Smooth symmetries of × a -invariant sets

We study the smooth self-maps f of × a -invariant sets X ⊆ [0 , 1] . Under various assumptions we show that this forces log f ′ ( x ) / log a ∈ Q at many points in X . Our method combines scenery ﬂow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery ﬂow which applies in the zero-entropy case.


Introduction
For an integer a ≥ 2 let T a denote the self-map of [0, 1], or R/Z, given by T a x = ax mod 1. Furstenberg famously proved that, although each of these maps individually admits a multitude of closed invariant sets, when a, b are non-commensurable, the only jointly invariant ones T a , T b are trivial [8,Theorem IV.1]. Here by trivial we mean either finite or all of [0, 1], and we say that real numbers a, b = 0 are commensurable, and write a ∼ b, if log |a|/ log |b| ∈ Q; otherwise they are non-commensurable, denoted a ∼ b.
Furstenberg's theorem has seen many generalizations. The main direction of generalization has been to commuting actions in algebraic settings: these include commuting automorphisms of compact abelian groups [1,2], and analogous (though still partial) measure-rigidity results in the automorphism setting as well as for higher-rank diagonal flows on homogeneous spaces [16,4,3]. Another generalization is to commuting diffeomorphisms on compact manifolds, see e.g. [15]. These results are distinct from the algebraic ones, although they share many common methods and are related by conjectures predicting that, in many cases, the only commuting smooth maps are those that are conjugate to algebraic ones. This paper deals with another related phenomenon, namely, that if X is T a -invariant and non-trivial, then very few smooth maps can map it onto (or even into) itself, and in fact, any such map must locally behave like T a , in the sense that it must satisfy f ′ ∼ a for "most" x ∈ X. We stress that no assumption of commutation is made between T a and f . We also show that if Y is another non-trivial T b -invariant set for b ∼ a, then X cannot be mapped to Y by a smooth map.
Results of this kind have a long history for "fractal" sets, sometimes even in the Lipschitz category. For example, when X, Y are "deleted-digit" Cantor sets, i.e. defined by restricting individual digits in non-commensurable bases a and b, Falconer and Marsh showed that there is no bi-Lipschitz map taking X to Y [6] (they have more general results too but in another context). More recent work on self-similar sets with similar flavor appears in the work of Elekes, Keleti and Máthé, and of Feng, Huang and Rao [5,7]. Our results have some overlap with these, but we emphasize that we deal with much more general sets, including sets of entropy (or dimension) zero.
Recall that we say a set X ⊆ [0, 1] is non-trivial if it is infinite and X = [0, 1].
For self-maps of a single T a -invariant set we obtain results under some additional dynamical and regularity assumptions. A set is perfect if it has no isolated points. If X is T a -invariant then a point x ∈ X is transitive if its orbit {T n a x} ∞ n=1 is dense in X, and X is transitive if it contains a transitive point. It is minimal if every point is transitive. Minimal infinite systems are perfect. Adapting Furstenberg's terminology, we say that X is self-restricted if X − X = [0, 1] mod 1, which holds in particular when X is minimal (see [8]), or dim X < 1/2. Finally, we say that f ∈ C 1 is piecewise curved if f ′ is piecewise strictly monotone (thus f is locally strictly concave or convex).
Theorem 2. Let X ⊆ [0, 1] be a closed, perfect, transitive and non-trivial T ainvariant set. Then there is no piecewise-curved f ∈ C 1 that is differentiable on X and maps X onto itself. Furthermore, without assuming curvedness, we have: for all x in a dense, relatively open subset of X.
In particular, in (2) and (3), if in addition f is real-analytic then f is affine.
We do not know whether f (X) ⊆ X for transitive X implies similar conclusions in general.
