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Dendrite P (DeP) systems are a new variant of neural-like P systems, abstracted by the information processing and feedback mechanisms of dendrites. In the variant, a global block is assumed to synchronize all of neurons, hence, DeP systems work in synchronous mode. This paper investigates sequential version of the variant, that is, sequential dendrite P (SDeP) systems. Based on maximum number of spikes in neurons, two sequential modes are distinguished: max-sequentiality and max-pseudo-sequentiality strategies. SDeP systems have two interesting and recognizable features: (i) it behaves as a firing-storing process; (ii) cooperative firing mechanism. The computational completeness of SDeP systems is discussed. We prove that SDeP systems can be used as Turing universal number generating/accepting devices for max-sequentiality and max-pseudo-sequentiality strategies. We also establish a small universal function computing device of SDeP systems with 91 neurons in max-sequentiality strategy.

Let S = S1, ..., Sm be a system of contracting similarities of ℝ2. The attractor K(S) of the system S is a non-empty compact set satisfying K = S1(K) ∪ ... ∪ Sm(K). We consider contractible polygonal systems S which are defined by a finite family of polygons whose intersection graph is a tree and therefore the attractor K(S) is a dendrite. We find conditions under which a deformation S′ of a contractible polygonal system S has the same intersection graph and therefore the attractor K(S′) is a self-similar dendrite which is isomorphic to the attractor K of the system S.

Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1) ω ( x , f ) ⊆ Per ( f ) ¯ \omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ; (2) ω(x, f ) is either a periodic orbit or a minimal Cantor set. In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3) Ω ( f ) = Per ( f ) ¯ \Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f. The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

Nowadays, the physical analogy in biology has become a desirable research domain for academics. Here, we discussed many exciting features regarding the biological system that could help us to understand the underlying mechanism inside it. This manuscript focuses on the soliton/Bloch wave concept along with the periodic active protein polarized dendrite. The velocity of the existent soliton wave is identified by considering the transmission line model of protein polarized dendrite as a dielectric waveguide. This paper aims to understand and explore the mechanisms of biological rhythms and the role of neural oscillations and synchrony in information processing.

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

Artificial neural networks (ANNs) were originally inspired by the brain; however, very few models use evolution and development, both of which are fundamental to the construction of the brain. We describe a simple neural model, called IMPROBED, in which two neural programs construct an artificial brain that can simultaneously solve multiple computational problems. One program represents the neuron soma and the other the dendrite. The soma program decides whether neurons move, change, die, or replicate. The dendrite program decides whether dendrites extend, change, die, or replicate. Since developmental programs build networks that change over time, it is necessary to define new problem classes that are suitable to evaluate such approaches. We show that the pair of evolved programs can build a single network from which multiple conventional ANNs can be extracted, each of which can solve a different computational problem. Our approach is quite general and it could be applied to a much wider variety of problems.

Let X be a compact connected metric space and $$f :~ X \rightarrow X$$ be a continuous map. In this paper, we prove that if f has a periodic point and $$\omega _f$$ is continuous then the almost periodic set is a finite union of cyclically permuted subcontinua of X. In particular, AP(f) is connected whenever f has a fixed point. Also we show that for dendrites with closed endpoint set, if $$\omega _f (a)$$ is infinite, then $$\omega _f$$ is continuous at a if, and only if, f is equicontinuous at a. We show that the later result fails whenever $$\omega _f (a)$$ is finite or the endpoints set is not closed. We give an example of a local dendrite map $$f :~ X \rightarrow X$$ for which $$\omega _f$$ is continuous, $$f_{|X_{\infty }}$$ is equicontinuous but f is not equicontinuous on the whole space X. Finally, we answer to an open question raised by Acosta and Fernandez, (Equicontinuous mappings on finite trees, Fund. Math) by providing a class of dendrites on which the equicontinuity of $$f_{|X_{\infty }}$$ imply the equicontinuity of f.

In this paper, we provide some equivalent conditions of local dendrite maps to be equicontinuous.

