Unique equilibrium states for systems without specification - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:40:38Z http://mathoverflow.net/feeds/question/48088 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48088/unique-equilibrium-states-for-systems-without-specification Unique equilibrium states for systems without specification Vaughn Climenhaga 2010-12-02T20:03:53Z 2011-07-04T04:32:18Z <p>Let $X$ be a compact metric space and let $f\colon X\to X$ be a continuous expansive map. Let $\mathcal{V}$ denote the space of H&ouml;lder continuous potential functions $\phi\colon X\to \mathbb{R}$, and let $\mathcal{W}$ denote the set of potential functions (not necessarily H&ouml;lder continuous) that have a unique equilibrium state.</p> <p>It is well known that if $f$ satisfies the specification property, then every $\phi\in \mathcal{V}$ has a unique equilibrium state, and hence $\mathcal{V} \subset \mathcal{W}$ (<a href="http://www.springerlink.com/content/uu2n4xh52463rn14/" rel="nofollow">Rufus Bowen, <em>Some systems with unique equilibrium states</em>, Math. Systems Theory <strong>8</strong> (1974/75), no. 3, 193–202</a>).</p> <p>There are many systems without the specification property for which something is known about the set of potentials $\mathcal{W}$. For example, intrinsic ergodicity (existence of a unique measure of maximal entropy), which is equivalent to the statement that $\mathcal{W}$ contains the constant functions, has been studied in a number of recent works (<a href="http://arxiv.org/abs/0903.3692" rel="nofollow">Buzzi-Fisher</a>, <a href="http://arxiv.org/abs/math/0703020" rel="nofollow">Bufetov-Gurevich</a>, <a href="http://arxiv.org/abs/1011.2780" rel="nofollow">Climenhaga-Thompson</a>), and there are stronger results for particular examples, such as $\beta$-shifts, for which $\mathcal{W}$ is known to contain the space of <em>Lipschitz</em> functions (<a href="http://www.springerlink.com/content/w2p858t306074612/" rel="nofollow">Peter Walters, <em>Equilibrium states for $\beta$-transformations and related transformations</em>, Math. Z. <strong>159</strong> (1978), no. 1, 65–88</a>).</p> <p>However, I do not know of any examples of expansive maps without specification for which the inclusion $\mathcal{V} \subset \mathcal{W}$ is known. Does anybody know of such an example?</p> <p>(Note that it should be not too difficult to adapt the answers to <a href="http://mathoverflow.net/questions/26094/a-topologically-mixing-subshift-with-multiple-measures-of-maximal-entropy" rel="nofollow">this question</a> to obtain a system for which $\mathcal{W}$ contains the constant functions, but does not contain all of $\mathcal{V}$.)</p> <p><strong>Edit</strong>: Of course if $f$ is uniquely ergodic then $\mathcal{W}$ contains <em>all</em> potential functions. The most obvious examples of uniquely ergodic systems, irrational circle rotations (or rather, their <a href="http://mathoverflow.net/questions/42201/what-are-the-zero-entropy-invariant-measures-for-an-anosov-geodesic-flow/42214#42214" rel="nofollow">symbolic counterparts</a>, which are expansive) possess a weak version of the specification property, but I don't know if this weak specification holds for every uniquely ergodic system.</p> <p>What I'd really like to know is if there is an expansive map that is not uniquely ergodic and does not have the specification property for which $\mathcal{V} \subset \mathcal{W}$. I'd also be interested in knowing whether unique ergodicity implies weak specification.</p> <p>(By "weak specification" I mean that orbits can be consecutively shadowed with uniformly bounded gaps, as in the usual specification property, but that we do not require the shadowing orbit to be periodic, and we allow the length of the gaps to vary.)</p> http://mathoverflow.net/questions/48088/unique-equilibrium-states-for-systems-without-specification/48112#48112 Answer by Ian Morris for Unique equilibrium states for systems without specification Ian Morris 2010-12-02T22:34:13Z 2010-12-03T18:41:30Z <p>Let $T_1 \colon X_1 \to X_1$ be an Anosov diffeomorphism and let $T_2 \colon X_2 \to X_2$ be a uniquely ergodic expansive homeomorphism which is not a periodic orbit. Let $T \colon X_1 \times X_2 \to X_1 \times X_2$ be given by the direct product of the two maps. Clearly $T$ is expansive, and $T$ does not have specification because it has no periodic orbits. The invariant measures of $T$ are precisely the products of the invariant measures of $T_1$ with the unique invariant measure of $T_2$ (right...?). So, calculating the equilibrium state(s) of a H&ouml;lder function defined on $X$ is the same problem as calculating the equilibrium state(s) of the function on $X_1$ defined by integrating $f$ along fibers against the unique invariant measure of $T_2$. The fiberwise integral has to be H&ouml;lder because $f$ is H&ouml;lder, and it follows that $f$ has a unique equilibrium state.</p> <p>Edit: the sentence beginning "The invariant measures of $T$ are precisely..." is probably wrong - see comments below.</p> http://mathoverflow.net/questions/48088/unique-equilibrium-states-for-systems-without-specification/69445#69445 Answer by Vaughn Climenhaga for Unique equilibrium states for systems without specification Vaughn Climenhaga 2011-07-04T04:32:18Z 2011-07-04T04:32:18Z <p>In the time since I asked this question, Dan Thompson and I have posted a <a href="http://arxiv.org/abs/1106.3575" rel="nofollow">preprint</a> showing, among other things, that every H&ouml;lder continuous potential function on a &beta;-shift has a unique equilibrium state. Since &beta;-shifts do not have specification for most &beta;>1, this gives an example of the sort I was after.</p>