Ruelle inequality on a noncompact space

Does someone have a reference where the Ruelle inequality would be proved in the following context.

Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, or smooth), whose differential is uniformly bounded ($\sup_{x\in M}\|T_xf\|<\infty$) on $M$.

Assume maybe that $M$ satisfies an additional assumption : [???? to complete ??]

Let $\mu$ be a $f$-invariant probability measure on $M$. Then $$h_\mu(f)\le \int_M \sum_{i:\chi_i(f)>0} \chi_i(x)\dim E_i(x) d\mu(x)$$ where the numbers $\chi_i(x)$ are the Lyapounov exponents, and $E_i(x)$ the corresponding spaces in the Oseledets decomposition.

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What is $h_\mu (f)$ ? – Alexander Chervov Jul 11 '12 at 9:39
It is the measure-theoretic (or Kolmogorov-Sinai) entropy of $\mu$. See any standard book in ergodic theory for a definition. Or also here : scholarpedia.org/article/Kolmogorov-Sinai_entropy – Barbara Schapira Jul 11 '12 at 11:28
Chapter 6 of the thesis of van Bargen seems to be relevant in the case M=R^d opus.kobv.de/tuberlin/volltexte/2010/2571/pdf/… – Thomas Sauvaget Jul 11 '12 at 13:49

Dear Barbara,

I don't whether this is useful in your case, but one can get a Ruelle inequality if $M$ admits a "nice compactification" and $f$ behaves "well" near the boundary of this compactification (i.e., at "infinity") because in this context the results in the book "Invariant manifolds, Entropy and Billiards" of A. Katok and J.-M. Strelcyn may be applied. More precisely, suppose that $M$ can be viewed as an open and dense subset of a compact metric space $N$ satisfying conditions (A), (B), (C) and (1.1) in Katok-Strelcyn's book, and $f$ is a $C^2$ diffeomorphism preserving a probability $\mu$ verifying conditions (1.3) and (1.4) in Katok-Strelcyn book and the usual integrability condition $\int \log^+\|df\|d\mu<\infty$. Then, the Ruelle inequality holds.

Of course, the integrability condition is true under your assumption of uniform bound on $\|df\|$, so that the main issue is to figure out if such a nice compactification of $M$ exists in your setting.

Best,

Matheus

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Hello I forgot to thank you earlier for this answer. But in Katok-Strelcyn, you need as you said a nice compactification. The example I had in mind is geodesic flows on complete non compact negatively curved manifolds, which are unbounded, and cannot be compactified without changing strongly the metric, which changes of course the Lyapounov exponents. Bests Barbara – Barbara Schapira Nov 8 '12 at 11:18

This is an oldie, but it is sufficient to assume that $df$ and $(df)^{-1}$ are uniformly bounded. Indeed, these are the only properties of $f$ required for the proof (say using the Ruelle argument, which at most involves some type of volume estimates depending on the derivatives, plus some initial argument involving the exponential map).

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