Related papers: Entropy and determinants for unitary representatio…
This paper studies properties of entropy functions that are induced by groups and subgroups. We showed that many information theoretic properties of those group induced entropy functions also have corresponding group theoretic…
We study the ergodic theory of a one-parameter family of interval maps T_alpha arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of T_alpha to be Hoelder-continuous in the…
Shannon and Khinchin showed that assuming four information theoretic axioms the entropy must be of Boltzmann-Gibbs type, $S=-\sum_i p_i \log p_i$. Here we note that in physical systems one of these axioms may be violated. For non-ergodic…
We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting…
The ability of many powerful machine learning algorithms to deal with large data sets without compromise is often hampered by computationally expensive linear algebra tasks, of which calculating the log determinant is a canonical example.…
We revisit the issue of defining the entropy of a spatial region in a broad class of quantum theories. In theories with explicit regularizations, working within an elementary but general algebraic framework applicable to matter and gauge…
We propose an abstract framework of a kind of representation theory for $C^*$-flows, i.e., $C^*$-algebras equipped with one-parameter automorphism groups, as a proper generalization of Olshanski's formalism of unitary representation theory…
The $f$-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.
We produce a probabilistic space from logic, both classical and quantum, which is in addition partially ordered in such a way that entropy is monotone. In particular do we establish the following equation: Quantitative Probability = Logic +…
We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree…
Entropy of measure preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are the ones given by the techniques developed by Ollagnier and Pinchon on the one hand and the…
We show that for every linear toral automorphism, especially the non-hyperbolic ones, the entropies of ergodic measures form a dense set on the interval from zero to the topological entropy.
We present some new nonparametric estimators of entropies and we establish almost sure consistency and central limit Theorems for some of the most important entropies in the discrete case. Our theorical results are validated by simulations.
Some general considerations on the notion of entropy in physics are presented. An attempt is made to clarify the question of the differentiation between physical entropy (the Clausius-Boltzmann one) and quantities called entropies…
Asymptotically entropy of chaotic systems increases linearly and the sensitivity to initial conditions is exponential with time: these two behaviors are related. Such relationship is the analogous of and under specific conditions has been…
This study focuses on conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, $(\eta, \psi)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $\eta$-averaging (\texttt{EAVG}) framework for $(\eta,…
We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to…
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carath\'{e}odory dimension characteristic, motivated by the work of Bowen and Pesin etc. We…
We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued…
We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy…