[Paper Review] Topological Entropy and bulging deformation of real projective structures on surface

Patrick Foulon, Инканг Ким|arXiv (Cornell University)|Aug 24, 2016

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper investigates the topological entropy of geodesic flows on strictly convex real projective structures on closed surfaces via bulging deformations—continuous deformations along simple closed curves that stretch the domain in the direction of the neutral eigenvector. The key result is a construction of a divergent sequence of such structures whose topological entropy converges to any prescribed value in (0,1), demonstrating dense entropy values in the interval via controlled deformation parameters.

ABSTRACT

In this paper we study the deformation of strictly convex real projective structures on a closed surface. Specially we study the deformation in terms of the entropy on bulging deformations. As a byproduct we construct a sequence of divergent structures whose topological entropy converges to a designated number between 0 and 1.

Motivation & Objective

• To analyze the dynamical behavior of geodesic flows on strictly convex real projective structures via topological entropy.
• To understand the limiting behavior of such structures under bulging deformations along separating and non-separating simple closed curves.
• To construct a sequence of divergent real projective structures whose topological entropy converges to a specified value in (0,1).
• To extend known results on entropy convergence by using bulging (vertical) parameters instead of internal parameters.

Proposed method

• Utilizes bulging deformation—defined by Goldman—as a continuous family of deformations along a pants decomposition of a surface, parameterized by a vertical (bulging) parameter s.
• Analyzes the Gromov-Hausdorff limit of the strictly convex domain as s → ±∞, showing convergence to non-strictly convex structures with attached half-open cylinders or full cylinders.
• Employs estimates on the growth rate of closed geodesic counting in pairs of pants to bound the topological entropy of the full surface.
• Uses the fact that lengths of curves in S\P remain invariant under bulging, while internal parameters (not varied here) can be used to adjust lengths to control entropy.
• Applies asymptotic estimates involving M_s, q_s, and r_s to show that entropy contributions from non-pants regions vanish in the limit.
• Constructs a diagonal sequence by combining paths from hyperbolic structures to low-entropy pairs of pants, leveraging continuity of entropy in the parameter space.

Experimental results

Research questions

• RQ1 What happens to the topological entropy of a strictly convex real projective structure as the bulging parameter s → ∞ along a separating curve?
• RQ2 Can one construct a divergent sequence of real projective structures whose topological entropy converges to any given value in (0,1)?
• RQ3 How does the limit structure of the domain behave under bulging deformation as s → ±∞, particularly in terms of convexity and geometry?
• RQ4 Does the topological entropy of the full surface converge to the maximum of the entropies on the individual pairs of pants in the decomposition as s → ∞?
• RQ5 Is the topological entropy monotonic during the bulging deformation process?

Key findings

• As s → ∞ along a separating curve γ, the strictly convex structure on one component of S\γ converges to a non-strictly convex structure of infinite volume with an attached half-open cylinder.
• When s → -∞, the limit structure has the axis of γ on its boundary, resulting in a finite cylinder attached to the surface.
• For non-separating curves, the limit structure involves attaching a cylinder along each copy of γ.
• The topological entropy of the full surface under bulging deformation converges to the maximum of the topological entropies on the individual pairs of pants in the pants decomposition as s → ∞.
• A divergent sequence of real projective structures exists such that the topological entropy converges to any prescribed value α ∈ (0,1), as shown via a diagonal argument combining low-entropy pairs of pants and bulging deformations.
• The contribution from geodesics not in the pants decomposition vanishes in the entropy estimate, confirming that the entropy is asymptotically determined by the pairs of pants.