TY - CONF AU - Krishnendu Chatterjee AU - Dasgupta, Pallab AU - Chakrabarti, Partha P ID - 3447 TI - Weighted quantified computation tree logic ER - TY - JOUR AB - For diffeomorphisms of smooth compact manifolds, we consider the problem of how fast the number of periodic points with period $n$grows as a function of $n$. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call ``prevalence'', the growth is not much faster than exponential. Specifically, we show that for each $\delta > 0$, there is a prevalent set of ( $C^{1+\rho}$ or smoother) diffeomorphisms for which the number of period $n$ points is bounded above by $\operatorname{exp}(C n^{1+\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of $n$. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity shows this to be a subtle and complex phenomenon, reminiscent of KAM theory. AU - Kaloshin, Vadim AU - Hunt, Brian R. ID - 8522 IS - 4 JF - Electronic Research Announcements of the American Mathematical Society KW - General Mathematics SN - 1079-6762 TI - A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I VL - 7 ER - TY - JOUR AB - We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period $n$. In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period $n$) an almost periodic point that is almost nonhyperbolic. We also formulated our results for $1$-dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the $1$-dimensional case. AU - Kaloshin, Vadim AU - Hunt, Brian R. ID - 8521 IS - 5 JF - Electronic Research Announcements of the American Mathematical Society KW - General Mathematics SN - 1079-6762 TI - A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II VL - 7 ER - TY - JOUR AB - A number α∈R is diophantine if it is not well approximable by rationals, i.e. for some C,ε>0 and any relatively prime p,q∈Z we have |αq−p|>Cq−1−ε. It is well-known and is easy to prove that almost every α in R is diophantine. In this paper we address a noncommutative version of the diophantine properties. Consider a pair A,B∈SO(3) and for each n∈Z+ take all possible words in A, A -1, B, and B - 1 of length n, i.e. for a multiindex I=(i1,i1,…,im,jm) define |I|=∑mk=1(|ik|+|jk|)=n and \( W_n(A,B ) = \{W_{\cal I}(A,B) = A^{i_1} B^{j_1} \dots A^{i_m} B^{j_m}\}_{|{\cal I|}=n \).¶Gamburd—Jakobson—Sarnak [GJS] raised the problem: prove that for Haar almost every pair A,B∈SO(3) the closest distance of words of length n to the identity, i.e. sA,B(n)=min|I|=n∥WI(A,B)−E∥, is bounded from below by an exponential function in n. This is the analog of the diophantine property for elements of SO(3). In this paper we prove that s A,B (n) is bounded from below by an exponential function in n 2. We also exhibit obstructions to a “simple” proof of the exponential estimate in n. AU - Kaloshin, Vadim AU - Rodnianski, I. ID - 8524 IS - 5 JF - Geometric And Functional Analysis SN - 1016-443X TI - Diophantine properties of elements of SO(3) VL - 11 ER - TY - JOUR AB - Epigenetic silenced alleles of the Arabidopsis SUPERMANlocus (the clark kent alleles) are associated with dense hypermethylation at noncanonical cytosines (CpXpG and asymmetric sites, where X = A, T, C, or G). A genetic screen for suppressors of a hypermethylated clark kent mutant identified nine loss-of-function alleles of CHROMOMETHYLASE3(CMT3), a novel cytosine methyltransferase homolog. These cmt3 mutants display a wild-type morphology but exhibit decreased CpXpG methylation of the SUP gene and of other sequences throughout the genome. They also show reactivated expression of endogenous retrotransposon sequences. These results show that a non-CpG DNA methyltransferase is responsible for maintaining epigenetic gene silencing. AU - Lindroth, A. M. AU - Cao, Xiaofeng AU - Jackson, James P. AU - Zilberman, Daniel AU - McCallum, Claire M. AU - Henikoff, Steven AU - Jacobsen, Steven E. ID - 9444 IS - 5524 JF - Science KW - Multidisciplinary SN - 0036-8075 TI - Requirement of CHROMOMETHYLASE3 for maintenance of CpXpG methylation VL - 292 ER -