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An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite groups assigns in a functorial way to a $G$-equivariant topological field theory an $H$-equivariant topological field theory, the pushforward theory. When $H$ is the trivial group, this yields an orbifold construction for $G$-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization., 21 pages, accepted for publication in the Journal of Pure and Applied Algebra

In these lectures, I pedagogically review some recent advances in the study of the non-equilibrium dynamics of isolated quantum systems. In particular I emphasise the role played by the reduced density matrix and by the entanglement entropy in the understanding of the stationary properties after a quantum quench. The idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics is introduced and used to provide quantitative predictions for the time evolution of the entanglement itself. The harmonic chain is studied as an elementary model in which the quench dynamics can be easily and exactly worked out. This example provides a useful playground where general concepts can be simply understood and later applied to more complex and realistic systems.

In this paper we compute topological entropy with prescribed accuracy for different economic models, showing the existence of a topologically chaotic regime for them. In order to make the paper self-contained, a general overview on the topological entropy of continuous interval maps is given. More precisely, we focus on piecewise monotone maps which often appear as dynamical models in economy, but also in population growth and physics. Our main aim is to show that when topological entropy can be approximated up to a given error, it is a useful tool which helps to analyze the chaotic dynamics in one dimensional models.

We use the modified spin wave theory of Takahashi to study the entanglement entropy in the quantum one-dimensional integer spin Heisenberg antiferromagnet. We calculate the entanglement entropy of this spin system that is well known to be a quantum wire, in the classical limit ( N → ∞ ). We obtain a decreasing the entanglement entropy with the temperature and we obtain none change in the entanglement in the point Δ = 1 at T = 0 where the system presents a quantum phase transition from a gapless phase in the spectrum Δ 1 to a gapped phase Δ ≥ 1 .

We investigate the topological properties of a ladder model of the Kitaev superconductor chains with a periodically modulated chemical potential. It is found that the system has two different topological classes, the class BDI characterized by the Z index and the class D characterized by the Z 2 index. The topological class of the system is determined by the relative phase θ between the inter- and intra-chain superconducting pairing. Based on the calculation of the topological numbers of the class BDI and the class D, we present the topological phase diagrams of the system for two different topological classes. It is shown that the topological phase transition accompanying the variation of the number of Majorana zero modes could be induced by periodical potential in both two classes. Finally, we calculate the differential conductance of normal metal-superconductor junction at zero bias, and find that it matches the topological phase diagrams in both class BDI and class D.

Left Bol loops with the A l -property or gyrogroups are generalization of groups which do not explicitly have associativity. In this work, we define topological gyrogroups and study some properties of them. In spite of having a weaker algebraic form, topological gyrogroups carry almost the same basic properties owned by topological groups. In particular, we prove that being T 0 and T 3 are equivalent in topological gyrogroups. Furthermore, a topological gyrogroup is first countable if and only if it is premetrizable. Finally, a direct product of topological gyrogroups is a topological gyrogroup.

Topological derivatives for quasilinear elliptic equations have not been studied yet. Such results are needed to apply topological asymptotic methods in shape optimization to nonlinear elasticity equations and in imaging to detect sets with codimensions ≥2 (e.g. points in 2D or curves in 3D). Our contribution is to provide for the first time topological derivatives for a class of quasilinear elliptic equations. The topological derivative can be split into a classical linear term and a new term which accounts for the nonlinearity of the principal part. Shifting from linear equations to nonlinear ones requires to significantly change the asymptotic analysis method. One core issue lies in the ability to approximate the variation of the direct state at scale 1 in R N . Accordingly we construct dedicated function spaces and rely on comparison techniques. Moreover one has to set up a duality scheme between the direct and adjoint states that applies in the nonlinear framework at each stage of the analysis.

We study holographic superconductor model with two scalar fields coupled to one single Maxwell field in the AdS soliton background away from the probe limit. We disclose properties of phase transitions mostly from the holographic topological entanglement entropy approach. With different sets of parameters, we observe various types of transitions, especially a new first order phase transition between the condensation of different order parameters in the insulator/superconductor system. Our results show that the entanglement entropy is a good probe to critical phase transition points and the order of phase transitions in the two-order model. We also conclude that the entanglement entropy is useful to some extent in determining the physically supported phases. In addition, we investigate properties of the condensation through the scalar operator and the charge density in the dual theory. As a summary, we draw the complete phase diagram of the effects of the scalar charge on phase transitions. At last, we give some qualitative understanding and obtain the analytical condition for the first order phase transition to occur., Comment: 12 pages, 7 figures. arXiv admin note: text overlap with arXiv:1512.08950

The goal of this paper is to establish connections between two entropy concepts, namely the fuzzy entropy of a fuzzy set, which appears in fuzzy set theory, and the topological entropy of Zadeh's extensions of continuous maps defined on metric compact spaces, which is a topological dynamics concept. We develop several concepts that allow us to simultaneously address the two entropy concepts. Except for stating several basic properties, we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments. Such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp, discrete dynamical systems.

