We prove rigidity results for a class of non-uniformly hyperbolic holomorphic maps. If a holomorphic Collet-Eckmann map$f$is topologically conjugate to a holomorphic map$g$, then the conjugacy can be improved to be quasiconformal. If there is only one critical point in the repeller, then$g$is Collet-Eckmann, too.

There is gained interest in understanding statistical inference under possibly non-sparse high-dimensional models. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform non-testability that allow the study of limitations of tests across a broad set of alternatives. Uniform non-testability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size of the test. Implications of the new constructions include new minimax testability results that in sharp contrast to the existing results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that in models with weak feature correlations minimax lower bound can be attained by a test whose power has the parametric rate regardless of the size of the model sparsity.

Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composition of qo with the conformal automorphisms of S". We let Vc (n, qg) be the maximum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonstrate that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of RP z is 6n and the conformal area of the square torus is 27c 2. We believe that the computation of the conformal area for general surfaces will be very important in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk [> 27c where k is its curvature. Then Milnor [12] r proved that 5lk]> 4~ z if cr is

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