Related papers: The entropy function of an invariant measure
Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < \lambda(A) < 1 < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, or the other way…
The problem of Shannon entropy estimation in countable infinite alphabets is addressed from the study and use of convergence results of the entropy functional, which is known to be discontinuous with respect to the total variation distance…
We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth…
We present a unified axiomatic approach to contextuality and non-locality based on the fact that both are resource theories. In those theories the main objects are consistent boxes, which can be transformed by certain operations to achieve…
For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite…
We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for…
A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers -- we call such a linear extension a…
This is an elementary introduction to infinite-dimensional probability. In the lectures, we compute the exact mean values of some functionals on C[0,1] and L[0,1] by considering these functionals as infinite-dimensional random variables.…
We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We…
Around the mean dimensions and rate-distortion functions, using some tools from local entropy theory this paper establishes the following main results: $(1)$ We prove that for non-ergodic measures associated with almost sure processes, the…
We study in this paper real-valued functions on the space of all sub-$\sigma$-algebras of a probability measure space, and introduce the notion of Kudo-continuity, which is an a priori strengthening of continuity with respect to strong…
For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure.…
In question-answering tasks, determining when to trust the outputs is crucial to the alignment of large language models (LLMs). Kuhn et al. (2023) introduces semantic entropy as a measure of uncertainty, by incorporating linguistic…
We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such…
We define a new quantitative measure for an arbitrary factorial language: the entropy of a random walk in the prefix tree associated with the language; we call it Markov entropy. We relate Markov entropy to the growth rate of the language…
The log-periodic equation for the entropy $S = - (k/a) \sum_{i=1}^{N} p_{i} \sin(a \ln p_{i})$, based on the forgotten Sharma-Taneja entropy measure, is studied for the first time with $N$ the total number of system states and $p_{i}$ the…
We define an entropy based on a chosen governing probability distribution. If a certain kind of measurements follow such a distribution it also gives us a suitable scale to study it with. This scale will appear as a link function that is…
We study the entropy rate of pattern sequences of stochastic processes, and its relationship to the entropy rate of the original process. We give a complete characterization of this relationship for i.i.d. processes over arbitrary…
Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi : X \to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all…
When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a…