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The intrinsic metric formula on the code set of the Sierpinski Gasket is explicitly given in Definition 1.1. In this paper, we obtain the intrinsic metric formula on the code set of the Sierpinski tetrahedron and we investigate some geometrical properties of this structure. Moreover, we define a dynamical system on the Sierpinski tetrahedron and then we give an algorithm to compute the periodic points of this dynamical system. Finally, we show that this dynamical system is both Devaney chaotic and Li-Yorke chaotic.

Let (X, d) be a compact metric space and f : X → X a continuous function. Consider the hyperspace ( K ( X ) , H ) of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d, and let ( F ( X ) , d ∞ ) be the metric space of all nonempty compact fuzzy set on X equipped with the supremum metric d∞ which is calculated as the supremum of the Hausdorff distances of the corresponding level sets. If f ¯ is the natural extension of f to ( K ( X ) , H ) and f ˆ is the Zadeh’s extension of f to ( F ( X ) , d ∞ ) , then the aim of this paper is to study the dynamics of f ¯ and f ˆ when f is turbulent (erratic, respectively).

In this paper we first explain the concept �-fuzzy metric spaces and in this sequel explain the nation of Cauchy sequence and convergent in �-fuzzy metric spaces and finally we prove a common fixed point theorem in two complete �-fuzzy metric space.

We extend some results in [Ghaemi MB, Razani A. Fixed and periodic points in the probabilistic normed and metric spaces. Chaos, Solitons & Fractals 2006;28:1181–7] and answer in the affirmative an open question raised in the above quoted paper.

In this paper, the results on fuzzy contractive mapping proposed by Dorel Mihet will be proved for B-contraction and C-contraction in the case of George and Veeramani fuzzy metric space. The existence of fixed point with weaker conditions will be proved; that is, instead of the convergence of subsequence, p-convergence of subsequence is used.

The purpose of this paper is to prove some common fixed point theorems for finite number of discontinuous, noncompatible mappings on noncomplete intuitionistic fuzzy metric spaces. Our results extend, generalize and intuitionistic fuzzify several known results in fuzzy metric spaces. We give an example and also give formulas for total number of commutativity conditions for finite number of mappings.

In this paper, we give some new definitions of compatible mappings of types (I) and (II) in fuzzy metric spaces and prove some common fixed point theorems for four mappings under the condition of compatible mappings of types (I) and (II) in complete fuzzy metric spaces. Our results extend, generalize and improve the corresponding results given by many authors.

Recently, Abu-Donia [Abu-Donia HM. Common fixed points theorems for fuzzy mappings in metric space under ϕ contraction condition. Chaos, Solitons & Fractals 2007;34:538–43] studied the Hausdorff metric between fuzzy subsets via its correspondence between the two classical sets and established two common fixed point theorems for two fuzzy mappings which are useful in geometric problems arising in high energy physics. However, the proofs of Abu-Donia’s main results are incorrect. The purpose of this paper is to present correct proofs of these results.

As a natural generalization of fuzzy metric spaces due to George and Veeramani [George A, Veeramani P. On some result in fuzzy metric space. Fuzzy Sets Syst 1994;64:395–9], the present author defined the notion of L -fuzzy metric spaces. In this paper we prove some known results of metric spaces including Uniform continuity theorem and Ascoli–Arzela theorem for L -fuzzy metric spaces. We also prove that every L -fuzzy metric space has a countably locally finite basis and use this result to conclude that every L -fuzzy metric space is metrizable.

In this article, the topological properties of the Menger probabilistic metric spaces and the mappings between these spaces are studied. In addition, contractive and k -contractive mappings are introduced. As an application, a new fixed point theorem in a chainable Menger probabilistic metric space is proved.

In this paper, the transitive property for a flow without sensitive dependence on initial conditions is studied and it is shown that a Lyapunov stable flow with the average-shadowing property on a compact metric space is topologically ergodic.

In this paper we investigate the pointwise dimension of self-similar measures in complete metric spaces. Due to the lack of the geometry property of general complete metric spaces, some new techniques will be needed.

This paper is concerned with chaos of discrete dynamical systems in complete metric spaces. Discrete dynamical systems governed by continuous maps in general complete metric spaces are first discussed, and two criteria of chaos are then established. As a special case, two corresponding criteria of chaos for discrete dynamical systems in compact subsets of metric spaces are obtained. These results have extended and improved the existing relevant results of chaos in finite-dimensional Euclidean spaces.

J. R. Norris proved that the small time asymptotic liml → 02t·logp(t,x,y) of a symmetric elliptic diffusion on R n (or, more general, on a Lipschitz manifold) is determined by the intrinsic metric defined in terms of the associated Dirichlet form. Here we ask the question: Is the Dirichlet form (or the diffusion process) determined uniquely by its intrinsic metric (i.e. by its small time asymptotic)? The answer is NO. For any symmetric elliptic diffusion there exists another one with the same small time asymptotic but with strictly smaller diffusion coefficients. However, the answer is YES if a priori we know that the diffusion coefficients are continuous.