Ruelle inequality on a noncompact space

Question

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.

Answer 1

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

Answer 2

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).