WHATIS... ? Rigidity? Manfred Einsiedler

Measure rigidity is a commonly used shorthand integral as the expected value for the outcome of term for rigidity of measures. Here the f . Then the invariance of µ is simply the require- term rigidity does not have a formal mathematical ment that the expected value of the outcome is the definition. Rather, it is an informal description of same now and one time unit later. The set M(T ) of the frequently appearing phenomenon that, for invariant probability measures depends crucially certain mathematical objects, the only examples on the transformation T . For many maps T this set have much more algebraic structure than was orig- M(T ) is rather large, and it is impossible to give inally demanded. A simple example would be the a reasonable description. However, sometimes we statement that we have rigidity of continuous ho- also have rigidity of invariant measures: the set of momorphisms of the . This is a shorthand invariant measures shows a surprising amount of way of saying that all continuous homomorphisms structure. We will give examples of both scenarios. of the additive group R to itselfarein fact linear.(A Let us start by studying the case of the circle ro- more subtle result is that one even has rigidity of tation defined by Rα(x) = x+α, where x ∈ T, α ∈ R measurable homomorphisms of the real line.) As is a given number, and addition is understood as the requirement to be a homomorphism is much modulo Z. Let us assume α ∈ R\ Q is irrational, as weaker than the requirement to be linear, this this makes the more interesting. is indeed an example where additional algebraic The standard substitution rule of the Riemann in- structure is forced. One may say that a linear map tegral shows in fact that the mT cannot be perturbed to a nonlinear continuous is invariant under Rα. We claim that mT is the only homomorphism or that linear maps are rigid. Rα-invariant measure. One way to see this uses 2πinx To explain what an invariant measure is, we the characters en(x) = e ∈ C(T) with n ∈ Z. need to introduce a transformation on a space—a Suppose µ is an unknown Rα-invariant probability dynamical system. Let X be a metric space. For measure. Then by definition R endµ = R en ◦ Rαdµ. example, we could set X = T = R/Z, which is often For n = 0 this contains no new information, as called the torus or the circle group and consists of e0 = 1T is constant. So assume n ≠ 0; then we 2πinα the cosets x = r + Z for r ∈ R. Let us agree that a have en(Rα(x)) = e en(x), which shows that 2πinα measure µ on a metric space X is simply a way of R en ◦ Rαdµ = e R endµ. As α ∈ R \ Q we have 2πiαn assigning to every continuous f ∈ C(X) e ≠ 1, and so R endµ = 0. This agrees with the a number—its integral R f (x)dµ(x) or R f dµ with value R endmT that the Lebesgue measure mT as- respect to µ—so that the usual properties of an signs to en. Hence we have R f dµ = R f dmT for any integral hold: we require that f → R f dµ is linear finite linear combination of characters en. As the and that f ≥ 0 implies R f dµ ≥ 0. In fact, we will latter can be used to approximate any other contin- be studying probability measures on X, and so uous function uniformly (by the Stone-Weierstrass µ = m the integral of the constant function 1X is one. theorem), one sees that T. This is the most 1 basic example of rigidity of invariant measures The Riemann integral R0 f (r +Z)dr for f ∈ C(R/Z) would in this sense define a , and also the strongest form of it: If M(T ) con- tains only one measure, then T is called uniquely which is called the Lebesgue measure mT. Now let T : X → X be a continuous map. ergodic. A probability measure µ on X is invariant if One may ask why one should care about rigidity R f dµ = R f ◦ T dµ. One should think of T as the of invariant probability measures. The answer lies → time evolution of the dynamical system, of f as in a simple construction. Let T : X X be an the outcome of a physical experiment, and of the arbitrary continuous map from a compact met- ric space X to itself. Then notice that for any Manfred Einsiedler is professor of mathematics at The f ∈ C(X), x ∈ X, and any large N, the ergod- 1 N−1 n Ohio State University. His email address is manfred@ ic (time) average N Pn=0 f (T x) is bounded by math.ohio-state.edu. kf k∞ = max{|f (x)| : x ∈ X} and is close to being

