Ruelle inequality on a noncompact space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:04:03Z http://mathoverflow.net/feeds/question/101917 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101917/ruelle-inequality-on-a-noncompact-space Ruelle inequality on a noncompact space Barbara Schapira 2012-07-11T06:43:53Z 2012-07-20T19:37:59Z <p>Does someone have a reference where the Ruelle inequality would be proved in the following context. </p> <p>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 ($\sup_{x\in M}\|T_xf\|&lt;\infty$) on $M$. </p> <p>Assume maybe that $M$ satisfies an additional assumption : [???? to complete ??]</p> <p>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. </p> http://mathoverflow.net/questions/101917/ruelle-inequality-on-a-noncompact-space/102766#102766 Answer by Matheus for Ruelle inequality on a noncompact space Matheus 2012-07-20T19:37:59Z 2012-07-20T19:37:59Z <p>Dear Barbara, </p> <p>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 $\int \log^+\|df\|d\mu&lt;\infty$. Then, the Ruelle inequality holds. </p> <p>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. </p> <p>Best, </p> <p>Matheus </p>