Significance To study the systematic variation of dendrite diameters, we established a superresolution method that allows us to resolve dendrite diameters in Drosophila class IV dendritic arborization neurons, a model cell for studying branching morphogenesis. Interestingly, the diameters do not follow any of the known scaling laws. We propose a different scaling law that follows from two concepts: Terminal branches have the smallest diameters, whose average is about 230 nm, and there is an incremental increase in cross-sectional area needed to support each additional terminal branch. The law is consistent with the growing dendritic tips making the primary metabolic demand, which is supplied by microtubule-based transport. If the law generalizes to other neurons, it may facilitate segmentation in connectomic studies., The systematic variation of diameters in branched networks has tantalized biologists since the discovery of da Vinci’s rule for trees. Da Vinci’s rule can be formulated as a power law with exponent two: The square of the mother branch’s diameter is equal to the sum of the squares of those of the daughters. Power laws, with different exponents, have been proposed for branching in circulatory systems (Murray’s law with exponent 3) and in neurons (Rall’s law with exponent 3/2). The laws have been derived theoretically, based on optimality arguments, but, for the most part, have not been tested rigorously. Using superresolution methods to measure the diameters of dendrites in highly branched Drosophila class IV sensory neurons, we have found that these types of power laws do not hold. In their place, we have discovered a different diameter-scaling law: The cross-sectional area is proportional to the number of dendrite tips supported by the branch plus a constant, corresponding to a minimum diameter of the terminal dendrites. The area proportionality accords with a requirement for microtubules to transport materials and nutrients for dendrite tip growth. The minimum diameter may be set by the force, on the order of a few piconewtons, required to bend membrane into the highly curved surfaces of terminal dendrites. Because the observed scaling differs from Rall’s law, we propose that cell biological constraints, such as intracellular transport and protrusive forces generated by the cytoskeleton, are important in determining the branched morphology of these cells.

Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace C(G) of a dendrite G, then, we give necessary and sufficient conditions so that the aforementioned result can be applied when G is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space X for which the anchored hyperspace Cp(X) is an arc for some p∈X.

Client application workloads for data systems are known to vary in load and access patterns over time. This variability can place undue stress on data systems, tying up resources and degrading performance. To meet this challenge, systems must adapt by adjusting resource allocation and processing techniques to ameliorate contention and to deliver stable performance. We demonstrate Dendrite, a system designed to bootstrap adaptivity for data systems through its widely-applicable approach for extracting metrics, developing adaption rules, and applying them through user-defined functions to effect system behaviour changes. We highlight Dendrite's features and capabilities through a proof-of-concept implementation with the popular PostgreSQL database system.

Let D be a dendrite and G be a finitely generated group. In this paper, we obtain some equivalent conditions of equicontinuous flow (D,G) under some conditions.

Let D 1 be the set of dendrites with the number of the accumulation points of branch points being finite, D ∈ D 1 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 ∈ D ω ( x , f ) and ω n + 1 ( f ) = ∪ x ∈ ω n ( f ) ω ( x , f ) and Ω n + 1 ( f ) = Ω ( f | Ω n ( f ) ) for any n ∈ N . In this paper, we show that Ω 4 ( f ) = R ( f ) ‾ and the depth of f is at most 4, and ω 4 ( f ) = ω 3 ( f ) . Furthermore, we show that there exist dendrites D 1 , D 2 ∈ D 1 and f 1 ∈ C 0 ( D 1 ) and f 2 ∈ C 0 ( D 2 ) such that Ω 3 ( f 1 ) ≠ R ( f 1 ) ‾ and ω 3 ( f 2 ) ≠ ω 2 ( f 2 ) .

We prove that the M\"obius disjointness conjecture holds for graph maps and for all monotone local dendrite maps. We further show that this also hold for continuous map on certain class of dendrites. Moreover, we see that there is a transitive dendrite map with zero entropy for which M\"obius disjointness holds., Comment: 18 pages

Let X be a local dendrite and let f : X → X be a monotone map. Denote by P ( f ) and Ω ( f ) the sets of periodic points and nonwandering points of f, respectively. We show that Ω ( f ) = P ( f ) ‾ , whenever P ( f ) is nonempty and Ω ( f ) is the unique minimal set included in a circle which is either a Cantor set or a circle, whenever P ( f ) is empty. In the case where the set of endpoints of X is countable, we show that Ω ( f ) = P ( f ) whenever P ( f ) is nonempty.

