Computing the topological entropy of maps of the interval with three monotone pieces

An algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces. The accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed. Some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces.     

Computing the topological entropy of maps

We give an algorithm for determining the topological entropy of a unimodal map of the interval given its kneading sequence. We also show that this algorithm converges exponentially in the number of letters of the kneading sequence.     

Calculating Topological Entropy

The attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algorithm introduced here is motivated by the fact that if f: [0, 1] → [0, 1] is a piecewise monotone map of the unit interval into itself, thenh(f)=lim_n→∞ (1/n) log Var(f ⁿ), where h(f) is the topological entropy off, and Var(f ⁿ) is the total variation off ⁿ. We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper. We prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the function is Markov, and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter families of test functions.     

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.     

Entropy spectrum of the D-dimensional massless topological black hole

There are exact solutions to Einstein’s equations with negative cosmological constant that represent black holes whose event horizons are manifolds of negative curvature, the so-called topological black holes. Among these solutions there is one, the massless topological black hole, whose mass is equal to zero. Hod proposes that in the semiclassical limit the asymptotic quasinormal frequencies determine the entropy spectrum of the black holes. Taking into account this proposal, we calculate the entropy spectrum of the massless topological black hole and we compare with the results on the entropy spectra of other topological black holes.     

Inner Structure of Entropy of Reissner–Nordström Black Holes

Using the relationship between the entropy and the Euler characteristic, and the usual decomposition of spin connection, an entropy density is introduced to describe the inner structure of the entropy of RN black holes. It is pointed out that the entropy of RN black holes is determined by the singularities of the timelike Killing vector field of RN spacetime, and that these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally, which are topological invariants. Taking account of the physical meaning of entropy in statistics, the entropy and its density of RN black holes are modified and they are given by the Hopf indices merely.     

Topological Structure of Entropy of 4-Dimensional Axisymmetric Black Holes

Using the relationship between the entropy and the Euler characteristic, an entropy density is introduced to describe the inner topological structure of the entropy of 4-dimensional axisymmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of spacetime, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, which are topological invariants. At last, Kerr–Newman black hole as an example of axisymmetric black holes is given. What’s more, the entropy and the latent heat of the topological phase transition of the black hole mentioned above are calculated and the latent heat just lies in the range of the energy of gamma ray bursts. This work is supported in part by the NSFs of China under Grant No. 10575068 and of Shanghai Municipal Committee of Science and Technology under Grant No. 04ZR14059 and Shanghai Leading Academic Discipline Project under Project Number: T0104.     

Scaling properties of the measure of constant topological entropy

The topological entropy for some families of one-dimensional unimodal maps is studied. By arranging the windows of constant topological entropy in a binary tree, we have obtained the total measure of these windows. The scaling properties of this measure are studied.     

Topological entropy for endomorphisms of localC *-algebras

A notion of topological entropy for endomorphisms of localC ^*-algebras is introduced as a generalisation of the topological entropy of classical dynamical systems. The basic properties are derived and a series of calculations are presented.     

Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval

A new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented. It uses a transition matrix associated with the transformation. For this matrix we give a spectral theorem. This can be used for an estimation of the accuracy of the algorithm.     

TOLOMEO,a Novel Machine Learning Algorithm to Measure Information and Order in Correlated Networks and Predict Their State

We present ToloMEo (TOpoLogical netwOrk Maximum Entropy Optimization), a program implemented in C and Python that exploits a maximum entropy algorithm to evaluate network topological information. ToloMEo can study any system defined on a connected network where nodes can assume N discrete values by approximating the system probability distribution with a Pottz Hamiltonian on a graph. The software computes entropy through a thermodynamic integration from the mean-field solution to the final distribution. The nature of the algorithm guarantees that the evaluated entropy is variational (i.e., it always provides an upper bound to the exact entropy). The program also performs machine learning, inferring the system’s behavior providing the probability of unknown states of the network. These features make our method very general and applicable to a broad class of problems. Here, we focus on three different cases of study: (i) an agent-based model of a minimal ecosystem defined on a square lattice, where we show how topological entropy captures a crossover between hunting behaviors; (ii) an example of image processing, where starting from discretized pictures of cell populations we extract information about the ordering and interactions between cell types and reconstruct the most likely positions of cells when data are missing; and (iii) an application to recurrent neural networks, in which we measure the information stored in different realizations of the Hopfield model, extending our method to describe dynamical out-of-equilibrium processes.     

