Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems. Part of this work was done at Université Paris-Sud, dép. de mathématiques, Orsay.     

Equilibrium states,pressure and escape for multimodal maps with holes

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.     

Computing the topological entropy of continuous maps with at most three different kneading sequences with applications to Parrondo’s paradox

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.     

Optimal transport maps on infinite dimensional spaces

We will give a survey on results concerning Girsanov transformations, transportation cost inequalities, convexity of entropy, and optimal transport maps on some infinite dimensional spaces. Some open Problems will be arisen.     

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.     

Poincaré recurrences and entropy of suspended flows

We prove a formula giving the metric entropy of an ergodic flow suspended over a weakly specified subshift with positive entropy by using recurrence-time properties of generic orbits. This is a partial generalization to suspended flows of a formula by Ornstein and Weiss established for maps.     

Liouville-type results for stationary maps of a class of functional related to pullback metrics

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress–energy tensor which is naturally linked to conservation law and yield the monotonicity formula via the coarea formula and the comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also, we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.     

Entropy and periodic points for transitive maps

The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the -star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

     

The coarea formula for real‐valued Lipschitz maps on stratified groups

We establish a coarea formula for real‐valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the Q – 1 dimensional spherical Hausdorff measure. The number Q is the Hausdorff dimension of the group with respect to its Carnot–Carathéodory distance. We construct a Lipschitz function on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multicriteria fuzzy decision-making method using entropy weights-based correlation coefficients of interval-valued intuitionistic fuzzy sets

A multicriteria fuzzy decision-making method based on weighted correlation coefficients using entropy weights is proposed under interval-valued intuitionistic fuzzy environment for the some situations where the information about criteria weights for alternatives is completely unknown. To determine the entropy weights with respect to a decision matrix provided as interval-valued intuitionistic fuzzy sets (IVIFSs), we propose two entropy measures for IVIFSs and establish an entropy weight model, which can be used to determine the criteria weights on alternatives, and then propose an evaluation formula of weighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked and the most desirable one(s) can be selected according to the values of the weighted correlation coefficients. Finally, two applied examples demonstrate the applicability and benefit of the proposed method: it is capable for handling the multicriteria fuzzy decision-making problems with completely unknown weights for criteria.     

Measuring complexity in a business cycle model of the Kaldor type

The purpose of this paper is to study the dynamical behavior of a family of two-dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this important topological invariant with the parameters of the system, allows us to distinguish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our understanding of higher dimensional economic models can be enhanced by the theory of dynamical systems.     

一簇Lorenz映射的混沌行为的符号动力学分析

首先从符号动力学的角度论证了一簇Lorenz映射且有的混沌性质:稠密的周期轨道,周期的集合,拓扑熵,几乎所有(关于Lebesgue测度)的点的Lyapunov指数;并从揉序列的分析给出了该簇映射的拓扑熵的一个下界及Lyapunov指数的一个下界与上界,在很大程度上反应了Lorenz系统的复杂程度.其次仍从符号动力学的角度论证了更一般的Lorenz映射,通过设立参数空间,穷尽了Lorenz映射中函数为直线段的所有情况,并得出同前述Lorenz映射相似的且较为复杂的性质.     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

First and second moments for self‐similar couplings and Wasserstein distances

We study aspects of the Wasserstein distance in the context of self‐similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self‐similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self‐similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non‐trivial upper and lower bounds for the 2nd Wasserstein distance for these self‐similar measures.     

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.     

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman?s reverse Brunn–Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman?s deep technology of M-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke–Ruzsa inequalities from additive combinatorics.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

On the topological entropy of the free semigroup action generated by proper maps

In this paper, we introduce the topological entropy of a free semigroup action generated by proper maps, which extends the notions of the topological entropy of the free semigroup actions defined by Bufetov in 1999 and topological entropy of the proper maps defined by Patrão in 2010. We then give some properties of these notions and discuss the relations between them. We also give a partial variational principle for locally compact separable metric spaces. Moreover, the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given. These results extend the results obtained by Patrão, Bufetov and Lin, Ma and Wang in 2018.