Schizophrenia (SCZ) is highly polygenic, and thousands of genes contribute to its risk. The 145 GWAS loci identified to date do not fully reveal SCZ genetic risk pathways. In this study, we explore a cost-effective strategy to increase power of inference of novel pathways, by expanding the analysis to include sub-threshold GWAS (subGWAS) loci. We identify 180 subGWAS loci (e.g., 5 x10-8 < P ≤ 10-6) based on SCZ summary statistics of 40,675 cases and 64,643 controls from CLOZUK and PGC datasets, and show that subGWAS loci contain substantial true genetic association signals. We merge GWAS (sigGWAS) and subGWAS loci and identify in total 304 high-confidence risk genes (HRGs) by jointly modeling the expanded set of loci. We identify dendrite development and morphogenesis (DDM, GO:0016358 and GO:0048813) as a novel category of biological processes implicated in SCZ genetic risk. SigGWAS loci fail to detect DDM, which is predominantly enriched in subGWAS loci. Further, DDM genes are significantly enriched for heritability of SCZ, as well as bipolar disorder and major depression. Genes in this functional process show cell type specificity in neurons in both fetal and adult brains, and their involvement in SCZ risk is further supported by eQTL analysis of SCZ risk alleles. We derived induced pluripotent stem cell (iPSC) lines from sporadic SCZ patients and normal controls and observe increased neurite lengths and soma sizes in patient-derived iPSC lines along multiple time points during neuronal development, further validating the genetic findings. We also find that the implicated genes are enriched in FDA-approved drug targets, suggesting a therapeutic potential for targeting the implicated biological processes for prevention and treatment. Our results showcase that expanding the analysis to include subGWAS loci is a valuable strategy for enhancing power of uncovering disease mechanisms, especially those of weak effect size, for SCZ and other complex diseases.

We characterize dendrites $D$ such that a continuous selfmap of $D$ is generically chaotic (in the sense of Lasota) if and only if it is generically $\varepsilon$-chaotic for some $\varepsilon>0$. In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with nonempty interiors under iterates of the map. A dendrite $D$ belongs to this class if and only if it is completely regular, with all points of finite order (that is, if and only if $D$ contains neither a copy of the Riemann dendrite nor a copy of the $\omega$-star)., Comment: accepted for publication in Ergodic Theory Dynam. Systems

We find the conditions under which the attractor \(K({\mathcal {S}}')\) of a deformation \({\mathcal {S}}'\) of a contractible P-polygonal system \({\mathcal {S}}\) in \(\mathbb {R}^2\) is a dendrite. The most important one is the parameter matching condition at the points where the images of the vertices of the polygon P meet.

Common optical pattern analysis can be characterized by diverse sophisticated mathematical methods, similar to the known mechanism of neural activities. It is known that living primitive creatures with small numbers of neurons can recognize objects, navigate in their environment and distinguish between similar objects. Based on our investigations regarding the neural coding, storing and retrieving information and topological structures of the dendrite trees, we developed a new theory about the pure neural-based optical signal analysis. Basic assumptions of this theory are that neurons only fire or not fire, that they are combined with each other and that their synaptic structures seems to follow topological structures. These assumptions have not been taken into account by other models. Together which the cybernetic model developed by Helmuth Benesch’s “Carrier-Pattern-Meaning” [1], we developed and tested a simple algorithm to find and highlight structures, textures, and hidden or disturbed objects from random pictures. We have found that the overall basic principle of our algorithm can be defined as a potential oriented approach to describe the neuronal activity and interaction.

Dendrite P systems (DeP systems) are a recently introduced neural-like model of computation. They provide an alternative to the more classical spiking neural (SN) P systems. In this paper, we present the first software simulator for DeP systems, and we investigate the key features of the representation of the syntax and semantics of such systems. First, the conceptual design of a simulation algorithm is discussed. This is helpful in order to shade a light on the differences with simulators for SN P systems, and also to identify potential parallelizable parts. Second, a novel simulator implemented within the PLingua simulation framework is presented. Moreover, MeCoSim, a GUI tool for abstract representation of problems based on P system models has been extended to support this model. An experimental validation of this simulator is also covered. Ministerio de Economía, Industria y Competitividad TIN2017-89842-P (MABICAP)

Dendrites are cellular structures essential for the integration of neuronal information. These elegant but complex structures are highly patterned across the nervous system but vary tremendously in their size and fine architecture, each designed to best serve specific computations within their networks. Recent in vivo imaging studies reveal that the development of mature dendrite arbors in many cases involves extensive remodeling achieved through a precisely orchestrated interplay of growth, degeneration, and regeneration of dendritic branches. Both degeneration and regeneration of dendritic branches involve precise spatiotemporal regulation for the proper wiring of functional networks. In particular, dendrite degeneration must be targeted in a compartmentalized manner to avoid neuronal death. Dysregulation of these developmental processes, in particular dendrite degeneration, is associated with certain types of pathology, injury, and aging. In this article, we review recent progress in our understanding of dendrite degeneration and regeneration, focusing on molecular and cellular mechanisms underlying spatiotemporal control of dendrite remodeling in neural development. We further discuss how developmental dendrite degeneration and regeneration are molecularly and functionally related to dendrite remodeling in pathology, disease, and aging.

