topological entropy
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R. Kazemi ◽  
M.R. Miri ◽  
G.R.M. Borzadaran

The category of metric spaces is a subcategory of quasi-metric spaces. It is shown that the entropy of a map when symmetric properties is included is greater or equal to the entropy in the case that the symmetric property of the space is not considered. The topological entropy and Shannon entropy have similar properties such as nonnegativity, subadditivity and conditioning reduces entropy. In other words, topological entropy is supposed as the extension of classical entropy in dynamical systems. In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified $(r,s)$-entropy. In this paper, we extend the notion of unified $(r, s)$-entropy for the continuous maps of a quasi-metric space via spanning and separated sets. Moreover, we survey unified $(r, s)$-entropy of a map for two metric spaces that are associated with a given quasi-metric space and compare unified $(r, s)$-entropy of a map of a given quasi-metric space and the maps of its associated metric spaces. Finally we define Tsallis topological entropy for the continuous map on a quasi-metric space via Bowen's definition and analyze some properties such as chain rule.

2021 ◽  
Vol 25 (1) ◽  
Jung-Chao Ban ◽  
Chih-Hung Chang ◽  
Yu-Liang Wu ◽  
Yu-Ying Wu

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 658-680
Xueting Tian ◽  
Weisheng Wu

Abstract In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36); Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56). Our results give new insights to the multifractal analysis for partially hyperbolic systems.

2021 ◽  
Vol 31 (15) ◽  
Jan Andres

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

2021 ◽  
pp. 1-35

Abstract Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

2021 ◽  
Chiara Vischioni ◽  
Valerio Giaccone ◽  
Paolo Catellani ◽  
Leonardo Alberghini ◽  
Riccardo Miotti Scapin ◽  

GenBank files contain genomic data of sequenced living organisms. Here, we present GBRAP (GenBank Retrieving, Analyzing and Parsing software), a tool written in Python 3 that can be used to easily download, parse and analyze viral and bacterial GenBank files, even when contain more than one genomic sequence for each species. GBRAP can analyze more files simultaneously through single command line parameters that give as output a single table showing the genomic characteristics of each organism. It is also able to calculate Shannon, LZSS (Lempel Ziv Storer Szymanski) and topological entropy for both the entire genome and its constitutive elements such as genes, rRNAs, tRNAs, tmRNAs and ncRNAs together with Chargaff's second parity rule scores obtained using different mathematical methods. Moreover, GBRAP can calculate, the number, the length and the nucleotides abundance of genomic components for each DNA strand and for the overlapping regions among the two complementary helixes. To our knowledge, this is the only software capable of providing this type of genomic analyses all together in a single tool, that, therefore can be used by the scientists interested in both genomics and evolutionary research.

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