The proofs split into two parts. The main new ingredient of this paper is a method to handle the case that X is self-restricted (though actually the main case of interested is the more special case when its entropy is zero). To explain this part it is useful to recall Furstenberg's original proof that there are no non-trivial jointly T a , T b -invariant sets. For such an X, the first observation is that X − X mod 1 is jointly T a , T b invariant (because both T a and T b are endomorphisms of R/Z). On the other hand, if X is infinite, then it has an accumulation point, whence 0 is an accumulation In contrast, in our setting the first stage of this argument already fails: we are assuming that X is invariant under T a and under another map f , and while X − X is still invariant under T a , it is generally not f -invariant. The main new ingredient in our proof is to use a local version of the difference set X − X which behaves well under smooth maps. This is developed in Section 2, and is motivated by the scenery flow and spectral arguments of [10,14]. These do not make a formal appearance here, but see the remark at the end of Section 2.
In the non-restricted case, and specifically when X has positive topological entropy, the theorems above are proved via analysis of invariant measures on X, adapting scenery flow methods from [10,14]. This is also the source of the piecewise curvedness requirement. 1 To state our result for positiveentropy measures, let us say that a probability measure µ has dimension t if µ(E) < 1 for all Borel sets E with dim E < t, but it is supported on some set of dimension t. We write f µ = µ • f −1 for the push-forward of µ by a map f , and note that when f is bi-Lipschitz, µ and f µ have the same dimension. Our result for measures is the following: 2 Theorem 3. For every T a -ergodic measure µ of dimension 0 < s < 1, there exists ε = ε(µ) such that the following holds. For every piecewise curved f , every weak-* accumulation point of the sequence 1 Combined with the variational principle, this implies that if X is a nontrivial T a -invariant set with positive dimension, then f (X) ⊆ X is impossible for a piecewise curved f . Thus for a piecewise curved function f , every jointly f -and T a -invariant ergodic probability measure is either Lebesgue or has dimension zero, and every jointly invariant set is either [0, 1] or has dimension 0. This and the other results stated earlier make the following conjecture seems reasonable: and assume that f is not affine. Then every jointly T a -and f -invariant set is trivial.
Of course one could also ask this for f with less smoothness, or make the same conjecture for measures.
The paper is organized as follows. We begin with the topological analysis of the zero entropy (or self-restricted) case: In Section 2 we define the local difference (or distance) set and discuss its properties, in Section 3 we discuss dimension and the relation between the local difference set and the original set, and in Sections 4 and 5 we prove Theorem 1 and (most of) Theorem 2. The last section is devoted to Theorem 3 and completing the proof of Theorem 2 (1). Acknowledgement. I am grateful to J.-P. Thouvenot for encouraging me to revisit these questions. I also thank Amir Algom and the anonymous referee for their comments on an early version of the paper.

Localizing the difference set
As explained in the introduction, we require a "localized" version of the difference set X − X. For this, define the a-adic fractional part of s > 0 by {s} a = − log a s mod 1 2 We recently learned of results by Eskin, related to work of Brown and Rodriguez-Hertz, which shows that certain "general position", expanding-on-average pairs of diffeomorphisms of a compact manifold can preserve nothing but Lebesgue measure. Their methods do not appear to apply in our setting, where both the expansion and invertability are absent.
where x = y mod 1 and X = Y mod 1, and the lift Y is chosen so that reduction modulo 1 is a bijection of neighborhoods of x and y. When X is T a -invariant, we view X as a subset of R/Z when defining F a,X (x), even if X is initially given as a subset of [0, 1]. This convention potentially enlarges F a,X (0) when 0, 1 ∈ X.
This defines a function F a,X : X → {subsets of R/Z}, but we sometimes think of the range as subsets of [0, 1], and make the identification whenever convenient. It will be convenient to write We state, mostly without proof, some elementary properties of F a,X (x).

(Locality
If the X i are closed and disjoint then F a,X (X) = F a,X i (X i ). 6. (Semi-continuity): If x n → x and t n ∈ F a,X (x n ), and if t n → t in R/Z, then t ∈ F a,X (x) (equivalently, in the space of compact subsets of R/Z, In particular, if X is compact, then F a,X (X) = x∈X F a,X (x) is closed.
7. (Linear transformation): For X ⊆ R and any 0 = b ∈ R, we have 3 F a,bX (bx) = F a,X (x) − log a |b| mod 1 3 We use the usual notation for arithmetic operations between sets: for u ∈ R and W ⊆ R, uW = {uw : w ∈ W } and W + u = {w + u : w ∈ W }.
