The $C^1$ property of convex carrying simplices for competitive maps

TL;DR

This paper proves that for certain competitive maps, the convexity of the carrying simplex guarantees it is a smooth $C^1$ manifold-with-corners, using Lyapunov exponents to establish neat embedding.

Contribution

It establishes the $C^1$ regularity of convex carrying simplices in competitive maps, linking convexity to smoothness via Lyapunov exponent inequalities.

Findings

Convexity implies the carrying simplex is a $C^1$ submanifold-with-corners.

Lyapunov exponents characterize neat embedding of the simplex.

The result applies to a class of competitive maps with invariant manifolds.

Abstract

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In the present paper it is shown that its convexity implies that it is a $C^1$ submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.