Dear Colleagues,

Ever since its creation by Adler, Konheim, and McAndrew in 1965 for dynamical systems on compact spaces and its later reformulation for compact metric spaces by Dinaburg and independently by Bowen, topological entropy has had a profound influence on dynamical systems theory and several of its multitudinous applications. Roughly speaking, the topological entropy of a dynamical system represents the exponential growth rate of distinct or distinguishable orbits and so naturally has a connection with complexity and chaos, which have been areas of intense research over the last half-century.

Topological entropy research has tended to follow several related paths. The complexity connection has stimulated extensive research on the chaotic aspects of topological entropy, which has produced some useful estimates and equalities involving Lyapunov exponents, largely through the efforts of Pesin, Ledrapier, and Young. There has also been considerable work on extensions to wider classes of spaces, including those that are not compact. Work on generalizations to cover more types of dynamical actions has also been greatly in evidence. There have been myriad investigations involving comparisons with other forms of entropy as well, such as Kolmogorov–Sinai (metric) entropy, fundamental group entropy, Shannon entropy, and algebraic entropy. In addition, very active research has continued on connections, largely by way of estimates, with other aspects of the spaces such as measurable dynamical system structure as in the variational principle, smooth manifold structures, Riemannian metrics (where Newhouse has played a major role), homotopy and homology—as in the Shub conjecture (which was confirmed for smooth maps by Yomdin).

The increasing activity in applications research involving topological entropy is most likely due to and catalyzed by its intimate association with complexity, communication efficiency, unpredictability, and dynamical chaos. For example, applications to fluid mechanics and granular dynamics related to mixing are now rather well established and still being intensely investigated. Other application areas have also seen intensifying activity in recent years. These include biological applications where, for instance, a generalized form of topological entropy has been used in the analysis of DNA sequences, industrial engineering, where manufacturing networks have been studied using topological entropy, communication engineering, control theory and engineering, condensed matter and quantum physics, group theory, and even in the social sciences. With the success of topological entropy as a tool in these efforts, it is likely that such research is bound to grow apace.

Inasmuch as topological entropy has proven so important in dynamical systems theory and numerous related application areas and is in general rather intricate, there have been numerous research efforts dedicated to finding efficient, effective schemes for its approximate or exact computation. Considerable progress has been made in these endeavors, especially in the development of algorithmic schemes for computing lower bounds in two and three dimensions, but much work remains to be done.

This Special Issue is dedicated to presenting significant advances in the theoretical, applications and computational aspects of topological entropy described above. In addition, we would like to include an updated survey article or two that covers all or most of the theoretical, applied and computational work featuring topological entropy done since the most recent such papers (several of which are excellent) appeared.

Prof. Denis L. Blackmore
Prof. Raymond Addabbo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• dynamical systems
• topological entropy
• complexity
• chaos
• generalizations and extensions
• metric entropy
• variational principle
• connections with homotopy and homology
• applications
• computational schemes