[Paper Review] Projection of Markov measures may be Gibbsian

Jean-René Chazottes, Edgardo Ugalde|arXiv (Cornell University)|Nov 29, 2002

Mathematical Dynamics and Fractals8 references4 citations

TL;DR

This paper investigates when the projection of a 1-step Markov measure under a factor map remains a Gibbs measure (in the sense of Bowen). Using a matrix-based approach involving infinite products of non-square matrices and a projective metric, the authors establish sufficient conditions under which the induced measure on the factor system is Gibbsian, even when the original system has forbidden blocks. The key result is that the induced potential exists and is Hölder continuous under these conditions, ensuring the projected measure is Gibbsian.

ABSTRACT

We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Holder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.

Motivation & Objective

• To determine under what conditions the factor map projection of a 1-step Markov measure on a topological Markov chain remains a Gibbs measure.
• To construct the induced potential on the projected system and prove its Hölder continuity when the factor system is also a topological Markov chain.
• To address the challenge introduced by forbidden blocks in the original system, which induce strong topological correlations in the projected process.
• To clarify the relationship between the structure of the original Markov measure and the nature of the induced measure, especially in cases with infinite-range potentials.
• To provide examples and counterexamples illustrating when the induced measure is or is not Gibbsian, particularly in the context of full shifts and periodic points.

Proposed method

• The authors use a matrix method to analyze infinite products of non-square matrices that arise from the factor map, modeling the transition probabilities of the projected system.
• A projective metric is introduced to control the convergence of these matrix products, enabling the construction of the induced potential at specific points.
• The existence of the induced potential is established by proving contractivity of positive non-square matrices over simplices, leveraging Perron-Frobenius theory.
• The potential is extended to the entire projected system by proving its Hölder continuity through asymptotic analysis of matrix powers and spectral properties.
• The method relies on an ansatz for the induced potential derived from the Gibbs property, using the relation between the original transition matrices and the projected dynamics.
• The analysis includes explicit computation of the potential at periodic points using matrix iteration and spectral decomposition.

Experimental results

Research questions

• RQ1 Under what conditions is the projection of a 1-step Markov measure on a topological Markov chain a Gibbs measure?
• RQ2 Can the induced potential on the projected system be constructed explicitly, and is it Hölder continuous?
• RQ3 How do forbidden blocks in the original system affect the Gibbsian nature of the projected measure?
• RQ4 Is the induced potential necessarily of finite range, or can it be infinite-range even when the original system is a full shift?
• RQ5 What characterizes the set of points where the induced potential is well-defined, and how does this relate to periodic orbits?

Key findings

• The induced potential exists and is Hölder continuous under sufficient conditions involving the structure of the transition matrices and the factor map, ensuring the projected measure is Gibbsian.
• Even when the original system is a full shift, the induced potential may be of infinite range, though it remains Gibbsian.
• The induced potential is not necessarily of finite range, as demonstrated by an explicit example where the potential has infinite range despite the original system being a full shift.
• For periodic points, the induced potential can be computed via a limit involving matrix powers and spectral radii, yielding a closed-form expression in terms of the dominant eigenvalue and associated projections.
• The induced potential is well-defined at all points of the projected system if the matrix products converge appropriately, which is guaranteed under the stated conditions.
• Counterexamples show that the induced measure may fail to be Gibbsian if the necessary matrix conditions are not met, particularly when the transition matrices lack sufficient rank or spectral properties.