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Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems. [ABSTRACT FROM AUTHOR]

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as X t = B V (t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the L q spectrum of V. We will show that if V is generated by a self-similar measure on R 1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its L q spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1 + 1 2 ⋅ log 2 log 3 . Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical computation of the harmonic-measure distribution function, or h -function for short, associated with a particular planar domain. This function describes the hitting probability of a Brownian walker released from some point with the boundary of the domain. We use a fast and accurate boundary integral method for the numerical calculation of the h -functions for symmetric multiply connected slit domains with high connectivity. In view of the fact that the middle-thirds Cantor set C is a special limiting case of these slit domains, the proposed method is used to approximate the h -function for the set C. We also numerically analyze some asymptotic features of the calculated h -functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized Cantor sequence c , which is an ℓ -automatic sequence. We prove that the abelian complexity of the 2-block sequence of c is ℓ -regular if the factor set of the sequence c is mirror invariant. As a consequence, we show that the 2-abelian complexity of a generalized Cantor sequence satisfying certain conditions is ℓ -regular. • We prove that abelian complexity of 2-block coding of generalized Cantor sequences satisfying certain conditions is regular. • We build a bridge between 2-abelian complexity of generalized Cantor sequences and abelian complexity of its 2-block coding. • We prove that 2-abelian complexity of generalized Cantor sequences satisfying certain conditions is regular. [ABSTRACT FROM AUTHOR]

Given a metric continuum X , a subset A of X is said to be T -closed, provided that T (A) = A , where T denotes the Jones's set function. This concept was introduced in [1]. In [6] , the hyperspace of T -closed subcontinua C T (X) and the hyperspace of T -closed sets 2 T X were introduced; the paper is focused in studying topological properties of C T (X) and many important examples are given. In this paper, we study general topological properties of 2 T X such as connectedness, compactness, etc.; also we present interesting examples, particularly we give a continuum X for which 2 T X is the Cantor set, also a continuum X for which 2 T X − { X } is homeomorphic to the arc (compare with [6, Problem 3.18] and conjecture thereafter). Finally, we introduce the concepts of T -finite type and T -big subset of a continuum and present interesting examples and results. Also, this work includes many open problems. [ABSTRACT FROM AUTHOR]

Given a positive integer M , a number q > 1 is called a univoque base if there is exactly one sequence ( c i ) = c 1 c 2 ⋯ with integer digits c i belonging to the set { 0 , 1 , … , M } , such that 1 = ∑ i = 1 ∞ c i q − i . The topological and combinatorial properties of the set of univoque bases U and their corresponding sequences ( c i ) have been investigated in many papers since a pioneering work of Erdős, Horváth and Joó 25 years ago. While in most studies the attention was restricted to univoque bases belonging to ( M , M + 1 ] , a recent work of Kong and Li on the Hausdorff dimension of unique expansions demonstrated the necessity to extend the earlier results to all univoque bases. This is the object of this paper. Although the general research strategy remains the same, a number of new arguments are needed, several new properties are uncovered, and some formerly known results become simpler and more natural in the present framework. [ABSTRACT FROM AUTHOR]

After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if T is a complete recursive L -theory in continuous logic, and v (φ) is the truth value of the L -sentence φ in models of T , then v (φ) is a recursive real uniformly recursive in φ. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex ⁎-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures. [ABSTRACT FROM AUTHOR]

Abstract Let C be the Cantor set. For each n ⩾ 3 we construct an embedding A : C × C → R n such that A (C × { s }) , for s ∈ C , are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's necklaces). This serves as a base for another new result proved in this paper: for each n ⩾ 3 and any non-empty perfect compact set X which is embeddable in R n − 1 , we describe an embedding A : X × C → R n such that each A (X × { s }) , s ∈ C , contains the corresponding A (C × { s }) , and is "nice" on the complement A (X × { s }) − A (C × { s }) ; in particular, the images A (X × { s }) , for s ∈ C , are ambiently incomparable pairwise disjoint copies of X. This generalizes and strengthens theorems of J.R. Stallings (1960), R.B. Sher (1968), and B.L. Brechner–J.C. Mayer (1988). [ABSTRACT FROM AUTHOR]

