[Paper Review] Computation of Topological Entropy via $ϕ$-expansion, an Inverse Problem for the Dynamical Systems $βx+α\mod 1$

Bastien Faller, Pfister, Charles-Edouard|Jun 5, 2008

Hermeneutics and Narrative Identity11 references3 citations

TL;DR

This paper presents an algorithm based on Parry's φ-expansion to compute the topological entropy of shift spaces generated by the dynamical system βx + α mod 1, solving an inverse problem to recover α and β from symbolic sequences u̱ and v̱. The key contribution is a constructive method to determine entropy via follower-set graphs and a novel characterization of the parameter pair (α, β) that reproduces the given symbolic dynamics.

ABSTRACT

We give an algorithm, based on the $ϕ$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $βx+α\mod1$.The first part is an exposition of the $ϕ$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $ϕ$-expansion, necessary and sufficient conditions, which are different from those in Parry's paper.

Motivation & Objective

• To develop an algorithm for computing topological entropy of shift spaces defined by symbolic constraints u̱ ≤ σⁿx ≤ v̱ for all n ≥ 0.
• To solve the inverse problem: given symbolic sequences u̱ and v̱, determine parameters α and β such that u̱ = u̱^{α,β} and v̱ = v̱^{α,β}.
• To extend φ-expansion theory for piecewise monotone dynamical systems with new necessary and sufficient conditions distinct from Parry’s original results.
• To characterize the structure of shift spaces associated with βx + α mod 1 maps beyond the classical β-shift case (α = 0).

Proposed method

• Utilizes Parry’s φ-expansion to represent real numbers in terms of symbolic sequences derived from the map T(x) = βx + α mod 1.
• Constructs a follower-set graph for the shift space Σ(u̱, v̱) to analyze its combinatorial structure and compute entropy.
• Applies an algorithm (Proposition 3.2) to compute real parameters β̄ and ᾱ such that the entropy of Σ(u̱, v̱) equals that of the shift space generated by β̄x + ᾱ mod 1.
• Employs lexicographic ordering and infinite φ-series φ∞^{α,β}(w̱) to characterize the boundary sequences u̱^{α,β} and v̱^{α,β}.
• Uses the condition φ∞^{α,β}(w̱) = 1 to identify the extremal sequences defining the shift space boundaries.
• Applies results from symbolic dynamics and Baire category theory to ensure the set X\S is dense and invariant under T.

Experimental results

Research questions

• RQ1 How can topological entropy be computed for shift spaces defined by two symbolic sequences u̱ and v̱ under lexicographic ordering?
• RQ2 What are the necessary and sufficient conditions for the validity of φ-expansion in piecewise monotone dynamical systems, differing from Parry’s original formulation?
• RQ3 Can the inverse problem—recovering α and β from symbolic sequences u̱ and v̱—be solved algorithmically for βx + α mod 1 systems?
• RQ4 Under what conditions does the shift space Σ(u̱, v̱) correspond to a βx + α mod 1 system with parameters α and β?
• RQ5 How does the periodicity of the boundary sequences u̱^{α,β} and v̱^{α,β} affect the solvability of the inverse problem?

Key findings

• The topological entropy of the shift space Σ(u̱, v̱) is given by log₂β̄, where β̄ is computed via the algorithm in Proposition 3.2.
• The shift space Σ(u̱, v̱) has the same topological entropy as the shift space generated by the map β̄x + ᾱ mod 1, with parameters ᾱ and β̄ derived from u̱ and v̱.
• For k = 2, the condition σu̱ ≤ σv̱ is required to ensure the existence of ᾱ and β̄ such that u̱ = u̱^{ᾱ,β̄} and v̱ = v̱^{ᾱ,β̄}.
• If h(Σ(u̱, ŵ)) > 1, then the existence of a sequence u̱* ≺ v̱ implies that u̱ = u̱^{ᾱ,β̄} and v̱ = v̱^{ᾱ,β̄} if and only if β̄ > β̃, where β̃ is derived from the φ-expansion of ŵ.
• When both u̱^{ᾱ,β̄} and v̱^{ᾱ,β̄} are periodic, the inverse problem may fail to have a solution unless additional lexicographic constraints are satisfied.
• The algorithm correctly identifies ᾱ and β̄ such that the φ-series φ∞^{ᾱ,β̄}(ŵ) = 1 for the sequence ŵ = σu̱, ensuring consistency with the boundary conditions.