| TWO PROBLEMS IN MEASURE RIGIDITY (2009) | |||||||||||||||
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| Theorem 1. [4] Let k ≥ 2, α be a C 1+ǫ, ǫ> 0 action of Z k on a k+1-dimensional manifold, µ an ergodic invariant measure of α with no proportional Lyapunov exponents and at least one element of α has positive entropy. Then µ is absolutely continuous. The only known model for such action is the algebraic Cartan action on the torus T k+1, i.e. the action by hyperbolic maps with real eigenvalues. All known examples are differentiably conjugate to a Cartan action on an invariant open set. Notice however that there are many manifolds which can carry such actions even if one requires topological transitivity in addition. Those manifolds are constructed by blowing up periodic orbits of a Cartan action and either glueing in projective spaces (a σ-process) or identifying boundary spheres of different holes. Problem 1. What compact manifolds carry actions satisfying assumptions of Theorem 1? The answer may be different for real-analytic actions where certain restrictions are plausible and smooth (C ∞ ) actions which are likely to exist on any compact manifold. The key case is that of the ball D k+1. In fact existence of an action on the ball which is sufficiently “flat ” at the boundary would imply existence on any compact manifold as in [1]. The most interesting problem concerns certain arithmetic structure present in such actions. It is motivated by the following result for the torus. Theorem 2. [3, 5] Let α be a C 1+ǫ, ǫ> 0, Z k action on T k+1 Cartan homotopy data i.e. each element is homotopic to the corresponding element of a linear Cartan action α0. Then • The set M consists of a single measure µ. • The measure µ is absolutely continuous. • The semi-conjugacy h is bijective on a set of full measure and thus effects a measurable isomorphism between (α, µ) and (α0, λ). • The semi-conjugacy is differentiable along almost every leaf of each Lyapunov foliation. Problem 2. What are possible values of entropy for elements of an action α satisfying | |||||||||||||||
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