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang-Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories of effective models are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems., This work will appear in a special Nuclear Physics issue "Memoriam Raymond Stora"

This paper deals with higher-order topological sensitivity analysis for the Laplace operator with respect to the presence of a Dirichlet geometry perturbation. Two main results are presented in this work. In the first one, we discuss the influence of the considered geometry perturbation on the Laplace solution. The second one is devoted to the higher-order topological derivatives. We derive a higher-order topological sensitivity analysis for a large class of shape functions.

The purpose of this study is to indicate fundamental propositions of the pre-image entropy related to a system proposed by Bufetov [Bufetov A. Topological entropy of free semigroup actions and skew-product transformations. J Dyn Control Syst 1999;5:137–143. 1.] for generating free semigroup actions. This study reveals the formula for the pre-image entropy of skew-product transformation with respect to the one-sided shift space. Finally, one example is presented to show how to obtain the pre-image entropy value for the skew-product transformation.

We study quantum phase transition of interacting fermions by measuring the local entanglement entropy in the one-dimensional Hubbard model. The reduced density matrices for blocks of a few sites are constructed from the ground state wave function in infinite systems by adopting the matrix product state representation where time-evolving block decimations are performed to obtain the lowest energy states. The local entanglement entropy, constructed from the reduced density matrices, as a function of the chemical potential shows clear signatures of the Mott transition. The value of the central charge, numerically determined from the universal properties of the local entanglement entropy, confirms that the transition is caused by the suppression of the charge degrees of freedom.

We introduce the notions of pseudo-orbit entropy, periodic pseudo-orbit entropy and pseudo-orbit point entropy for continuous maps on compact uniform spaces, and study their relationships with the topological entropy.

Induced by the Hagedorn instability, weakly-coupled $U(N)$ gauge theories on a compact manifold exhibit a confinement/deconfinement phase transition in the large-$N$ limit. Recently we discover that the thermal entropy of a free theory on $\mathbb{S}^3$ gets reduced by a universal constant term, $-N^2/4$, compared to that from completely deconfined colored states. This entropy deficit is due to the persistence of Gauss's law, and actually independent of the shape of the manifold. In this paper we show that this universal term can be identified as the topological entangle entropy both in the corresponding $4+1 D$ bulk theory and the dimensionally reduced theory. First, entanglement entropy in the bulk theory contains the so-called "particle" contribution on the entangling surface, which naturally gives rise to an area-law term. The topological term results from the Gauss's constraint of these surface states. Secondly, the high-temperature limit also defines a dimensionally reduced theory. We calculate the geometric entropy in the reduced theory explicitly, and find that it is given by the same constant term after subtracting the leading term of ${\mathcal O}(\beta^{-1})$. The two procedures are then applied to the confining phase, by extending the temperature to the complex plane. Generalizing the recently proposed $2D$ modular description to an arbitrary matter content, we show the leading local term is missing and no topological term could be definitely isolated. For the special case of ${\mathcal N}=4$ super Yang-Mills theory, the results obtained here are compared with that at strong coupling from the holographic derivation., Comment: Relation between the thermal entropy and entanglement entropy clarified, employing the Bisognano-Wichmann theorem, journal version

We prove recurrence and multiple recurrence results for topological dynamical systems indexed by an arbitrary directed partial semigroup with respect to a coideal basis suitable for this semigroup, but otherwise arbitrary. These results are then applied to topological dynamical systems indexed by semigroups possessing digital representation. Our theory includes recurrence and multiple recurrence results for topological dynamical systems, indexed by natural numbers, or by finite non-empty subsets of natural numbers, proved by Furstenberg and Weiss, and analogous results for topological dynamical systems indexed by words.

We extend the notion the topological entropy of a free semigroup action defined by Bufetov [6] to the case of a free semigroup action on a metric space not necessarily compact, provide some properties of this extended topological entropy, extend the topological analogue of the classical Abramov–Rokhlin formula for the entropy of a skew product transformations with respect to a metric space not necessarily compact and give some bounds for the entropy for some particular systems such as a free semigroup with m generators of affine transformations on p -dimensional torus and smooth maps on a Riemannian manifold.