Abstract: For subspaces X and Y of the notation means that X is homeomorphic to a subspace of Y and means . The resulting set of equivalence classes is partially-ordered by the relation if . In a previous paper by the author it was established that this poset is essentially determined by considering only the scattered of finite Cantor–Bendixson rank. Results from that paper are extended to show that this poset is computable. [Copyright &y& Elsevier]

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z 2 -invariant universal unfolding N b c k v = a y 2 2 − (λ + b I 1) x 2 2 + c x 4 4 with respect to Z 2 -equivariant BCKV-restricted morphisms of the planar singularity a 2 y 2 + c 4 x 4 (the coefficients a , b , c ≠ 0 , the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z 2 -invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z 2 -symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding N b c k v with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB. • We prove a KAM theorem in a quasi-periodic Hamiltonian pitchfork bifurcation. • We obtain the structural stability of a bifurcation in a phenomenological model. • We verify some of the previous numerical phenomena of Litvak-Hinenzon et al. [ABSTRACT FROM AUTHOR]

In this paper we develop the theory of Artin-Wraith glueings for topological spaces. It means that we are glueing two topological spaces to each other without identifying points on them. These constructions are far from unique and this theory is used to deal with all of them. As an application, we show that some categories of compactifications of coarse spaces that agree with the coarse structures are invariant under coarse equivalences. As a consequence, if X and Y are some well behaved metric spaces that are coarse equivalent, then they have the same space of ends (generalizing the well known fact that works on quasi-isometric proper geodesic metric spaces). As another application, we show that for every compact metrizable space Y , there exists only one, up to homeomorphisms, compactification of the Cantor set minus one point such that the remainder is homeomorphic to Y. [ABSTRACT FROM AUTHOR]

Given x ∈ ( 0 , 1 ] , let U ( x ) be the set of bases q ∈ ( 1 , 2 ] for which there exists a unique sequence ( d i ) of zeros and ones such that x = ∑ i = 1 ∞ d i ∕ q i . Lü et al. (2014) proved that U ( x ) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U ( x ) + λ U ( x ) and product U ( x ) ⋅ U ( x ) λ contain an interval for all x ∈ ( 0 , 1 ] and λ ≠ 0 . As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017). [ABSTRACT FROM AUTHOR]

Let C be a Cantor set in the real line. Tymchatyn and Walker in [3] proved that every embedding h of C × [ 0 , 1 ] into the plane R 2 is tame, i.e., there is a homeomorphism H : R 2 → R 2 such that H (h (C × [ 0 , 1 ])) ⊂ C × [ 0 , 1 ]. In this paper we answer a question of L. Oversteegen [2] by proving that every embedding of a closed subset of C × [ 0 , 1 ] into the plane is tame. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of points u ∈ A N that are such that the omega-limit set ω ( u , σ ) is precisely X , where X ⊆ A N is an internally transitive shift space. We then use those results to study the possible structures of the omega-limit set of the turning point for a unimodal map. Examples are provided of unimodal maps f where no iterate of the turning point c is recurrent and ω ( c , f ) is either a minimal Cantor set or properly contains a minimal Cantor set. [ABSTRACT FROM AUTHOR]

Nowadays, numerical modelling is increasingly used to assess the stability of tunnels and underground caverns. However, an analysis of the mechanical behaviour of existing brick-lined tunnels remains challenging due to the complex material components. In order to study the mechanical behaviour of the masonry in brick-lined tunnels, this paper reports a series of small scale physical tunnel model tests to represent the true behaviour of a real tunnel under extreme loading. Advanced monitoring techniques of laser scanning and photogrammetry are used to record tunnel deformation and lining defects. This investigation shows how these techniques may substitute or supplement the conventional monitoring procedures. Moreover, numerical analyses based on continuum and discontinuum approaches are carried out. The numerical results are compared with physical model tests to assess the overall stability of these tunnels. Predictions using numerical models under various conditions have also been carried out to show the mechanical behaviour of masonry tunnel and to quantify the influence of the boundary and loading conditions. [ABSTRACT FROM AUTHOR]

In this paper, we give a precise formula for the permutation complexity of Cantor-like sequences, which are non-uniformly recurrent automatic sequences. Since the sequences are automatic, as it was proved by Charlier et al. in 2012, the permutation complexity of each of them is a regular sequence. We give a precise recurrence relation and a generalized automaton for it. [ABSTRACT FROM AUTHOR]

