Null Entropy Research Articles

The topological entropy of powers on Lie groups

AbstractIn this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$ , where G is a Lie group, given by some power $\psi (g) = g^k$ , with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$ , where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math.234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$ , where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

Cosmology of the Symmetrical Relativity versus Spontaneous Creation of the Universe Ex Nihilo

The cosmology of "Spontaneous Creation of the Universe Ex Nihilo" and the cosmology of the Symmetrical Relativity offer proposals to explain the creation and evolution of the universe. In essence they are still very distinct. However, we will argue that there was an antecedent to the big bang. Thus, we will penetrate a trans-Planckian regime, where we find an effective Planck length $L_P^{\prime}\rightarrow 0$. This will lead to more fundamental physical reflections about {\it nothing} in the cosmology of spontaneous creation. From the point of view of spontaneous creation, the step backwards went back to another principle, based on the {\it information}, which led to the big bang. While the spontaneous creation refers to the virtual pre-existence of {\it information}, which would have emerged randomly from {\it nothing}, the cosmology of Symmetrical Relativity does not stop there: we go back to one's own origin by projecting it before the creation of one's own time. From the present perspective, {\it nothing} is a primordial vacuum, whose {\it information} has made the universe by condensing and igniting. It was not randomly created, since the entropy had to vary from infinite (chaos) to zero (big bang) by violating the 2nd.law of thermodynamics in a trans-Planckian regime. {\it Nothing} or chaos with infinite entropy precedes the big bang (null entropy) within a trans-Planckian scenario and it determines the whole plot until the total extinction of the universe. We will show that {\it information}, besides not being born of the universe, also does not develop from it like the computational idea of Artificial Intelligence (AI). Thus, the universe is not simply self-taught as defended by the spontaneous creation. This could shed light on the problem of Penrose's Weyl curvature hypothesis, which considers a null Weyl curvature due to a null entropy in the big bang.

Joint predictiveness in inflectional paradigms

This paper contributes to addressing the Paradigm Cell Filling Problem (PCFP) in inflectional paradigms, as defined by Ackerman et al. (2009) . We define a method for extending the use of conditional entropy to address the PCFP to prediction based on multiple paradigm cells. We apply this method to French and European Portugese and show that, on average, knowledge of multiple paradigm cells is dramatically more predictive than knowledge of a single cell. Moreover, this new entropy measure proves useful in studying principal parts systems, which correspond to sets of predictors yielding a null entropy. Using a graded measure allows us to highlight the relevance of non-categorical or “good enough” principal parts systems.

Automorphism groups of positive entropy on projective threefolds

We prove two results about the natural representation of a group $G$ of automorphisms of a normal projective threefold $X$ on its second cohomology. We show that if $X$ is minimal, then $G$, modulo a normal subgroup of null entropy, is embedded as a Zariski-dense subset in a semi-simple real linear algebraic group of real rank $\le 2$. Next, we show that $X$ is a complex torus if the image of $G$ is an almost abelian group of positive rank and the kernel is infinite, unless $X$ is equivariantly non-trivially fibred.

Finiteness of solvable automorphisms with null entropy on a compact Kähler manifold

Let G be a solvable subgroup of the automorphism group Aut(X) of a compact Kähler manifold X of complex dimension n, and let N(G) be the normal subgroup of G consisting of elements with null entropy. Let us denote by G* the image of G under the natural map from Aut(X) to GL(V, R), where V is the Dolbeault cohomology group H1, 1(X, R). Assume that the Zariski closure of G* in GL(VC) is connected. The main aim of this paper is to show that, when the rank r(G) of the quotient group G/N(G) is equal to n − 1 and the identity component of Aut(X) is trivial, then the normal subgroup N(G) of G is finite. This affirmatively answers a question in Invent. Math. posed by D.-Q. Zhang.

Automorphism groups and anti-pluricanonical curves

We show the existence of an anti-pluricanonical curve on every smooth projective rational surface $X$ which has an infinite group $G$ of automorphisms of either null entropy or of type $\Z \ltimes \Z$, provided that the pair $(X, G)$ is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.

The H-theorem in κ-statistics: influence on the molecular chaos hypothesis

We rediscuss recent derivations of kinetic equations based on the Kaniadakis' entropy concept. Our primary objective here is to derive a kinetical version of the second law of thermodynamics in such a κ-framework. To this end, we assume a slight modification of the molecular chaos hypothesis. For the H κ -theorem, it is shown that the collisional equilibrium states (null entropy source term) are described by a κ-power law extension of the exponential distribution and, as should be expected, all these results reduce to the standard one in the limit κ → 0 .

Relativity, nonextensivity, and extended power law distributions

A proof of the relativistic theorem by including nonextensive effects is given. As it happens in the nonrelativistic limit, the molecular chaos hypothesis advanced by Boltzmann does not remain valid, and the second law of thermodynamics combined with a duality transformation implies that the parameter lies on the interval [0,2]. It is also proven that the collisional equilibrium states (null entropy source term) are described by the relativistic power law extension of the exponential Juttner distribution which reduces, in the nonrelativistic domain, to the Tsallis power law function. As a simple illustration of the basic approach, we derive the relativistic nonextensive equilibrium distribution for a dilute charged gas under the action of an electromagnetic field . Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the space-time ideas contained in the special relativity theory.

Algorithmic information for interval maps with an indifferent fixed point and infinite invariant measure.

Measuring the average information that is necessary to describe the behavior of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider a class of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We show that the average information that is necessary to describe the behavior of the orbits increases with time n approximately as nalpha, where alpha < 1 depends only on the asymptotic behavior of the map near the origin.