Indeed, for small r > 0 the map T b acts on B r (x) as an affine map of expansion b, and sends X ∩ B r (x) into a subset of T b X ∩ B br (T b x). The inclusion above follows by locality and monotonicity.
For X ⊆ [0, 1], the same holds except possibly at points The converse inclusion is shown similarly by considering f −1 .
Definition 5 is motivated by the spectral analysis of the scenery flow of ×a-invariant measures that was carried out in [10]. Indeed, property (7) means that the function t → F a,tX (0) is periodic (in log t), and so defines an eigenfunction for the scenery flow (with values in the space of closed subsets of R/Z).

Dimension
We write dim X for the Hausdorff dimension of a set X ⊆ R or R/Z, and dim B X for its box dimension, defined as the limit as r ց 0 of log N (X, r)/ log(1/r), N (X, r) is the minimal number of r-balls needed to cover X (in the cases we apply this, the limit exists). We have the following standard facts.
2. dim and dim B are non-decreasing under Lipschitz maps.
5. If X is closed and T a -invariant then dim B X exists and dim B X = dim X = h top (X, T a )/ log a, where h top (X, T a ) is the topological entropy of (X, T a ) [8, Section III].
It follows from these properties that if Y ⊆ [0, 1] is closed and T a -invariant, then dim X × Y = dim X + dim Y for all sets X.
. We can assume that t = 0, since if it is then b 0 ∈ Y − Y mod 1 trivially. By definition, there exist points y ′ n , y ′′ n ∈ Y such that {y ′ n − y ′′ n } b → t. This means that log b (y ′ n − y ′′ n ) → t mod 1. Assuming without loss of generality that y ′ n ≥ y ′′ n . there exist k n ∈ N and 0 ≤ t n < 1 such that y ′ n − y ′′ n = b −kn−tn and t n → t mod 1. Since t = 0, this means convergence is also in R once we identify t n , t with elements of [0, 1), which we do henceforth. Since y ′ n −y ′′ Passing to a subsequence we can assume bi-Lipschitz to its image, so it preserves dimension. In particular its image, which is a subset of Y − Y , has the same dimension as F b,Y (Y ). This gives A real function f is to piecewise satisfy a property P if there is a discrete set E such that for all x ∈ dom f \ E, the function satisfies P in some neighborhood of x.
Writing π(y, t) = y − t mod 1, we have F b,X (X) = π(Z), and since π is Lipschitz, where in the first line we used the fact that log preserves dimension, the next inequality used the fact that g ′ is C 1 , hence piecewise Lipschitz, so dim B g ′ (Y ) ≤ dim B Y , and the last inequality is because g is bi-Lipschitz, and hence dim Y = dim g(X) = dim X. Combining the inequalities gives the first claim. For the second statement, consider a finite partition of R into intervals is a C 2 -diffeomorphism except possibly at the endpoints, which then lie outside of X. Let X i = X ∩ I i and Y i = f (X i ). Since X = X i and Y = Y i and since there are finitely many sets in these unions, we have , and again the box dimension of each set is given by the maximum of the dimensions of the sets in the union. Thus, Suppose first that dim X > 0. Then h top (X, T a ) = log a · dim X > 0. By the variational principle we can find a T a -invariant and ergodic probability measure µ on X with h(µ, T a ) > 0. Then by [14,Theorem 1.10], for µ-a.e.
x, the point f (x) equidistributes under T b for Lebesgue measure, and in particular the T b -orbit of f (x) is dense. Since f (X) ⊆ Y this means that Y = [0, 1]. Since f is a diffeomorphism, X must also be an interval, and the only interval in [0, 1] invariant under T a is [0, 1].
It remains to show that dim X > 0.