In the definition of a set-open topology on C ( X ) , the set of all real-valued continuous functions on a Tychonoff space X , we use a certain family of subsets of X and open subsets of R . But instead of using this traditional way to define topologies on C ( X ) , in this paper, we adopt a different approach to define two interesting topologies on C ( X ) . We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies. [ABSTRACT FROM AUTHOR]

Abstract: The usual Tychonoff product space of arbitrary many compact (ω-bounded) spaces is well-known to be also compact (ω-bounded). In this paper, we compare the lexicographic ordered topologies on some products of ordinals with the Tychonoff product topologies. We see: [•] The lexicographic ordered space is ω-bounded. [•] The lexicographic ordered space is not ω-bounded. [•] If α and β are ordinals with , then the lexicographic ordered space is a subspace of the lexicographic ordered space , thus the lexicographic ordered space is a subspace of the lexicographic ordered space . [•] The lexicographic ordered space is not a subspace of the lexicographic ordered space . [•] For all with , the lexicographic ordered space is homeomorphic to the Cantor set. [•] The lexicographic ordered space is not metrizable. [•] The lexicographic ordered spaces and are not homeomorphic. [•] The lexicographic ordered topology on coincides with its usual Tychonoff product topology. [•] The lexicographic ordered topology on is strictly weaker than its usual Tychonoff product topology. [•] The lexicographic ordered topology on is strictly weaker than its usual Tychonoff product topology. [•] The lexicographic ordered topology on coincides with its usual Tychonoff product topology. [Copyright &y& Elsevier]

Given a positive integer M and a real number q ∈ (1 , M + 1 ] , an expansion of a real number x ∈ [ 0 , M / (q − 1) ] over the alphabet A = { 0 , 1 , ... , M } is a sequence (c i) ∈ A N such that x = ∑ i = 1 ∞ c i q − i. Generalizing many earlier results, we investigate in this paper the topological properties of the set U q consisting of numbers x having a unique expansion of this form, and the combinatorial properties of the set U q ′ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [3] by adapting the method given in [12] for the case M = 1. [ABSTRACT FROM AUTHOR]

Abstract: Let f be an R-closed homeomorphism on a connected orientable closed surface M. In this paper, we show that if M has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If and f is neither minimal nor periodic, then either each minimal set is a finite disjoint union of essential circloids or there is a minimal set which is an extension of a Cantor set. If and f is not periodic but orientation-preserving (resp. reversing), then the minimal sets of f (resp. ) are exactly two fixed points and a family of circloids and . [Copyright &y& Elsevier]

Abstract: This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a class of Cantor sets, which can be put into a one-to-one correspondence with the continued fraction expansions of irrational numbers. By using the recursive relations of the continued fractions, we get the exact Hausdorff dimensions of the Cantor sets. As an example, we exhibit a sequence of sets whose Hausdorff dimensions are an elementary function of the Fibonacci number. [Copyright &y& Elsevier]

Abstract: The main results of the paper are: [(1)] If X is metrizable but not locally compact topological space, then contains a closed copy of , and hence does not have the property AP; [(2)] For any zero-dimensional Polish X, the space is sequential if and only if X is either locally compact or the derived set is compact; and [(3)] All spaces of the form , where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the st. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, by the three kinds of basic components (BCs) of three-component Thue–Morse (3CTM) sequence we construct a type of interesting optical basic-structural-units (BSUs) and propose multi-component generalized Thue–Morse (mCGTM) model. Based on the conventional electromagnetic wave theory we investigate the optical transmission vertically through the one-dimensional (1D) mCGTM superlattices. It is found that the optical transmission possesses an interesting pseudo-constant characteristic at the central wavelength. mCGTM sequence exhibits a cantor-set structure which results in the system possessing certain kinds of effective component pairs (ECPs), and each kind of ECP brings about certain contribution towards the optical transmission. The cantor-set structure is the reason that mCGTM multilayers exhibit the optical transmission pseudo-constant property. For the pseudo-constant optical transmission of mCGTM superlattices, there would be a potential application in the designing of some complex optical devices. [Copyright &y& Elsevier]

