Perturbing a product of stable flows

Abstract:

Suppose that $f$ and $f’$ are axiom A flows with attractors $A$ and $A’$. Then the attractor $A \times A’$ for the product flow $g_t =f_t \times f’_t$ on the product manifold is no longer hyperbolic (although there is a hyperbolic action of $\mathbb {R}^2$). It is easy to see that the attractor cannot explode but we show here that it cannot implode: for any flow $(h_t)$ sufficiently close to $(g_t)$ any attractor whose basin is not too thin is $\varepsilon$-dense in $A \times A’$.