In this paper, a three-dimensional palaeoclimate dynamical model by Saltzman is revisited. We find that chaotic behavior displays in this three-dimensional dynamical system in numerical studies. To verify the chaoticity, we resort to Poincare section and Poincare map technique to present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoe theory, and give an estimation of the topological entropy.

We introduce an algorithm to compute the topological entropy of piecewise monotone maps with at most three different kneading sequences, with prescribed accuracy. As an application, we compute the topological entropy of 3-periodic sequences of logistic maps, disproving a commutativity formula for topological entropy with three maps, and analyzing the dynamics Parrondo’s paradox in this setting.

We present a unified approach to the asymptotic topological indices of various lattices.We propose the topological indices per vertex problem for lattice systems.The explicit asymptotic values of Laplacian energies for various lattices are obtained.We deduce the Laplacian energies of many types of lattices are independent of various boundary conditions. In this paper, we present a unified approach to the asymptotic topological indices of various lattices. Moreover, we propose the various topological indices per vertex problem for lattice systems and show that the various topological indices per vertex of lattices are independent of the toroidal, cylindrical, and free boundary conditions. Our result is a generalization of some earlier results.

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials. This study presents a variational inequality for sub-additive topological conditional pressure on a closed subset, which is the other form of the variational principle for the sub-additive topological pressure presented by Cao, Feng, and Huang in [9]. Moreover, under some additional assumptions, this result can be generalized to the non-compact case.

The topological embedding of a topological semigroup S, commutative with the property of cancelation, into the group G = S × S / R , (R the equivalence ( a , b ) R ( a ′ , b ′ ) ⇔ ab ′ = a ′ b ) to which S is algebraically embedded, was the subject of the search for the mathematicians of a long period. It was based on the fact that S must naturally be a uniform topological space, as every topological group was. The present paper is devoted to the fact that a quasi-uniformity is defined to any topological space, thus to any topological semigroup.

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadeh's extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadeh's extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

We study the properties of the quantum states in the one-dimensional system with a shifted periodic potential in both the discrete model and the continuous model. With open boundary conditions, the edge states appear in the energy gaps which indicate non-trivial topological structures. The Chern numbers with respect to the Bloch vector and the potential shift angle are computed. In the limit of the continuous model, the Chern number of each band is exactly one. We demonstrate the particle number pumped by the adiabatically shift of the potential is directly related to the topological invariants., 5 pages, 3 figures, 1 table

Boolean control networks?(BCNs) are discrete-time dynamical systems with Boolean state-variables and inputs that are interconnected via Boolean functions. BCNs are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs.The topological entropy of a BCN with? m inputs is a nonnegative real number in the interval 0 , m log 2 . Roughly speaking, a larger topological entropy means that asymptotically the control is "more powerful". We derive a necessary and sufficient condition for a BCN to have the maximal possible topological entropy. Our condition is stated in the framework of Cheng's algebraic state-space representation of BCNs. This means that verifying this condition incurs an exponential time-complexity. We also show that the problem of determining whether a BCN with? n state variables and? m = n inputs has a maximum topological entropy is NP-hard, suggesting that this problem cannot be solved in general using a polynomial-time algorithm.

In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models.

We have obtained the equation of the extremal hypersurface by considering the Jacobson-Myers functional and computed the entanglement entropy. In this context, we show that the higher derivative corrected extremal surfaces can not penetrate the horizon. Also, we have studied the entanglement temperature and entanglement entropy for low excited states for such higher derivative theories when the entangling region is of the strip type., 1+39 pages; v2) Typos fixed and refs added; v3) Improved presentation; v4) Typos fixed; v5) A figure added, typos fixed and the discussion on complex valued area removed

Let f denote a continuous map of the compact interval to itself. Suppose that f is topologically transitive. We show that if f exhibits a pattern and has the same topological entropy as the pattern, then f is topologically conjugate to the linearization of the pattern.

Based on an analytical model of topological insulator, we present the quantum phase transitions of topological insulators with different symmetries by calculating their phase diagrams and edgestates. In particular, we show the symmetry protection nature of the topological quantum phase transitions. Topological quantum phase transitions cannot be classified by symmetries. However, the symmetry of the system plays an important role, where different topological quantum phase transitions are protected by different global symmetries.