Abstract: Global functions of cellular automata on state spaces equipped with the Cantor topology are well characterized by the Curtis–Hedlund–Lyndon theorem. In this paper, we develop a characterization of global functions of cellular automata on , if the state space is equipped by Weyl and Besicovitch topology. The necessary and sufficient condition for a function to be the global map of a cellular automaton is (1) a strong localization property, a condition that strengthen Lipschitz continuity, (2) the set of (Cantor) periodic states are positively invariant and (3) the function commutes (in the Weyl/Besicovitch sense) with the shift operator. [Copyright &y& Elsevier]

Abstract: A discrete dynamical system on a compact metric space is called universal (with respect to -limit sets) if, among its -limit sets, there is a homeomorphic copy of any -limit set of any dynamical system on . By a result of Pokluda and Smítal the unit interval admits a universal system. In this paper, we study the problem of the existence of universal systems on Cantor spaces, graphs, dendrites and higher-dimensional spaces. [Copyright &y& Elsevier]

Abstract: A maximal consistent theory is a maximal theory with respect to its consistency. The present paper is divided into two parts. The first one is devoted to characterize the maximality of a consistent theory in the formal deductive system (which is a logic system equivalent to the nilpotent minimum logic). It is proved that each maximal consistent theory in this logic must be the deductive closure of a collection of simple compound formulas. Hence, it follows that there is a one-to-one correspondence between the set of all maximal consistent theories and the set of evaluations e assigning to each propositional variable p its truth degree . The Satisfiability Theorem and Compactness Theorem of are obtained. The second part is to investigate the topological structure of the set of all maximal consistent theories over , and the results show that this topological space is a Cantor space. [Copyright &y& Elsevier]

Secure multi-party computation (SMC) is a research focus in international cryptographic community. At present, there is no SMC solution to the intersection problem of sets. In this paper, we first propose a SMC solution to this problem. Applying Cantor encoding method to computational geometry problems, and based on the solution to set-intersection problem, we further propose solutions to points inclusion problem and intersection problem of rectangles and further prove their privacy-preserving property with widely accepted simulation paradigm. Compared with the known solutions, these new solutions are of less computational complexity and less communication complexity, and have obvious superiority m computational and communication complexity. [ABSTRACT FROM AUTHOR]

Abstract: The important concept of identifying a fuzzy subset of a set R with its corresponding chain of level sets is widely used in the literature. We investigate in this paper the problem of characterization of all fuzzy subsets of R that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of . Although our main focus is on the problem of uniqueness, we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets. We continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed. To further support our results, we present some examples which we hope will shed more light on the above mentioned problem of identification. We finish by raising a few natural questions that are left as open problems for further investigation. [Copyright &y& Elsevier]

In this paper, we study the Hausdorff centered measure of certain linear Cantor sets. We establish the relationship between the Hausdorff centered measure of this set and the maximum centered density of the corresponding self- similar measure. From this relationship, the Hausdorff centered measure of certain sets is obtained. In particular, we consider the linear iterated function system consisting of three maps with the same contraction ratios. Under some technical restrictions, we determine the exact Hausdorff centered mea- sure of its attractor. [ABSTRACT FROM AUTHOR]

In [R. Man˜e´, Trans. Amer. Math. Soc. 252 (1979) 313–319], R. Man˜e´ proved that minimal sets of expansive homeomorphisms are 0-dimensional. More generally, minimal sets of continuum-wise expansive homeomorphisms are 0-dimensional (see [H. Kato, Canad. J. Math. 45 (1993) 576–598]). Also, for each continuum-wise expansive homeomorphism f :X→X of a compactum X with dimX>0, there is an f-invariant closed subset Y of X such that dimY>0 and f&z.sfnc;Y :Y→Y is weakly chaotic in the sense of Devaney (see [H. Kato, Lecture Notes in Pure and Appl. Math., Vol. 170, Dekker, New York, 1995, pp. 265–274]). In this paper, we prove the following result: If f :X→X is a continuum-wise expansive homeomorphism of a compactum X with dimX=1, then there is a Cantor set Z in X such that for some natural number N, fN(Z)=Z and fN&z.sfnc;Z :Z→Z is semiconjugate to the shift homeomorphism σ :Σ→Σ, where Σ is the Cantor set {0,1}Z. As a corollary, there is a family {Cα∣α∈Λ} of minimal sets Cα of f such that each Cα is a Cantor set, Cl(⋃{Cα∣α∈Λ})=Y is 1-dimensional and f&z.sfnc;Y :Y→Y is weakly chaotic in the sense of Devaney. [Copyright &y& Elsevier]

