[Paper Review] A continuous, piecewise affine surface map with no measure of maximal entropy

Jérôme Buzzi|ArXiv.org|Feb 16, 2009

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper constructs a continuous, piecewise affine surface map on the unit square with topological entropy log 2, demonstrating that no invariant probability measure achieves this entropy—thus proving the existence of a continuous, piecewise affine map without a measure of maximal entropy, contrary to known results for homeomorphisms. The construction exploits unstable dynamics near a fixed point and a symbolic extension to show that entropy-supremizing measures cannot exist due to unbounded growth in a coordinate.

ABSTRACT

It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discontinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of maximal entropy.

Motivation & Objective

• To resolve the open question of whether continuous, piecewise affine surface maps always admit a measure of maximal entropy.
• To construct a counterexample where topological entropy is finite but no invariant probability measure achieves it.
• To extend the understanding of entropy maximization in piecewise affine dynamics beyond homeomorphisms and smooth maps.
• To demonstrate that the variational principle for entropy can fail even in the presence of global continuity and affine structure.
• To explore the role of dynamical complexity and coordinate growth in obstructing the existence of maximal entropy measures.

Proposed method

• Define a piecewise affine map T on a parallelogram Q ⊂ [0,1]^2 with fixed points at N and S, and partition Q into 26 triangles where T is affine.
• Establish that the vertical cone field Cs is invariant under the inverse of T on the critical regions ABS and CDS, ensuring stable dynamics.
• Use a symbolic extension γ: K₀ → Σ (the 2-shift) defined by tracking whether iterates land in ABS (0) or CDS (1), with γ preserving entropy via Bowen’s result.
• Construct a nested family of finite extensions F_M of subshifts Σ_M with entropy approaching log 2, embedded into the dynamics near S.
• Prove that any hypothetical maximal entropy measure μ would require unbounded growth in the y-coordinate along orbits, contradicting the existence of a probability measure.
• Leverage the fact that y_{k+1} ≥ 2^{sign(x_k)} y_k to show that ∑ sign(x_k) must diverge for entropy-maximizing measures, which is impossible under a finite invariant measure.

Experimental results

Research questions

• RQ1 Can a continuous, piecewise affine surface map fail to admit a measure of maximal entropy, even when topological entropy is finite?
• RQ2 What dynamical features prevent the existence of a measure achieving the topological entropy in such maps?
• RQ3 How does the interplay between affine structure, continuity, and coordinate growth affect entropy maximization?
• RQ4 To what extent does the failure of entropy maximization in this class contrast with known results for C^∞ maps or piecewise monotone interval maps?
• RQ5 Is there a structural obstruction—such as unbounded growth in a coordinate—that can prevent the existence of a maximal entropy measure in piecewise affine systems?

Key findings

• The constructed map T on [0,1]^2 has topological entropy h_top(T) = log 2.
• Despite this finite entropy, no invariant probability measure μ satisfies h(T, μ) = log 2, so no measure of maximal entropy exists.
• The absence of a maximal entropy measure arises from the requirement that any such measure would need to assign positive mass to orbits with unbounded y-coordinate growth.
• The symbolic extension γ: K₀ → Σ shows that entropy of invariant measures on K₀ is bounded by log 2, and equality would require the (1/2,1/2)-Bernoulli measure on the shift.
• However, such a measure would imply that ∑ sign(x_k) diverges along almost every orbit, which forces y_n → ∞, contradicting the finiteness of a probability measure.
• The result extends to all dimensions d ≥ 2 via direct product with the identity on a cube, yielding similar counterexamples in higher dimensions.