Limits of intrinsically ergodic systems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:14Z http://mathoverflow.net/feeds/question/87905 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87905/limits-of-intrinsically-ergodic-systems Limits of intrinsically ergodic systems Rafael Alcaraz 2012-02-08T17:20:31Z 2012-02-09T01:13:07Z <p>Let $(X_i)$ be a sequence of compact metric spaces and $(f_i)$ a sequence of transitive transformations $f_i:X_i \to X_i$ with $0 &lt; h_{top}(f_i) &lt; \infty$.</p> <p>The sequence of dynamical systems satifies:</p> <ul> <li>$X_i \subset X_{i+1}$, $h_{top}(f_i) &lt; h_{top}(f_{i+1})$;</li> <li>$X_i$ converges to a compact metric space $X$;</li> <li>$f_{i+1}\mid_{X_i} = f_i$ for every $i$;</li> <li>Besides, there is a transformation $f:X \to X$ such that f is transitive, $0 &lt; h_{top}(f) &lt; \infty$ and $f\mid_{X_i} = f_i$.</li> <li>$h_{top}(f_i)$ converges to $h_{top}(f)$</li> </ul> <p>Assume now that the system $(X_i, f_i)$ is intrinsically ergodic for all $i\ge0$, i.e., it has a unique measure of maximal entropy. </p> <p>QUESTION. Is $(X,f)$ intrinsically ergodic? </p> <p>(If it helps, each $(X_i,f_i)$ in my set-up is a transitive subshift of finite type (SFT), but $(X,f)$ is not an SFT.) </p> <p>If the answer is yes, does there exist a natural way to project the (unique) measure of maximal entropy $\mu$ on $X$ onto $X_i$ so that the projection of $\mu$ is the measure of maximal entropy $\mu_i$ on $X_i$?</p> http://mathoverflow.net/questions/87905/limits-of-intrinsically-ergodic-systems/87927#87927 Answer by Anthony Quas for Limits of intrinsically ergodic systems Anthony Quas 2012-02-08T19:39:50Z 2012-02-09T01:13:07Z <p>The answer is no. It's based on a (un?)published example of Crannell, Rudolph and Weiss.</p> <p>The example is the following shift: $X$ is the subset of $\lbrace 0,\pm 1\rbrace ^{\mathbb Z}$ with the property that $x_k\cdot x_{k+2^n}$ is not allowed to be $-1$ for any values of $k$ and $n$.</p> <p>What they prove is that there are 2 measures of maximal entropy for $X$: one the Bernoulli (1/2,1/2) measure living on sequences of 0's and 1's; the other the Bernoulli (1/2,1/2) measure living on sequences of 0's and $-1$'s. In fact I showed with Ayse &#350;ahin that these are the unique measures of maximal entropy.</p> <p>Now if you let $X_i$ be the subset of $X$ where you can't have $i$ consecutive $-1$'s, then $X_i$ is intrinsically ergodic, but $X$ is not.</p>