# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

December 2015 , Volume 20 , Issue 10

Special issue on entropy, entropy-like quantities, and applications

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2015, 20(10): i-iii doi: 10.3934/dcdsb.2015.20.10i +[Abstract](576) +[PDF](136.8KB)
Abstract:
It is our pleasure to thank Prof. Peter E. Kloeden for having invited us to guest edit a special issue of Discrete and Continuous Dynamical Systems - Series B on Entropy, Entropy-like Quantities, and Applications. From its inception this special issue was meant to be a blend of research papers, showing the diversity of current research on entropy, and a few surveys, giving a more systematic view of lasting developments. Furthermore, a general review should set the framework first.

2015, 20(10): 3301-3343 doi: 10.3934/dcdsb.2015.20.3301 +[Abstract](1140) +[PDF](707.1KB)
Abstract:
This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical entropies and some newer, related notions to a broad audience with an intermediate background in dynamical systems and ergodic theory. Due to the breadth and depth of the subject, we have opted for a compact exposition whose contents are a compromise between conceptual import and instrumental relevance. The intended technical level and the space limitation born furthermore upon the final selection of the topics, which cover the three items named in the title. Specifically, the first part is devoted to the avatars of entropy in the traditional contexts: many particle physics, information theory, and dynamical systems. This chronological order helps present the materials in a didactic manner. The axiomatic approach will be also considered at this stage to show that, quite remarkably, the essence of entropy can be encapsulated in a few basic properties. Inspired by the classical entropies, further akin quantities have been proposed in the course of time, mostly aimed at specific needs. A common denominator of those addressed in the second part of this review is their major impact on research. The final part shows that, along with its profound role in the theory, entropy has interesting practical applications beyond information theory and communications technology. For this sake we preferred examples from applied mathematics, although there are certainly nice applications in, say, physics, computer science and even social sciences. This review concludes with a representative list of references.
2015, 20(10): 3345-3362 doi: 10.3934/dcdsb.2015.20.3345 +[Abstract](902) +[PDF](404.6KB)
Abstract:
This is a survey on recent developments of the dimension theory of flows, with emphasis on hyperbolic flows. In particular, we describe various results of the dimension theory and multifractal analysis of flows, including the dimension of hyperbolic sets, the dimension of invariant measures, the multifractal analysis of equilibrium measures, conditional variational principles, multidimensional spectra, and dimension spectra taking both into account past and future. The dimension theory and the multifractal analysis of dynamical systems grew out exponentially during the last two decades, but for various reasons flows have initially been given less attention than maps. We emphasize that this is not the case anymore and the survey is also an invitation to the theory.
2015, 20(10): 3363-3374 doi: 10.3934/dcdsb.2015.20.3363 +[Abstract](885) +[PDF](406.7KB)
Abstract:
In this paper we give a brief view on the topological entropy. The results here presented are well known to the people working in the area, so this survey is mainly for non--experts in the field.
2015, 20(10): 3375-3383 doi: 10.3934/dcdsb.2015.20.3375 +[Abstract](985) +[PDF](302.3KB)
Abstract:
In recent years there has been a great deal of progress in the study of actions of countable groups. In particular, the concept of the entropy of an action has been extended to all sofic groups following the seminal work of Lewis Bowen. This survey is an invitation to these new developments. It includes a new proof of the analogue of Kolmogorov's theorem for sofic groups, namely that isomorphic Bernoulli shifts have the same base entropy.
2015, 20(10): 3385-3401 doi: 10.3934/dcdsb.2015.20.3385 +[Abstract](751) +[PDF](281.7KB)
Abstract:
We introduce and study the notion of a directional complexity and entropy for maps of degree $1$ on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map.
2015, 20(10): 3403-3413 doi: 10.3934/dcdsb.2015.20.3403 +[Abstract](1084) +[PDF](331.8KB)
Abstract:
It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not continuous.
