A C1 Anosov diffeomorphism with a horseshoe that attracts almost any point

We present an example of a C1 Anosov diffeomorphism with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.


Introduction.
Consider a diffeomorphism F that preserves a probability measure ν . The basin of ν is the set of all points x such that the sequence of measures converges to ν in the weak- * topology. A measure is called physical if its basin has positive Lebesgue measure. It is well known (see [1,Sec. 1.3]) that any transitive C 2 Anosov diffeomorphism (note that all known Anosov diffeomorphisms are transitive) has a unique physical measure and • the basin of this measure has full Lebesgue measure, • the support of this measure coincides with the whole phase space. However, in C 1 dynamics there are many phenomena that are impossible in C 2 . For example, Bowen [2] constructed an example of a C 1 diffeomorphism of the plane with a "thick" horseshoe (i.e., a horseshoe with positive Lebesgue measure). Using an analogue of this construction, we prove the following theorem. In our construction the horseshoe H will be semithick, i.e., H will be the product of a Cantor set with positive measure in the unstable direction and a Cantor set with zero measure in the stable direction. The dynamics on H in the unstable direction will be as in Bowen's thick horseshoe, and in the stable direction, as in a linear horseshoe. Let us also mention that the last assertion of the theorem is not a corollary of the former two. Indeed, δ n x → ν does not imply that ω(x) ⊂ supp ν , since the orbit of x may stay near supp ν almost all the time but still move far away from this set infinitely many times.
The Milnor attractor of a map F (denoted by A M (F )) is the minimal (by inclusion) closed set that contains the ω-limit sets of almost all points with respect to the Lebesgue measure. This definition was introduced by Milnor in [4]. It was also proved in [4] that the Milnor attractor always exists. Since the definition of a Milnor attractor uses the Lebesgue measure, the Milnor attractor may change after conjugation by a homeomorphism. Let us say that the Milnor attractor of a C 1 diffeomorphism F is topologically invariant in C 1 if, for any C 1 -diffeomorphism G such that G

Corollary 2.
For any C 1 diffeomorphism that is sufficiently close to F H , the Milnor attractor is not topologically invariant in C 1 .

Proof.
Since F H is structurally stable, it has a C 1 neighborhood U such that all maps in U are conjugate to each other. Since C 2 diffeomorphisms are dense in the space of C 1 diffeomorphisms, one may find a

The semithick horseshoe.
The thick horseshoe constructed in [2] can be embedded in a C 1 Anosov diffeomorphism of T 2 (see [5]). We need a similar example but with a semithick horseshoe, i.e., a horseshoe that is the product of a Cantor set with positive measure in the unstable direction and a Cantor set with zero measure in the stable direction. Consider any linear Anosov diffeomorphism F L . In the sequel, we will refer to the unstable direction of F L as vertical and to the stable direction as horizontal.

Lemma 3.
There exists a C 1 Anosov diffeomorphism F Init : T 2 → T 2 with the following properties: * 1. F Init preserves the unstable foliation of F L (we will call it the vertical foliation). We do not present the proof of this lemma, because it is straightforward but has many technical details. Let us represent UK as the product of a vertical segment V and a horizontal segment H . Let S = H × C thick denote the union of the stable leaves of the points of H for the restriction of F Init to UK . The set S is foliated by the horizontal stable leaves of the points of the horseshoe, and so every point of S is attracted to H .

The restriction of F Init to some rectangle UK is a horseshoe (see the picture). The boundary of UK is formed by two vertical and two horizontal segments. The intersection of UK with
3. Plan of the proof of Theorem 1. We will construct a special class C 1 of C 1 Anosov diffeomorphisms, and then we will prove that a generic map in this class has the properties claimed in Theorem 1.
The auxiliary class of homeomorphisms C consists of all homeomorphisms F of the torus with the following properties: (1) F coincides with F Init on S ; (2) F preserves the vertical foliation and expands the vertical leaves; (3) F is bi-Lipschitz with constant L, where L is a large number, the same for all maps in C ; (4) F satisfies some technical requirements that we do not state here.
The special class of Anosov diffeomorphisms C 1 is defined as the intersection of C with the closure of some small C 1 -neighborhood of F Init .
To prove Theorem 1, we need the following three facts. Let us endow C with the metric induced from Homeo(T 2 ) and C 1 with the metric induced from Diff 1 (T 2 ).

