Let $D$ be an $n$-dimensional disk (considered a $C^{\infty}$ manifold-with-boundary), and let $\partial D$ denote its boundary. Assume that $f \colon D \to D$ is a $C^1$ diffeomorphism having the following property:

For each $x \in D \setminus \partial D$ its $\omega$-limit set, $\omega(x)$, is not fully contained in $\partial D$ (the possibility that $\omega(x) \cap \partial D \ne \emptyset$ is not excluded).

Let $\mu$ be an ergodic invariant probability measure supported on $\partial D$. By the Oseledets theorem, there are $n-1$ (counting multiplicity) Lyapunov exponents on the tangent bundle of $\partial D$, and the "remaining" Lyapunov exponent (corresponding to directions transverse to $\partial D$; call it the external Lyapunov exponent for $\mu$).

The question is:

Is the external Lyapunov exponent nonnegative, for each $\mu$?

Comment. When $f$ is $C^{1 + \alpha}$ with $\alpha > 0$, Pesin's theory of invariant measurable families of submanifolds gives a positive answer. I do not know what happens if $f$ is $C^1$ only.


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