Proof. X is closed and infinite it contains an accumulation point To prove dim B X > 0, suppose by way of contradiction that dim B X = 0. Now, Y = f (X), so by Proposition 7,

Proof of Theorem 2
Proof of part (2): Let X ⊆ [0, 1] be an infinite, minimal T a -invariant set. Given x, y ∈ X there is a sequence n k → ∞ such that we have T n k a x → y. Since F a,X (x) ⊆ F a,T n k a X (T n k a x) = F a,X (T n k a x) for all k, by semi-continuity of F a,X (·) we conclude that F a,X (x) ⊆ F a,X (y). Since x, y ∈ X were arbitrary, F a,X (x) is independent of x ∈ X. We denote this set by E. Since X is infinite there is an accumulation point x 0 ∈ X, so E = F a,X (x 0 ) = ∅.
Suppose that I is a non-empty open interval and f : I → R a C 1embedding such that f (X ∩ I) ⊆ X. Let x ∈ X ∩ I, we must show that α = log a |f ′ (x)| ∈ Q. Indeed suppose α / ∈ Q. Then, writing y = f (x), by the last paragraph we have Thus E is closed, non-empty and invariant under t → t + α mod 1. Since α is irrational this implies E = [0, 1]. Now, for x ∈ X we have F a,X (x) ⊆ log a (X − X) by Proposition 6. Thus log b (X − X) has non-empty interior, so the same is true of X − X, and since X − X mod 1 is T a -invariant this implies that But, since X is minimal, this is impossible by [8,Theorem III.1].
Proof of part (3): We recall that a set in a complete metric space is called residual it it contains a dense G δ subset. By Baire's theorem the intersection of countably many residual sets is residual, and a residual set which is also an F σ set contains a dense open set.
Suppose that X is perfect and transitive, and self-restricted, and that f (X) = X for some local diffeomorphism f ∈ C 1 . The proof is very similar to the minimal case. Indeed, since X is perfect, F a,X (x) = ∅ for all x ∈ X, in particular at transitive points. By the same argument as above, F a,X (x) takes the same value for all transitive points x ∈ X. Let W ⊆ X denote the set of transitive points, which is well-known to be is a residual set in X.
Let I ⊆ [0, 1] be a non-trivial interval such that, writing J = f (I), the restriction f : X ∩ I → X ∩ J is bijective. Clearly W ∩ I is residual in X ∩ I. At the same time, f : X ∩ I → X ∩ J is a homeomorphism, and By the last paragraph, the set of x ∈ X such that f ′ (x) ∈ {a s : s ∈ Q} is residual, i.e. contains a dense G δ . On the other hand this set is just (f ′ ) −1 ({a s : s ∈ Q}) = s∈Q (f ′ ) −1 (a s ), and by continuity of f ′ this is an F σ -set. By Baire's theory applied in the compact metric space X, the set in question must contain a dense and open (relative to X) set.
Real-analytic case of (2) and (3): It remains to note that if f is real-analytic, then our conclusion shows that f ′ (x) belongs to the countable set {a s : s ∈ Q} for an uncountable number of x, hence f ′ takes on some rational power a s of a on a convergent sequence, and being itself real-analytic, f ′ ≡ a s . Thus f is affine, and has the stated form.
Proof of main statement of theorem: Let X ⊆ [0, 1] be a closed perfect, transitive and non-trivial T a -invariant set and f ∈ C 1 piecewise curved. If dim X = 0 then X is self-restricted, so f (X) = X implies that f ′ (x) ∈ {a s : s ∈ Q} for uncountably many x ∈ X; this is impossible because f ′ is piecewise strictly monotone, and so takes every value at most countably many times.
It remains to deal with the case that dim X > 0. By the variational principle, there exists a T a -ergodic probability measure µ on X with dim µ = dim X. If f (X) = X then f µ is also supported on X, and by Theorem 3, we obtain (by averaging T n a f µ along some subsequence of times) a measure on X of dimension > dim X, which is impossible.
Proof of part (1): Let f (x) = rx + t with r = 0. If dim X = 0, let x ∈ X with F a,X (x) = ∅. Then F a,X (f n x) = F a,X (x) − n log a |r| mod 1, and if log a |r| / ∈ Q this means that F a,X (X) contains a dense subset of [0, 1] and so is equal to [0, 1]. By Proposition 6 this is inconsistent with X being self-restricted. Thus log a |r| ∈ Q as claimed.
