New integrable cases of a Cremona transformation: a finite-order orbits analysis

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters ( and ), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of = 0 when , besides the already known values = 0 and = −1, and also arbitrary when = 0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable, …).     

Factorization properties of birational mappings

We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations.

It is shown that factorization properties do exist, even for birational transformations associated with noninvolutive permutations of entries of q × q matrices, and even for more general transformation which are rational transformations but no longer birational. The existence of factorization relations independent of q, the size of the matrices, is underlined.

The relations between the polynomial growth of the complexity of the iterations, the existence of recursions in a single variable and the integrability of the mappings, are sketched for the permutations yielding these properties.

All these results show that permutations of the entries of the matrix yielding factorization properties are not so rare. In contrast, the occurrence of recursions in a single variable, or of the polynomial growth of the complexity are, of course, less frequent but not completely exceptional.     

Zero Entropy Systems

This paper introduces the notion of entropy dimension to measure the complexity of zero entropy dynamical systems, including the probabilistic and the topological versions. These notions are isomorphism invariants for measure-preserving transformation and continuity. We discuss basic propositions for entropy dimension and construct some examples to show that the topological entropy dimension attains any value between 0 and 1. This paper also gives a symbolic subspace to achieve zero topological entropy, but with full entropy dimension.     

Complexity for Extended Dynamical Systems

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.     

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map is not upper-semi continuous.     

Algebraic Entropy of Birational Maps with Invariant Curves

The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this Letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.     

The Atiyah–Hitchin bracket and the open Toda lattice

The dynamics of the finite nonperiodic Toda lattice is an isospectral deformation of the finite three-diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one-to-one correspondence with their Weyl functions. These are rational functions mapping the upper half-plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic-geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear-fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.     

Tunnelling with a negative cosmological constant

The point of this paper is to see what light new results in hyperbolic geometry may throw on gravitational entropy and whether gravitational entropy is relevant for the quantum origin of the universe. We introduce some new gravitational instantons which mediate the birth from nothing of closed universes containing wormholes and suggest that they may contribute to the density matrix of the universe. We also discuss the connection between their gravitational action and the topological and volumetric entropies introduced in hyperbolic geometry. These coincide for hyperbolic 4-manifolds, and increase with increasing topological complexity of the 4-manifold. We raise the question of whether the action also increases with the topological complexity of the initial 3-geometry, measured either by its 3-volume or its Matveev complexity. We point out, in distinction to the non-supergravity case, that universes with domains of negative cosmological constant separated by supergravity domain walls cannot be born from nothing. Finally we point out that our wormholes provide examples of the type of Perpetual Motion machines envisaged by Frolov and Novikov.     

Pure Point Diffraction Implies Zero Entropy for Delone Sets with Uniform Cluster Frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy zero if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy vanishes whenever the repetitivity function satisfies a certain growth restriction.      

Topological Entanglement Entropy from the Holographic Partition Function

We study the entropy of chiral 2+01-dimensional topological phases, where there are both gapped bulk excitations and gapless edge modes. We show how the entanglement entropy of both types of excitations can be encoded in a single partition function. This partition function is holographic because it can be expressed entirely in terms of the conformal field theory describing the edge modes. We give a general expression for the holographic partition function, and discuss several examples in depth, including abelian and non-abelian fractional quantum Hall states, and $p+ip$ superconductors. We extend these results to include a point contact allowing tunneling between two points on the edge, which causes thermodynamic entropy associated with the point contact to be lost with decreasing temperature. Such a perturbation effectively breaks the system in two, and we can identify the thermodynamic entropy loss with the loss of the edge entanglement entropy. From these results, we obtain a simple interpretation of the non-integer ‘ground state degeneracy’ which is obtained in 1+1-dimensional quantum impurity problems: its logarithm is a 2+1-dimensional topological entanglement entropy.     

Subshifts on Infinite Alphabets and Their Entropy

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.     

On the estimation of topological entropy

We study a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using(n, )-spanning sets.     

Space-Time Invariant Measures, Entropy, and Dimension for Stochastic Ginzburg–Landau Equations

We consider a randomly forced Ginzburg–Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the corresponding Markov process and we define the spatial densities of topological entropy, of measure-theoretic entropy, and of upper box-counting dimension. We prove inequalities relating these different quantities. The proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square-integrable functions and a priori bounds on the semi-flow in these spaces. The bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution. Received: 21 June 2000 / Accepted: 28 September 2001     

Numerical and experimental investigation of the effect of filtering on chaotic symbolic dynamics

Motivated by the practical consideration of the measurement of chaotic signals in experiments or the transmission of these signals through a physical medium, we investigate the effect of filtering on chaotic symbolic dynamics. We focus on the linear, time-invariant filters that are used frequently in many applications, and on the two quantities characterizing chaotic symbolic dynamics: topological entropy and bit-error rate. Theoretical consideration suggests that the topological entropy is invariant under filtering. Since computation of this entropy requires that the generating partition for defining the symbolic dynamics be known, in practical situations the computed entropy may change as a filtering parameter is changed. We find, through numerical computations and experiments with a chaotic electronic circuit, that with reasonable care the computed or measured entropy values can be preserved for a wide range of the filtering parameter.     

A note on minimization of rational surfaces obtained from birational dynamical systems

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.     

Topological invariants in the study of a chaotic food chain system

The study of ecological systems has generated deep interest in exploring the complexity of chaotic food chains. The role of chaos in ecosystems is not entirely understood. One approach to have a better comprehension of ecological chaos is by analyzing it in mathematical models of basic food chains. In this article it is considered a classical chaotic food chain model from the literature. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of kneading sequences associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The topological entropy allows us to distinguish different chaotic states in some realistic system parameter region. Another numerical invariant is introduced in order to characterize isentropic dynamics. Studying a set of maps with the same topological entropy, we exhibit numerical results about the relation between the second topological invariant and each of the control parameters in consideration. This work provides an illustration of how our understanding of ecological models can be enhanced by the theory of symbolic dynamics.     

RLm子区拓扑熵等值性的证明

本文证明了RL^m子区整个子区内拓扑熵等值,同时,揭示了高级混沌带对拓扑熵无贡献的性质。 关键词：     

Dynamical quantities and their numerical analysis by saddle periodic orbits

We study estimators for dynamical quantities such as the topological entropy and the topological pressure which are based on numerical computations on periodic orbits. For the particular case of the Hénon family for some parameter ranges we find a reasonable convergence of the entropy, the pressure, and Birkhoff averages of test functions. However, pointing out possible pitfalls of such an analysis, we show that the evaluation by means of saddle orbits alone can be misleading if, for example, chaotic saddles and attractors coexist.     

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq ⁴-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.     

Braiding matrices,modular transformations and topological field theories in 2+1 dimensions

Relations between 3D topological field theories and rational conformal field theories are discussed. In the former framework, we can define the generalized Verlinde operators. Using these operators, we find modular transformations for conformal blocks of one point functions and two point functions on the torus. The result is generalized to higher genus. The correctness of our formulae is illustrated by some examples. We also emphasize the importance of the fusion algebra.Addresses after October 1, 1989: Institute of Theoretical Physics, Academia Sinica, Beijing, P. R. China