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω-limit points of x under f, respectively. Write ω(f)=∪x∈Dω(x,f) and ωn+1(f)=∪x∈ωn(f)ω(x,f) and Ωn+1(f)=Ω(f|Ωn(f)) for any n∈N. In this paper, we show that Ω4(f)=R(f)‾ and the depth of f is at most 4, and ω4(f)=ω3(f). Furthermore, we show that there exist dendrites D1,D2∈D1 and f1∈C0(D1) and f2∈C0(D2) such that Ω3(f1)≠R(f1)‾ and ω3(f2)≠ω2(f2).

ω-Chaos Without Infinite LY-Scrambled Set on Gehman Dendrite

We answer the last question left open in [Kočan, 2012] which asks whether there is a relation between an infinite LY-scrambled set and [Formula: see text]-chaos for dendrite maps. We construct a continuous self-map of a dendrite without any infinite LY-scrambled set but containing an uncountable [Formula: see text]-scrambled set.

Möbius disjointness conjecture for local dendrite maps

We prove that the Möbius disjointness conjecture holds for graph maps with zero topological entropy and for all monotone local dendrite maps. We further show that this also holds for continuous maps on certain class of dendrites. Moreover, we see that there is an example of transitive dendrite map with zero entropy for which Möbius disjointness conjecture holds.

Dendrites and Chaos

We answer the two questions left open in [Kočan, 2012] i.e. whether there is a relation between [Formula: see text]-chaos and distributional chaos and whether there is a relation between an infinite LY-scrambled set and distributional chaos for dendrite maps. We construct a continuous self-map of a dendrite without any DC3 pairs but containing an uncountable [Formula: see text]-scrambled set. To answer the second question we construct a dendrite [Formula: see text] and a continuous dendrite map without an infinite LY-scrambled set but with DC1 pairs.

On strong sensitive points of continuous maps on dendrites

Let f : I → I be a continuous map of a closed interval I. It is known that the positiveness of topological entropy of f, the existence of a horseshoe and the existence of a strong sensitive point from are mutually equivalent. The aim of the paper is to investigate some relations between the mentioned properties for continuous maps of dendrites (locally connected continua without subsets homeomorphic to the circle).

Special $$\alpha $$ α -Limit Points and $$\gamma $$ γ -Limit Points of a Dendrite Map

Let (X, d) be a compact metric space and f be a continuous map from X to X. Denote by R(f), \({ SA}(f)\) and \(\Gamma (f)\) the set of recurrent points, the set of special \(\alpha \)-limit points and the set of \(\gamma \)-limit points of f, respectively. It is well known that for an interval map f, the following three statements hold: (1) \(R(f)\subset { SA}(f)\cap \Gamma (f)\); (2) \({ SA}(f)= \Gamma (f)\); (3) \({ SA}(f)\cup \Gamma (f)\subset \overline{R(f)}\). The aim of this paper is to show that the above statement (1) holds for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\) (the cardinal number of the set of positive integers) and the above statements (2) and (3) do not hold for maps of dendrites with the number of endpoints being \(\aleph _{\mathbf{0}}\). Besides, we also study unilateral \(\gamma \)-limit points for maps of dendrites with the number of branch points being finite.

Negative limit sets of graph maps

Let $G$ be a graph and $f:G\rightarrow G$ be a continuous map. A sequence $\theta=(x_0,x_{-1},\ldots,x_{-n},\ldots)$ of points in $G$ is called a negative orbit through $x_0$ of $f$ if $f(x_{-n})=x_{-n+1}$ for every positive integer $n$. The $\alpha$-limit set of $\theta$ is the set $\alpha(\theta,f)$ of all limit points of $\theta$. In this paper, we show that the $\alpha$-limit set of every negative orbit of $f$ is an $\omega$-limit set of some point in $G$. Besides, we illustrate that the above result does not hold for a dendrite map.

A note on periodic points of dendrite maps and their induced maps

Let X be a compact metric space and 2X be the hyperspace of all nonempty closed subsets of X endowed with the Hausdorff metric. It is well known that for each continuous map f:X→X, the density of periodic points of f implies the density of periodic points of the induced map 2f:2X→2X. Méndez (2010) conjectured in [6] that the converse is true when the phase space X is a dendrite. In [8], Ŝpitalský (2015) constructed a continuous transitive map F:X→X on a dendrite X with only two periodic points. We prove in this note that the set of periodic points of its induced map 2F is dense in 2X. This answers in the negative Méndez's conjecture.

Sensitivity of dendrite maps

Suppose that X is a dendrite and f:X→X is a sensitive continuous map. We show that (a) (X,f) contains a bilaterally transitive subsystem with nonempty interior; (b) the system (X,f) satisfies only one of the following two conditions: (b1) (X,f) contains a topologically transitive subsystem with nonempty interior; (b2) there exists an f-invariant nowhere dense closed subset A of X such that the attraction basin Basin(A,f) contains a residual subset B of an open set and the strong attraction basin Sbasin(A,f) is dense in B; (c) if X is completely regular, then (X,f) contains a relatively strongly mixing subsystem with nonempty interior, dense periodic points and positive topological entropy. Unlike for interval maps, we construct a sensitive dendrite map with zero topological entropy.

Non-wandering sets of the powers of dendrite maps

Let (X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f) and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold: (1) If x ∈ Ω(f) − Ω(fn) for some n ≥ 2, then x ∈ EP(f). (2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = N0 (the cardinal number of the set of positive integers).

Transitive dendrite map with zero entropy

Hoehn and Mouron [Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys.34 (2014), 1897–1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn–Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Entropy estimates for transitive maps on trees. Topology40(3) (2001), 551–569].

Counterexamples of continuous maps on dendrites

We take a list of properties of continuous maps on compact metric spaces, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and others. Our aim is to find the relations between the properties in the case of dendrite maps. To achieve the aim at least partially, we consider several examples of dendrite maps and check which of the properties particular examples have. It also helps us to understand the dynamics of dendrite maps.

Formula omitted]-Limit sets for monotone local dendrite maps

We show that any ω-limit set of a monotone local dendrite map is a minimal set which is either finite, or a Cantor set, or a circle.