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Shannon Weights for binary dynamical recurrent sources of zero entropy

Ali Akhavi, Eda Cesaratto, Frédéric Paccaut, Pablo Rotondo, Brigitte Vallée

TL;DR

The paper develops a comprehensive framework to analyze Shannon weights $\underline m_\mu(n)$ for binary dynamical sources with zero entropy by combining analytic combinatorics with transfer-operator methods. It introduces a block-structured dynamical setup in the Good Class and derives invariant densities, renewal-type relations among generating functions, and singularity-based asymptotics that connect the waiting-time distribution to Shannon weights. A central achievement is showing how to construct explicit sources with prescribed growth of Shannon weights, including polynomial-logarithmic forms $n^{\beta}(\log n)^{\delta}$, with the Farey system as a key exemplar. The results illuminate the intricate link between intermittent dynamics (indifferent fixed points) and information-theoretic quantities, providing a rigorous path to tailor sources with desired information growth while enriching the connection to Infinite Ergodic Theory through renewal equations.

Abstract

A probabilistic source is defined as the set of infinite words (over a given denumerable alphabet) endowed with a probability $μ$. The paper deals with general binary sources where the distribution of any symbol (0 or 1) may depend on an unbounded part of the previous history. The paper studies Shannon weights: whereas the classical Shannon entropy ${\cal E}_μ$ is the average amount of information brought by one symbol of the emitted word, the Shannon weight sequence deals with the average amount of information $m_μ(n)$ that is brought by the emitted prefix of length $n$. For a source with a non zero entropy, the estimate $m_μ(n)\sim{\cal E}_μ \cdot n$ thus holds. The paper considers the model of dynamical sources, where a source word isemitted as an encoded trajectory of a dynamical system of the unit interval, when endowed with probability $μ$. It focus on sources with zero entropy and gives explicit constructions for sources whose Shannon weight sequence satisfies $m_μ(n)=o(n)$, with a prescribed behaviour. In this case, sources with zero entropy lead to dynamical systems built on maps with an indifferent fixed point. This class notably contains the celebrated Farey source, which presents well-known intermittency phenomena. Methods are based on analytic combinatorics and generating functions, and they are enlarged, in the present dynamical case, with dynamical systems tools (mainly transfer operators).

Shannon Weights for binary dynamical recurrent sources of zero entropy

TL;DR

The paper develops a comprehensive framework to analyze Shannon weights for binary dynamical sources with zero entropy by combining analytic combinatorics with transfer-operator methods. It introduces a block-structured dynamical setup in the Good Class and derives invariant densities, renewal-type relations among generating functions, and singularity-based asymptotics that connect the waiting-time distribution to Shannon weights. A central achievement is showing how to construct explicit sources with prescribed growth of Shannon weights, including polynomial-logarithmic forms , with the Farey system as a key exemplar. The results illuminate the intricate link between intermittent dynamics (indifferent fixed points) and information-theoretic quantities, providing a rigorous path to tailor sources with desired information growth while enriching the connection to Infinite Ergodic Theory through renewal equations.

Abstract

A probabilistic source is defined as the set of infinite words (over a given denumerable alphabet) endowed with a probability . The paper deals with general binary sources where the distribution of any symbol (0 or 1) may depend on an unbounded part of the previous history. The paper studies Shannon weights: whereas the classical Shannon entropy is the average amount of information brought by one symbol of the emitted word, the Shannon weight sequence deals with the average amount of information that is brought by the emitted prefix of length . For a source with a non zero entropy, the estimate thus holds. The paper considers the model of dynamical sources, where a source word isemitted as an encoded trajectory of a dynamical system of the unit interval, when endowed with probability . It focus on sources with zero entropy and gives explicit constructions for sources whose Shannon weight sequence satisfies , with a prescribed behaviour. In this case, sources with zero entropy lead to dynamical systems built on maps with an indifferent fixed point. This class notably contains the celebrated Farey source, which presents well-known intermittency phenomena. Methods are based on analytic combinatorics and generating functions, and they are enlarged, in the present dynamical case, with dynamical systems tools (mainly transfer operators).

Paper Structure

This paper contains 63 sections, 35 theorems, 200 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sources, Shannon weights, block sources, waiting time
  3. General sources
  4. Fundamental domains, fundamental probabilities and Kolmogorov Property.
  5. A recursive point of view on the source.
  6. Costs.
  7. Shannon weights
  8. Binary sources and number of ones.
  9. Recurrence, waiting time and block sources.
  10. Recurrence.
  11. Waiting time.
  12. Block source.
  13. Distribution of the waiting time and generating function $Q$.
  14. Dynamical sources of various types.
  15. A class of dynamical systems.
  16. ...and 48 more sections

Key Result

Lemma 1

For any increasing cost $c$, the generating function $(1-v) C_\mu(v)$ has positive coefficients and the partial sums of these coefficients coincide with $\underline c_\mu(n)$.

Figures (3)

  • Figure 1: A tent-shaped dynamical system with its two direct branches (left) and its two inverse branches (right).
  • Figure 2: The Farey map and the Gauss map
  • Figure 3: Five pairs $(\gamma, \delta)$, their corresponding Shannon pairs $(\beta_M, \delta_M)$, and their corresponding graphs

Theorems & Definitions (77)

  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Definition 5
  • Definition 6
  • Proposition 7
  • proof
  • ...and 67 more