600 Notices of the AMS Volume 56, Number 5 invariant under T in the sense that the differ- measure mT is again invariant under T2, and so ence between the average for f and for f ◦ T is is the measure defined by R f dδ0 = f (0). More- 2kf k∞ over, we can in fact apply our above construction bounded by N . Fixing a function f , we can choose a subsequence Nk along which the ergodic to produce many invariant probability measures. average for f converges—we may think of the Represent x ∈ [0, 1) by its binary expansion, limit as R f dµ for some µ ∈ M(T ). To completely and notice that application of T2 corresponds to define µ on all functions f ∈ C(X), one has to shifting the binary expansion of x. Choosing this continue picking subsequences until one finally infinite sequence in the digits 0 and 1, we may arrives at a subsequence for which the ergodic ensure, for example, that we never see the finite average converges for all functions. In a sense, sequence 000 or 111. Then the above construction the measure µ describes the statistical behavior will lead to T2-invariant probability measures that 1 1 2 − + Z ⊂ T of the orbit x, T (x), T (x), . . . , at least for cer- have support disjoint from ( 8 , 8 ) . For tain very long stretches of time. Combined with this transformation the abundance of T2-orbits measure rigidity this can have very interesting con- of different types (any sequence in 0s and 1s sequences. If T is uniquely ergodic and µ ∈ M(T ), defines an orbit) is reflected in the abundance of then no matter which subsequence (of a subse- T2-invariant measures. quence, etc.) one may have picked, the ergodic A highly interesting question was raised by average for that subsequence must converge to Furstenberg around 1967. He showed that the or- k ℓ R f dµ because µ is the only T -invariant measure. bit set {2 3 (x) : k, ℓ ≥ 1} is dense in T whenever However, this independence of the subsequence the starting point x ∈ T \ Q is irrational; here 1 N−1 n the orbit is taken with respect to the semigroup means that limN→∞ P f (T x) = R f dµ, both N n=0 generated by T and the times-three map T . As we for every f ∈ C(X) and for every x ∈ X. One may 2 3 have discussed, there is often a correspondence summarize this by saying that unique between orbits and invariant measures. Hence implies equidistribution (with respect to µ) of the it is natural to ask the following: What are the orbit x, T (x), T 2(x), . . . for any x ∈ X. Not only probability measures on T that are at the same does this help to describe the closure of the orbit time invariant under T and under T ? Certain (optimally as being equal to the support of µ), but 2 3 rational numbers r ∈ Q are periodic for both T it also asymptotically describes (in terms of µ) the 2 and T , and with these one can easily define in- amount of time the orbit will spend in various 3 variant probability measures. Also, we know that parts of the space. the Lebesgue measure is invariant. Are these (and Another example of a uniquely ergodic trans- their convex combinations) the only ones? The formation is S : (x , x ) → (x + 2α, x + x ) on 1 2 1 2 1 best-known result towards this conjecture is due T2 α ∈ R \ Q where . Furstenberg proved the to Dan Rudolph, and several generalizations have unique ergodicity of S and used the orbit of (α, 0) been obtained by Kalinin, A. Katok, Lindenstrauss, to derive the equidistribution of the sequence Spatzier, and me. Similar to Raghunathan’s con- 2 α, 4α, 9α, . . . , n α, . . . in T, which re-proved a the- jecture, these generalized conjectures are phrased orem of Weyl. The examples above are still quite for dynamical systems defined on quotients of simple, but a similar understanding of invariant Lie groups. However, in this case the dynamical probability measures for more general classes of system is defined by (several commuting) diago- transformations can be a very powerful tool. In nal matrices instead of unipotent matrices. Even fact, Marina Ratner proved a very general mea- though Furstenberg’s question and its generaliza- sure classification theorem (concerning unipotent tions are still open, the partial results have already group actions of quotients of Lie groups) and found several applications. The most striking of derived from it equidistribution and orbit closure these may be Lindenstrauss’s proof of the equidis- theorems; the orbit closure theorem is known as tribution of the arithmetic Laplace-eigenfunctions Raghunathan’s conjecture. The measure classifi- (Hecke-Maaß cusp forms) on certain quotients of cation is indeed a good example of rigidity: by the hyperbolic plane. assumption the measure is known to be invariant under a small subgroup, and in the end it is known References to be a highly structured measure (called algebraic [1] M. Einsiedler and T. Ward, : With or Haar) for which the support is the orbit of a a View towards Number Theory, in preparation, bigger group. However, unlike the above cases, http://www.mth.uea.ac.uk/ergodic/. in general there will be many different invariant [2] M. Einsiedler, Ratner’s theorem on SL(2, R)- probability measures. These theorems and their invariant measures, Jahresber. Deutsch. Math.- 108 extensions by Dani, Eskin, Margulis, Mozes, Rat- Verein. (2006), no. 3, 143–164. D. Witte Morris ner, Shah, Tomanov, and others have found many [3] , Ratner’s Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University applications, in particular in number theory. of Chicago Press, Chicago, IL, 2005. Another transformation on T is the times-two map T2(x) = 2x for x ∈ T. Here the Lebesgue

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