A typical feature of hyperbox-based dendrite morphological neurons (DMN) is the generation of sharp and rough decision boundaries that inaccurately track the distribution shape of classes of patterns. This feature is because the minimum and maximum activation functions force the decision boundaries to match the faces of the hyperboxes. To improve the DMN response, we introduce a dendritic model that uses smooth maximum and minimum functions to soften the decision boundaries. The classification performance assessment is conducted on nine synthetic and 28 real-world datasets. Based on the experimental results, we demonstrate that the smooth activation functions improve the generalization capacity of DMN. The proposed approach is competitive with four machine learning techniques, namely, Multilayer Perceptron, Radial Basis Function Network, Support Vector Machine, and Nearest Neighbor algorithm. Besides, the computational complexity of DMN training is lower than MLP and SVM classifiers.

Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Wa\.zewski dendrite $D_\infty$ is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group $G_\infty$, we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of $G_\infty$. This allows us to prove that point-stabilizers in $G_\infty$ are amenable and to describe the universal Furstenberg boundary of $G_\infty$., Comment: Slight modifications about the exposition

In this paper, we prove that if X is a local dendrite different from the circle and $$f:X\rightarrow X$$ is a continuous map then f is pointwise recurrent if and only if it is a homeomorphism such that every cut point is periodic. Meanwhile, we show in the case of dendrites that the set of almost periodic points is a continuum and equals to the union of all $$\omega $$-limit sets provided that the omega limit map is continuous and we show that it is only a necessary condition for the continuity of the omega limit map. We end the paper by suggesting some new open problems.

This chapter considers the relationship between learning and memory, the types of learning and their characteristics and the neurophysiological processes underlying them.

Dendritic spines, which are small protrusions on the dendrites of a neuron, are of interest in neuroscience as they are related to cognitive processes such as learning and memory. We analyse the distribution of spine locations on six different dendrite trees from mouse neurons using point process theory for linear networks. Besides some possible small-scale repulsion, we find that two of the spine point pattern data sets may be described by inhomogeneous Poisson process models, while the other point pattern data sets exhibit clustering between spines at a larger scale. To model this we propose an inhomogeneous Cox process model constructed by thinning a Poisson process on a linear network with retention probabilities determined by a spatially correlated random field. For model checking we consider network analogues of the empirical F -, G -, and J -functions originally introduced for inhomogeneous point processes on a Euclidean space. The fitted Cox process models seem to not only catch the clustering of spine locations between spines, but also possess a large variance in the number of points for some of the data sets causing large confidence regions for the empirical F - and G -functions.

According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to a repelling fixed point., 21 pages, 13 figures

Let G be the homeomorphism group of a dendrite. We study the normal subgroups of G. For instance, there are uncountably many non-isomorphic such groups G that are simple groups. Moreover, these groups can be chosen so that any isometric G-action on any metric space has a bounded orbit. In particular they have the fixed point property (FH)., Comment: Modifications according to referee's comments

Let G be a group acting by homeomorphisms on a local dendrite X with countable set of endpoints. In this paper, it is shown that any minimal set M of G is either a finite orbit, or a Cantor set or a circle. Furthermore, we prove that if G is a finitely generated group, then the flow ( G , X ) is a pointwise recurrent flow if and only if one of the following two statements holds: (1) X = S 1 , and ( G , S 1 ) is a minimal flow conjugate to an isometric flow, or to a finite cover of a proximal flow; (2) ( G , X ) is pointwise periodic.

Dendrite morphological neurons are a type of artificial neural network that can be used to solve classification problems. The major difference with respect to classical perceptrons is that morphological neurons create hyperboxes to separate patterns from different classes, while perceptrons use hyperplanes. In this paper, we introduce an improved version of dendrite morphological neural networks, which we have called dendrite ellipsoidal neuron that employs hyperellipsoids instead of hyperboxes. This ellipsoidal neuron is presented with a new training algorithm, to set the covariance matrix and the centroid of each hyperellipsoid based on k-means++, by applying hill climbing to search for an optimum number of hyperellipsoids. The main advantage of this approach is that dendrite ellipsoidal neuron creates smoother decision boundaries. The proposed neural model was tested on synthetic and real datasets from the UCI machine learning repository (in a paired t-test) achieving an average accuracy of 80.7%, while multi-layer perceptrons gave 78.4%, support vector machines obtained 74.2%, and radial basis networks 72.7%. Lastly, to test the proposed method performance in solving real practical problems, our model was used to detect lane lines on an urban highway, for classifying figures with a Nao robot and for traffic detection.

