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We know for a differentiable map $f:M\rightarrow M$ equipped with an $f$-invariant probability measure $\mu$, given a point $x\in M$, and a tangent vector $v$ at $x$, the Lyapunov exponent is defined as $$\lambda(x,v)=\overline{lim}_{n\rightarrow \infty}\frac{1}{n}log|Df^n_x(v)|.$$ I want to prove if $(f,\mu)$ is ergodic, then $\lambda$ is constant almost everywhere.

My attempt: We know that if $\mu$ is ergodic, then for any measurable function $U$ which is $f$-invariant, $U$ is constant almost everywhere. Can we check the $\lambda(x,v)$ is a $f$-invariant function?

Alp Uzman
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monell20
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  • Probably you understand that it doesn't make sense to ask whether $\lambda(x,v)$ is $f$-invariant. How about the second component? The proof is a bit more technical: note that $v\mapsto \lambda(x,v)$ takes only finitely many values. – John B Sep 15 '21 at 19:01
  • @JohnB I guess the correct statement should be to ask whether $\lambda (x,v)$ is $f$-invariant almost everywhere. Since it is constant on orbits, hence they are constant µ-almost everywhere if µ is ergodic. – monell20 Sep 16 '21 at 03:17

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