This paper considers the topological classification of non-autonomous dynamic equations on time scales. In this paper we show, by a counterexample, that the trivial solutions of two topologically conjugated systems may not have the same uniform stability. This is contrary to the expectation that two topologically conjugated systems should have the same topological structure and asymptotic behaviors. To counter this mismatch in expectation, we propose a new definition of strong topological conjugacy that guarantees the same topological structure, and in particular the same uniform stability, for the corresponding solutions of two strongly topologically conjugated systems. Based on the new definition, a new version of the generalized Hartman–Grobman theorem is developed. We also include some examples to illustrate the feasibility and effectiveness of the new generalized Hartman–Grobman theorem.

We investigate the duality structure of quantum lattice systems with topological order, a collective order also appearing in fractional quantum Hall systems. We define electromagnetic (EM) duality for all of Kitaev's quantum double models based on discrete gauge theories with Abelian and non-Abelian groups, and identify its natural habitat as a new class of topological models based on Hopf algebras. We interpret these as extended string-net models, whereupon Levin and Wen's string-nets, which describe all intrinsic topological orders on the lattice with parity and time-reversal invariance, arise as magnetic and electric projections of the extended models. We conjecture that all string-net models can be extended in an analogous way, using more general algebraic and tensor-categorical structures, such that EM duality continues to hold. We also identify this EM duality with an invertible domain wall. Physical applications include topology measurements in the form of pairs of dual tensor networks., v2: material added, 24 pages, 7 figures

With the help of the theory of Caratheodory structure, the topological pressure for flows on non-compact sets is introduced, and the measure-theoretic pressure for flows is defined as well. It is shown that the topological pressure for flows on arbitrary sets is equal to the one of its time one map. Moreover, given an ergodic measure for a flow, there exists a certain set on which the topological pressure is equal to the measure-theoretic one. A variational principle for the topological pressure defined above is also established. At the end, the pressures for a frame flow are studied.

We show that a quantized Berry phase in Bloch momentum space can serve as a topological order parameter to the quantum phases of a gapped spin chain system with time-reversal invariance. Specifically, we study this approach analytically in a class of XY spin-1/2 chain with multiple sites interactions in a transverse field. In order to derive a proper definition of the Berry curvature in a two-dimensional parameter space, we performed a local gauge transformation to the spin chain system by a twist operator, which endows the Hamiltonian of the system with the topology of a torus T 2 without changing its energy spectrum. We show that a topological Z 2 order parameter can be obtained as a quantized Berry phase by a loop integral of the Berry gauge potential along quarter of the Brillouin zone, which determines the zero-temperature phase diagram of the system.

This paper presents the properties of topological pressure dimension, which is an extension of the entropy dimension. Specifically, this paper studies the relationships among different types of topological pressure dimension and identifies an inequality relating them. This analysis calculates analogs of many known results of topological pressure. In particular, we will show the value of the pressure dimension is always equal to or greater than 1 for any positive constant potential function.

We consider the refinement of the holographic entanglement entropy for the holographic dual theories to the AdS solitons and AdS black holes, including the corrected ones by the Gauss-Bonnet term. The refinement is obtained by extracting the UV-independent piece of the holographic entanglement entropy, the so-called renormalized entanglement entropy which is independent of the choices of UV cutoff. Our main results are (i) the renormalized entanglement entropies of the AdS$_{d+1}$ soliton for $d=4,5$ are neither monotonically decreasing along the RG flow nor positive definite, especially around the deconfinement/confinement phase transition; (ii) there is no topological entanglement entropy for AdS$_5$ soliton even with Gauss-Bonnet correction; (iii) for the AdS black holes, the renormalized entanglement entropy obeys an expected volume law at IR regime, and the transition between UV and IR regimes is a smooth crossover even with Gauss-Bonnet correction; (iv) based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state., 48 pages, 24 figures; v2: few typos corrected; v3: mistake on the choice of dominant phase is corrected, differential subtraction scheme is introduced to remove the UV cutoff-ambiguity, some of the conclusions on RG flow are changed; v4: statement about C theorem revised; v5 Final version to NPB

The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations. The topological derivative has been successfully applied in the treatment of problems such as topology optimization, inverse analysis and image processing. In this paper, the calculation of the topological derivative for a general class of shape functionals is presented. In particular, we evaluate the topological derivative of a modified energy shape functional associated to the steady-state heat conduction problem, considering the nucleation of a small circular inclusion as the topological perturbation. Several methods were proposed to calculate the topological derivative. In this paper, the so-called topological-shape sensitivity method is extended to deal with a modified adjoint method, leading to an alternative approach to calculate the topological derivative based on shape sensitivity analysis together with a modified Lagrangian method. Since we are dealing with a general class of shape functionals, which are not necessarily associated to the energy, we will show that this new approach simplifies the most delicate step of the topological derivative calculation, namely, the asymptotic analysis of the adjoint state.