The curvilinear structure detection is widely applied in many real tasks, such as the fiber classification, river finding, blood vessel detection, and so on. In this paper, we proposed to use the ridge-based curvilinear structure detection (RCSD) for the road extraction from the remote sensing images. First, we employed the morphology trivial opening operation to filter out almost all the small clusters of noise and the small paths. Then RCSD was used to find the road from the remote sensing images. The experiments showed that our proposed method is efficient and give better results than the current existing road-detection methods. Considering the similar structure between backbone in the neuron dendrite images and the road in remote sensing images, we extended the application of RCSD to the backbone detection in neuron dendrite images. The results on backbone detection also proved the efficiency of RCSD.

We build an example of a system S of similarities in R^2 whose attractor is a plane dendrite K\supset [0,1]K⊃[0,1] which satisfies one point intersection property, while the post-critical set of the system S is a countable set whose natural projection to K is dense in the middle-third Cantor set.

Iterated monodromy groups of postcritically finite rational maps form a rich class of self‐similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non‐polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpinski carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non‐renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth.

Statistical distributions of dendrite arm spacing have not been addressed in the literature. Two histograms for primary dendrite arm spacing in a cast nickel-based superalloy, and four histograms f...

An ω-limit set of a continuous self-mapping of a compact metric space X is said to be totally periodic if all of its points are periodic. In [3] Askri and Naghmouchi proved that if f is a one-to-one continuous self mapping of a regular curve, then every totally periodic ω-limit set of f is finite. This also holds whenever f is a monotone map of a local dendrite by Abdelli in [1] . In this paper we generalize these results to monotone maps on regular curves. On the other hand, we give some remarks related to expansivity and totally periodic ω-limit sets for every continuous map on compact metric space.

Given a metric continuum X, let C ( X ) and F 1 ( X ) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D ( X ) be the collection of all regular subsets in X belonging to C ( X ) and let M ( X ) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F 1 ( X ) = M ( X ) , D ( X ) = D ( X ) − F 1 ( X ) , D ( X ) = C ( X ) − M ( X ) and C ( X ) = D ( X ) ∪ M ( X ) . We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D 1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f : X → [ 0 , 1 ] . We show that the conditions that X is a λ-type continuum and every t in [ 0 , 1 ] is a cohesion point are equivalent to the equalities D ( X ) = { A ∈ C ( X ) : f ( A ) ∈ D ( [ 0 , 1 ] ) } and M ( X ) = { A ∈ C ( X ) : f ( A ) ∈ M ( [ 0 , 1 ] ) } . Throughout this paper we pose open problems.

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.

A space X is a dimensionally stepwise space if dim ⁡ X ∞ and for any 1 ≤ m ≤ dim ⁡ X there is an open set U m of X such that dim ⁡ U m = m . In this paper, for given inverse sequence { X i , f i , i + 1 } i = 1 ∞ of compacta with upper semi-continuous set-valued functions, we introduce new indexes I ˜ ( { X i , f i , i + 1 } ) and W ˜ ( { X i , f i , i + 1 } ) , and by use of the indexes we investigate topological structures of inverse limits of graphs with upper semi-continuous set-valued functions. Especially, we prove the following theorems.

Let (G, D) be a flow such that D is a dendrite and G is a finitely generated group. Denote by E(D) the set of endpoints of D. In this paper, it is shown that if E(D) is closed and countable then the following properties are equivalent: (1) (G, D) is pointwise periodic; (2) (G, D) is pointwise almost periodic; (3) The orbit closure relation is closed; (4) (G, D) is equicontinuous. In addition, we show that if E(D) is countable, then (G, D) is not a minimal flow. We also show that if (G, D) is a pointwise periodic flow and if D does not contain any homeomorphic copy of special dendrites, then the action of G can factor through a finite group action.

Let X be a compact metric space and 2 X 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 2 f : 2 X → 2 X . Mendez (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 2 F is dense in 2 X . This answers in the negative Mendez's conjecture.

A continuum is a nondegenerate compact connected metric space. A dendrite is a locally connected continuum containing no simple closed curves. A continuum X is said to be 1 3 -homogeneous if there exist three nonempty and mutually disjoint subsets O 1 , O 2 and O 3 of X such that X = O 1 ∪ O 2 ∪ O 3 and for each x , y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y if and only if x , y ∈ O i for some i ∈ { 1 , 2 , 3 } . In 2006 V. Neumann-Lara, P. Pellicer-Covarrubias, and I. Puga showed that a dendrite X is 1 2 -homogeneous if and only if X is an arc. The purpose of this paper is to extend this result and classify all 1 3 -homogeneous dendrites.

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points x k ∈ T and a sequence of positive integers n 1 < n 2 < ··· such that lim k→∞ x k = x and lim k→∞ $$f^{n_{k}}$$ (x k ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩ =1 ∞ f n (T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T.