2015, 20(10): 3415-3434 doi: 10.3934/dcdsb.2015.20.3415 +[Abstract](723) +[PDF](511.5KB)
Abstract:
In this paper, a new formula for the topological entropy of a multimodal map $f$ is derived, and some basic properties are studied. By a formula we mean an analytical expression leading to a numerical algorithm; by a multimodal map we mean a continuous interval self-map which is strictly monotonic in a finite number of subintervals. The main feature of this formula is that it involves the min-max symbols of $f$, which are closely related to its kneading symbols. This way we continue our pursuit of finding expressions for the topological entropy of continuous multimodal maps based on min-max symbols. As in previous cases, which will be also reviewed, the main geometrical ingredients of the new formula are the numbers of transversal crossings of the graph of $f$ and its iterates with the so-called "critical lines". The theoretical and practical underpinnings are worked out with the family of logistic parabolas and numerical simulations.
2015, 20(10): 3435-3459 doi: 10.3934/dcdsb.2015.20.3435 +[Abstract](720) +[PDF](469.6KB)
Abstract:
We show that for any concave positive function $f$ defined on $[0,\infty)$ with $\lim_{x\rightarrow\infty}f(x)/x=0$ there exists a rank one system $(X_f,T_f)$ such that $\limsup_{n\rightarrow\infty} H(\alpha_0^{n-1})/f(n)\ge 1$ for all nontrivial partitions $\alpha$ of $X_f$ into two sets and that there is one partition $\alpha$ of $X_f$ into two sets for which the limit superior of $H(\alpha_0^{n-1})/f(n)$ is equal to one whenever the condition $\lim_{x\rightarrow\infty}\ln x/f(x)=0$ is satisfied. Furthermore, for each system $(X_f,T_f)$ we also identify the minimal entropy growth rate in the limit inferior.
2015, 20(10): 3461-3474 doi: 10.3934/dcdsb.2015.20.3461 +[Abstract](1466) +[PDF](369.0KB)
Abstract:
In this paper we define and study the topological entropy of a set-valued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the single-valued case will be obtained.
2015, 20(10): 3475-3485 doi: 10.3934/dcdsb.2015.20.3475 +[Abstract](983) +[PDF](355.6KB)
Abstract:
We introduce several notions of specification for iterated function systems and exhibit some of their dynamical properties. In particular, we show that topological entropy and algebraic pressure [4] of systems with specification are approximable by the corresponding expressions for finitely generated iterated function systems.
2015, 20(10): 3487-3505 doi: 10.3934/dcdsb.2015.20.3487 +[Abstract](817) +[PDF](459.9KB)
Abstract:
We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics:
1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.
2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair.
Our proofs are new and yield conclusions stronger than what was known before.
2015, 20(10): 3507-3524 doi: 10.3934/dcdsb.2015.20.3507 +[Abstract](903) +[PDF](435.5KB)
Abstract:
The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.
2015, 20(10): 3525-3545 doi: 10.3934/dcdsb.2015.20.3525 +[Abstract](863) +[PDF](334.9KB)
Abstract:
Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar dynamical invariants for homeomorphisms, like entropy and periodic point data, become more complex and permit multiple definitions. We briefly survey some of these and other related invariants in the setting of algebraic $\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy as a $\mathbb{Z}^d$-action vanishes, a powerful theory of directional entropy and periodic points can be built. An underlying theme is uniformity in dynamical invariants as the direction changes, and the connection between this theory and problems in number theory; we explore this for several invariants. We also highlight Fried's notion of average entropy and its connection to uniformities in growth properties, and prove a new relationship between this entropy and periodic point growth in this setting.
2015, 20(10): 3547-3564 doi: 10.3934/dcdsb.2015.20.3547 +[Abstract](885) +[PDF](449.4KB)
Abstract:
In the paper we provide exact lower bounds of topological entropy in the class of transitive and mixing maps preserving the Lebesgue measure which are nowhere monotone (with dense knot points).
2015, 20(10): 3565-3579 doi: 10.3934/dcdsb.2015.20.3565 +[Abstract](855) +[PDF](399.2KB)
Abstract:
We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of $g$-measures, it allows to assess the continuity of the entropy at $g$-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite $g$-functions, to the preservation at the limit, of certain ergodic properties for the associate $g$-measures.

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