Proposition 4. C 1 is a nonempty complete metric space.
Proof. C 1 is a closed subspace of the complete metric space Diff 1 (T 2 ). It is nonempty, because it contains F Init .
For G ∈ C , let B(G) = n 0 G −n (S) denote the union of the stable leaves of all points of H . For ε > 0, we use A ε to denote the set of all G ∈ C with Leb(B(G)) > 1 − ε, where Leb denotes the probability Lebesgue measure on T 2 .

Lemma 5. For any
Sketch of the proof. Take F ∈ A ε ; then Leb(B(F )) > 1 − ε. Let us approximate B(F ) = n 0 F −n (S) by a set F −N (Ŝ), where N is a large number andŜ ⊃ S is a finite union of rectangles that approximates S well (i.e., Leb(Ŝ \ S) is small). If we slightly perturb F , then the boundary of the set F −N (Ŝ) will shift slightly, so that the measure of this set will remain almost the same. Using that this set approximates B(F ) well, one can show that Leb(B(F )) will remain almost the same after a small perturbation of F . We use the bi-Lipschitz continuity of maps in C to estimate from above the measure of F −N (Ŝ \ S) before and after the perturbation.
Lemma 6. For any ε > 0, the set A ε ∩ C 1 is dense in C 1 .
Sketch of the proof. For a fixed map in C , we define the segments of level 0 as the connected components of the intersections of T 2 \ UK with the vertical leaves and the segments of level n as their nth preimages. If, for some N , a map G is linear on all segments of level greater than N , then Leb(B(G)) = 1. Let us sketch the proof of this fact.
• Each segment of level N after a bounded number of iterations is expanded onto some segment of a vertical leaf that cuts all the way through S (i.e., intersects all horizontal leaves that constitute S ).
• Thus, there is a constant c > 0 such that the proportion of points of B(G) in each segment of level N is at least c. This also holds for the segments of level greater than N , because they are linearly expanded to segments of level N .
• This leads to a contradiction with the fact that the restriction of T 2 \ B(G) to some vertical leaf has a density point. Hence Leb(B(G)) = 1.
Let us prove that A ε is dense. For each F 0 ∈ C 1 , we need to find a map F ∈ (A ε ∩ C 1 ) that is C 1 -close to F 0 . We will do this in two steps. First, we pick large N and build a nonsmooth map F P L ∈ C that is linear on all segments of level greater than N . Second, we pick small δ > 0 and, smoothing F P L , obtain a diffeomorphism F ∈ C 1 that is δ-close to F P L in the topology of Homeo(T 2 ). Since F P L ∈ A ε and this set is open, for small δ, we have F ∈ (A ε ∩ C 1 ). It is possible to show that, for large N and small δ, the map F is C 1 -close to F 0 .
Proof of Theorem 1. Consider the set A 0 := ε=1/n (A ε ∩ C 1 ). By Lemmas 5 and 6 A 0 is a residual subset of C 1 . By the Baire theorem it is nonempty. Let us take any diffeomorphism in A 0 for F H . It follows from the definition of A 0 that Leb(B(F H )) = 1.
Let χ : {0, 1} Z → H denote the symbolic encoding map of the horseshoe. The physical measure ν will be the χ-image of the (1/2, 1/2)-Bernoulli measure. Clearly, supp ν = H . It follows from Bowen's construction of a thick horseshoe that the projection of ν to the vertical axis coincides (up to a multiplicative constant) with the projection of the Lebesgue measure on S . Using this property and Birkhoff's ergodic theorem, we can easily prove that Leb-almost any point of S lies in the basin of ν . Thus, this basin also contains almost any point of F −n H (S) for any n and, therefore, almost any point in B(F H ). Since Leb(B(F H )) = 1, this means that the basin of ν has full Lebesgue measure.
Let us prove that ω(x) = H for almost any x. Indeed, for any x in the basin of ν , we have ω(x) ⊃ H , while for any x ∈ B(F H ), we have ω(x) ⊂ H .