Topological causality analysis of horizontal gas-liquid flows based on cross map of phase spaces

Nonlinear systems are always characterized by the interactions between constituents which yield data in the form of time series. Exploration of the causality between the times series is beneficial for understanding the dynamics of the system. We introduce a topological causality method to explore the dynamics of horizontal gas-liquid flows. First, the principle of the topological causality algorithm is illustrated and validated using the Lorenz system and transfer entropy. Then, we conducted an experiment of gas-liquid flows in a horizontal pipe, during which a wire-mesh sensor (WMS) was used to capture the flow structures. The WMS data at different time frames are embedded in high-dimension phase spaces. Through building a cross map between coupled phase spaces, a cross map smoothness was employed to derive the topological causality index. The causality index enables us to understand the mechanism of the flow pattern transition and the intrinsic dynamics of the transient gas-liquid flows.     

Euler Characteristic and Topological Phase Transition of NUT-Kerr-Newman Black Hole

From the Gauss-Bonnet-Chern theorem, the Euler characteristic of NUT-Kerr-Newman black hole is calculated to be some discrete numbers from 0 to 2. We find that the Bekenstein-Hawking entropy is the largest entropy in topology by taking into account of the relationship between the entropy and the Euler characteristic. The NUT-Kerr- Newman black hole evolves from the torus-like topological structure to the spherical structure with the changes of mass, angular momentum, electric and NUT charges. In this process, the Euler characteristic and the entropy are changed discontinuously, which give the topological aspect of the first-order phase transition of NUT-Kerr-Newman black hole. The corresponding latent heat of the topological phase transition is also obtained. The estimated latent heat of the black hole evolving from the star just lies in the range of the energy of gamma ray bursts.     

Topological order,entanglement, and quantum memory at finite temperature

We compute the topological entropy of the toric code models in arbitrary dimension at finite temperature. We find that the critical temperatures for the existence of full quantum (classical) topological entropy correspond to the confinement–deconfinement transitions in the corresponding ${\mathbb{Z}}_2$ gauge theories. This implies that the thermal stability of topological entropy corresponds to the stability of quantum (classical) memory. The implications for the understanding of ergodicity breaking in topological phases are discussed.     

Computing Topological Entropy in a Space of Quartic Polynomials

This paper adds a computational approach to a previous theoretical result illustrating how the complexity of a simple dynamical system evolves under deformations. The algorithm targets topological entropy in the 2-dimensional family P ^Q of compositions of two logistic maps. Estimation of the topological entropy is made possible by the correspondence between P ^Q and a subfamily of sawtooth maps P ^T , and is based on the well-known fact that the kneading-data of a map determines its entropy. A complex search for kneading-data in P ^T turns out to be computationally fast and reliable, delivering good entropy estimates. Finally, the algorithm is used to produce a picture of the entropy level-sets in P ^Q , as illustration to theoretical results such as Hu (Ph.D. thesis, CUNY, 1995) and Radulescu (Discrete Cont. Dyn. Syst. 19(1):139–175, 2007).     

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.     

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：     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

The Generalized Milnor–Thurston Conjecture and Equal Topological Entropy Class in Symbolic Dynamics of Order Topological Space of Three Letters

This paper presents an answer to an open problem in the dynamical systems of three letters: the generalized Milnor–Thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two-dimensional parameter plane. The concept of equal topological entropy class is introduced by the dual star product which is a generalization of the Derrida–Gervois–Pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval. The algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the Milnor–Thurston characteristic polynomials. Moreover, the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space Σ₃ of three letters. Received: 4 February 1998 / Accepted: 1 March 2000     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holes

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem,it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime.By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.