In this paper, we present a unified treatment on the linear, differentiable and topological equivalences for time-invariant linear control systems governed by ordinary differential equations (ODEs). Especially, we give a new solution to the topological classification problem for linear control systems by means of an elementary analytic approach, which differs considerably from J.C. Willems’ algebraic method developed in (Willems, 1980) [17] . More precisely, based on P. Brunovsky’s results in (Brunovsky, 1970) [2] , we determine all topological invariants for the linear control system. As an interesting by-product, we show that there exist only finitely many topological equivalence classes for any stabilizable system with given numbers of state and control variables, which might be useful for some engineering applications.

Using the notion of topological pressure for non-compact sets, we prove a relation for two topological pressures with a factor map. We also provide an application in symbolic dynamics and conformal repellers. These results are generalized to the cases of BS-dimensions.

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of physical phenomenon modeled by partial differential equations, considering homogeneous and isotropic constitutive behavior. In fact, only a few works dealing with heterogeneous and anisotropic material behavior can be found in the literature, and, in general, the derived formulas are given in an abstract form. In this work, we derive the topological derivative in its closed form for the total potential energy associated to an anisotropic and heterogeneous heat diffusion problem, when a small circular inclusion of the same nature of the bulk phase is introduced at an arbitrary point of the domain. In addition, we provide a full mathematical justification for the derived formula and develop precise estimates for the remainders of the topological asymptotic expansion. Finally, the influence of the heterogeneity and anisotropy are shown through some numerical examples of heat conductor topology optimization.

In this paper, we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network is obtained via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks. The result is quite general; in particular, nonlinear coupling is allowed and no assumptions of hyperbolicity of the local dynamics are made.

We show that the topological entropy of a continuous endomorphism of a compact abelian group coincides with the algebraic entropy of the dual endomorphism of the (discrete) Pontryagin dual group. As an application a relation is given between the topological Pinsker factor and the algebraic Pinsker subgroup.

The generalized entropy (and its properties) of maps f : exp ( X ) → exp ( X ) , usual functions and multifunctions in a generalized topological space X are studied. Moreover, the connection between generalized entropy and standard entropy for usual functions in a compact metric space is shown.

Topological phase transitions represent a new class of quantum critical phenomena. Although they cannot be described within the usual framework of Landau theory, one can still identify a diverging length and time at these transitions, which make them amenable to a scaling approach. Also, the renormalisation group with the flows of parameters to dissimilar attractors allows to distinguish the different phases, beyond their topological properties.

In this paper, we introduce a new type of fuzzy topological entropy of a fuzzy continuous function ψ : X → X , where X is an arbitrary fuzzy topological space, and investigate several of its properties. We also establish that, for fuzzy compact spaces, this new concept of fuzzy topological entropy coincides with the fuzzy topological entropy of Ismail Tok (2005) [11] .

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.

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.

International audience; We discuss an analogy between topological quantum field theories and the theory of Markov processes, which both rely on the combination of a notion of transition and a notion of locality. We assume no prior knowledge of topological quantum field theory (TQFT) and devote the first section to a motivation of its definition, which was originally given by M. Atiyah. We then discuss 1- and 2-dimensional TQFT's, and a mild generalization of them which incorporates a notion of time and is suited to the parallel with Markov processes.This text is a written version of a talk whose emphasis was on explaining and illustrating ideas. There are few rigorous statements in it, and no proof.

We discuss the correlation and entanglement of a three-level atom with a single-mode quantized field in a coherent state inside a phase-damped cavity. We analyze the influence of dissipation on the quantum and classical entropy. It has been shown that the quantum, classical and nonextensive entropy are sensitive to any change in the initial state setting of the atom and the quantized field. The relation between the long lived entanglement and dissipation is observed. On the other hand, a short disentanglement can be generated through special values of the atomic motion parameter.

We provide a class of triangular maps of the square, (x,y)↦(f(x),gx(y)) of type 2∞, i.e., such that the periods of periodic points are the powers of 2, which has a minimal set supporting positive topological entropy. This improves the famous example by S. Kolyada from 1992 and contributes to the solution of an old problem by A.N. Sharkovsky.

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.