Conditions for the Existence of SBR Measures for “Almost Anosov” Diffeomorphisms

Abstract

Abstract. A diffeomorphism f of a compact manifold M is called "almost Anosov" if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure µ that has absolutely continuous conditional measures on unstable manifolds. The measure µ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore,

Equilibrium states of almost Anosov diffeomorphisms

“…Among the similarities between Anosov and almost Anosov diffeomorphisms is the fact that the tangent bundle has a splitting T M = E u ⊕ E s into stable and unstable subspaces, and except at the singularities of the almost Anosov diffeomorphism, this splitting is continuous. This was proven in [3] as part of the proof that almost Anosov diffeomorphisms admit SRB measures. In particular, the geometric t-potentials φ t (x) = −t log Df x | E u (x) are well-defined for almost Anosov diffeomorphisms, but may not be Hölder continuous at the indifferent fixed points.…”

“…Furthermore, the decomposition T M = E u ⊕ E s is continuous everywhere except possibly on S. Remark 3. The proof of this theorem in [3] gives tangency of W u (x) to E u (x). The author notes that W s (x) is tangent to E s x , and the same argument can be used to prove this as was used to prove the fact for W u (x) and E u…”

“…Introduction. In [3], Hu gave conditions on the existence of Sinai-Ruelle-Bowen (SRB) measures for a class of surface diffeomorphisms that are hyperbolic everywhere except at a finite set of indifferent fixed points (that is, fixed points p of the map f : M → M such that Df p = id). Maps that are uniformly hyperbolic away from finitely many points are known as "almost Anosov" diffeomorphisms, and they provide a class of nonuniformly hyperbolic diffeomorphisms that are, in a sense, as close to being uniformly hyperbolic as one can ask for.…”

“…Rather, this map admits what is referred to as an "infinite SRB measure", or a measure µ whose conditional measures on unstable leaves are absolutely continuous with respect to Lebesgue, µ(M \ U ) < ∞ for any neighborhood U of the set of non-hyperbolic fixed points, and µ(M ) = ∞. In [3], the author makes the heuristic observation that even for almost Anosov maps with indifferent fixed points, if the differentials exhibit stronger contraction than expansion as points approach indifferent singularities, one may expect to find this map admits an infinite SRB measure rather than an SRB probability measure. On the other hand, when an almost Anosov map with an indifferent fixed point has differentials exhibiting stronger expansion than contraction as points approach the singularity, one may expect this map to admit an SRB probability measure.…”

“…From here, we assume M = T 2 , f : T 2 → T 2 is almost Anosov with singular set S = {0}, and that Df 0 = Id. It is shown in [3] that nondegeneracy of f implies D 2 f 0 = 0, so there is a coordinate system around 0 for which f is expressible as…”