In this paper, we establish and discuss basic properties of ϵ-connected sets embedded in an arbitrary complete space. These properties are used to establish that X is ϵ-connected for all ϵ>0 if, and only if, X is connected in the original sense of Cantor [Math. Ann. 23 (1884) 453]. Such sets are “virtually connected” in a finite precision machine and in some circumstances can be used as efficient estimators of connected geometric figures. [Copyright &y& Elsevier]

In this paper we show that, when we iteratively add Sacks reals to a model of ZFC we have for every two reals in the extension a continuous function defined in the ground model that maps one of the reals to the other. [Copyright &y& Elsevier]

The paper is devoted to the study of the following question: when does a k-dimensional subset of Rn (0) contain a set homeomorphic to a k-dimensional disk. [Copyright &y& Elsevier]

In this paper, we study inverse limits with one upper semi-continuous function which is the union of mappings. Using the concept of D o m (F) , we give sufficient conditions so that, the inverse limit is either a continuum or a Cantor set. [ABSTRACT FROM AUTHOR]

W. C. Lang determined wavelets on Cantor dyadic group by using Multiresolution analysis method. In this paper we have given characterization of wavelet sets on Cantor dyadic group which provides another method for the construction of wavelets. All the wavelets originating from wavelet sets are not necessarily associated with a multiresolution analysis. We have also established relation between multiresolution analysis and wavelets determined from wavelet sets. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of the kneading sequences of unimodal maps for which the omega-limit set of the turning point is a Cantor set and the map restricted to that omega-limit set is a minimal homeomorphism. We provide several characterizations of unimodal maps with these homeomorphic restrictions in terms of the kneading sequences and the associated shift spaces generated by the kneading sequences. [ABSTRACT FROM AUTHOR]

In 1994, John Cobb asked: given N > m > k > 0 , does there exist a Cantor set in R N such that each of its projections into m -planes is exactly k -dimensional? Such sets were described for (N , m , k) = (2 , 1 , 1) by L. Antoine (1924) and for (N , m , m) by K. Borsuk (1947). Examples were constructed for the cases (3 , 2 , 1) by J. Cobb (1994), for (N , m , m − 1) and in a different way for (N , N − 1 , N − 2) by O. Frolkina (2010, 2019), for (N , N − 1 , k) by S. Barov, J.J. Dijkstra and M. van der Meer (2012). We show that such sets are exceptional in the following sense. Let C (R N) be the set of all Cantor subsets of R N endowed with the Hausdorff metric. It is known that C (R N) is a Baire space. We prove that there is a dense G δ subset P ⊂ C (R N) such that for each X ∈ P and each non-zero linear subspace L ⊂ R N , the orthogonal projection of X into L is a Cantor set. This gives a partial answer to another question of J. Cobb stated in the same paper (1994). [ABSTRACT FROM AUTHOR]

The concept of T -closed set of a continuum was defined and studied by D.P. Bellamy et al. in [1] , where T denotes the Jones's set function. Given a continuum X , in this paper we define the hyperspace C T (X) as the collection of all T -closed subcontinua of X. We present examples of continua X for which this hyperspace has n elements for each positive integer n or it is the closure of the harmonic sequence or it is the Cantor set. We study the connectedness, compactness and density of C T (X). Finally, we relate the continuity of T with the structure of this hyperspace. [ABSTRACT FROM AUTHOR]

Abstract: Many examples exist of one-dimensional systems that are topologically conjugate to the shift operator on and are thus chaotic. Most of these examples which have invariant Cantor subsets, have Cantor subsets of measure zero. In this paper we outline the formulation of a map on a closed interval that has an invariant Cantor subset of positive Lebesgue measure. We also survey techniques used to analyze the dynamics of one-dimensional systems. [Copyright &y& Elsevier]