Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ an open subset, with $0\in U$ a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of gradient of $f$ converges to $0$ as $t\to \infty$, then the unit tangent vector $\dot{x}(t)/\| \dot{x}(t)\| $ has a unique limit as $t\to \infty$. This conjecture was proved by

 http://arxiv.org/abs/math.AG/9906212

I want to know the case for smooth functions. Is there any counterexample or not known yet? Or under what condition may one expect the same result as Thom conjectured?

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ is an open subset and $0 \in U$ is a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of gradient of $f$ converges to $0$ as $t\to \infty$, then the unit tangent vector $$\frac{\dot{x}(t)}{\| \dot{x}(t)\|}$$ has a unique limit as $t\to \infty$. This conjecture was proved by  Kurdyka, Mostowski & Parusinski.

I want to know the case for smooth functions. Is there any counterexample or not known yet? Or under what condition may one expect the same result as Thom conjectured?