In the case dim X > 0, we provide a similar proof in the next section using measure theoretic tools.

Proof of Theorem 3 and Theorem 2 part (1)
We provide a proof sketch, since a full proof would be lengthy, and the results in [17,18] can be used to give alternative proofs of the application to Theorems 1 and 2. We rely heavily on the scenery flow methods from [13,14] and additive combinatorics methods from [11] and refer the reader to those papers for definitions and notation.
Let µ be a T a -ergodic measure µ of dimension 0 < s < 1 and f ∈ C 1 piecewise curved. Let ν = f µ and let ν ′ = lim 1 It is well known (e.g. [10]) that µ generates an ergodic fractal distribution (EFD; [9, Definition 1.2]) P supported on measures of dimension s, which are supported (up to a bounded scaling and translation) on X. See e.g. [10, Section 2.2]. Because X is porous, P -a.e. measure is (1 − ε ′ ) -entropy porous along any sequence [n 1+τ ] of scales, in the sense of [12, Section 6.3], for some ε ′ depending only on dim X.
Since f is a piecewise diffeomorphism, for µ-a.e. x the measure ν = f µ log a-generates S * log f ′ (x) P at y = f (x) (see e.g. the proof of [14,Lemma 4.16]).
Let π denote the (partially defined) operation of restricting a measure to [0, 1] and normalizing it to a probability measure. It now follows, as in [14,Theorem 5 T n a ν → ν ′ then there is an auxiliary probability space (Ω, F, Q) and functions y, t : Ω → R and η : Ω → P([0, 1]), such that t ω , ω ∼ Q is distributed like log f ′ (x), x ∼ µ, and η ω , ω ∼ Q is distributed according to P , and such that ν ′ = π(S tω (η ω * δ yω ))dQ(ω) (the formula above differs from [14,Theorem 5.1] in the scaling S tω of the integrand; the scaling comes from the fact that at ν-a.e. point y = f (x), the measure ν generates S * t P , where t = log f ′ (x). Also, the theorem in [14] refers to ν ′ arising from the orbit of a single ν-typical point, not as above, but in fact the averaged version above follows from the pointwise one).
Re-interpreting the last equation and the properties of t, η stated before it, we find that ν ′ can be generated in the following way: choose x according to µ ω , independently choose a P -typical measure η, scale η by f ′ (x), and translate by a random amount y (whose distribution depends on x, η). Changing the order with which we choose x and η, we find that for P -typical η there is a probability measure θ η on the group of affine maps of R such that we can represent ν ′ as ν ′ = π(T η) dθ η (T ) dP (η) = π(θ η * η) dP (η) Furthermore, choosing T ∼ θ, the distribution of the contraction ratio of T is the same as the distribution of f ′ (x) for x ∼ µ. Since f was assumed piecewise curved, f ′ is locally bi-Lipschitz, so the image of µ under x → f ′ (x) has the same dimension as µ, so by the above, dim θ η ≥ dim ν ω = s. Also, recall that η has uniform entropy dimension s (in the sense of [11,Definition 5.1]; this is an immediate consequence of the definition of the measures η, of the definition of Kolmogorov-Sinai entropy, and of the ergodic theorem). It follows from [12,Theorem 9] that dim H θ η * η ≥ dim η + ε = s + ε for some ε = ε(s), hence which is what we wanted to show. We now turn to the case that f (x) = rx + t is affine, f (X) ⊆ X and dim X > 0. Fix a dimension-maximizing T a -invariant and ergodic measure µ on X. Choose a typical point x and consider the EFD P n log a-generated at f n x. Evidently this is P n = S n log a r P 0 . Assuming log a r / ∈ Q, we can average and pass to a weak* limit, and find that X supports a measure of the form ν = log a 0 π(S t (η * δ yη,t ))dtdP (η) We now again apply [12,Theorem 9] to conclude that dim ν > dim µ = dim X, a contradiction, which shows that log a r ∈ Q.