Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

We consider two numerical entropy-type invariants for actions of \({\mathbb{Z}^k}\) , invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. \({\mathbb{Z}^k,\,k \geq 2}\) actions on k + 1-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of \({{\mathbb Z}}\) actions the entropies of ergodic maximal rank actions take only countably many values. Our main result is the dichotomy that is best expressed under the assumption of weak mixing or, equivalently, no periodic factors: either both invariants vanish, or their values are bounded away from zero by universal constants. Furthermore, the lower bounds grow with dimension: for the first invariant (the Fried average entropy) exponentially, and for the second (the slow entropy) linearly.     

Heegner points,cycles and Maass forms

We derive for Hecke-Maass cusp forms on the full modular group a relation between the sum of the form at Heegner points (and integrals over Heegner cycles) and the product of two Fourier coefficients of a corresponding form of half-integral weight. Specializing to certain cycles we obtain the nonnegativity of theL-function of such a form at the center of the critical strip. These results generalize similar formulae known for holomorphic forms. Partially supported by NSF grant # DMS-9096262. Partially supported by NSF grant # DMS-9102082.     

Dictator game giving: Rules of fairness versus acts of kindness

In both dictator and impunity games, one player, the dictator, divides a fixed amount of money between himself and one other, the recipient. Recent lab studies of these games have produced seemingly inconsistent results, reporting substantially divergent amounts of dictator giving. Also, one prominent explanation for some of these differences, the impact of experimenter observation, displayed weak explanatory power in a different but related lab game. Data from the new experiment reported here offers some explanations. We find that dictators determine how much they will give on the basis of the total money available for the entire experimental session, not on the basis of what is available per game. This explains the reported differences between impunity and dictator studies. When distributing a gift among several recipients, individual dictators show little tendency towards equal treatment. Also, we find no evidence for the experimenter observation effect. Comparison with earlier experiments suggests that differences in the context of the game, affected by differences in written directions and independent of experimenter observation, account for differences across dictator studies. We propose a hypothetical decision procedure, based on the notion that dictator giving originates with personal and social rules that effectively constrain self-interested behavior. The procedure provides a link between dictator behavior and a broader class of laboratory phenomena. Received August 1993/Final version April 1994     

Global rigidity results for lattice actions on tori and new examples of volume-preserving actions

Any action of a finite index subgroup in SL(n,ℤ),n≥4 on then-dimensional torus which has a finite orbit and contains an Anosov element which splits as a direct product is smoothly conjugate to an affine action. We also construct first examples of real-analytic volume-preserving actions of SL(n,ℤ) and other higher-rank lattices on compact manifolds which are not conjugate (even topologically) to algebraic models. This work was partially supported by NSF grant DMS9017995.     

Differentiability and analyticity of topological entropy for Anosov and geodesic flows

Summary In this paper we investigate the regularity of the topological entropyh _top forC ^k perturbations of Anosov flows. We show that the topological entropy varies (almost) as smoothly as the perturbation. The results in this paper, along with several related results, have been announced in [KKPW].Partially supported by NSF Grant DMS85-14630     

Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

Topological transitivity of billiards in polygons

Consider a billiard in a polygon QR² having all angles commensurate with . For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also proved is the typicality of polygons for which some billiard trajectory is dense in phase space.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 291–300, August, 1975.     

2-Isoxazolines with an electron-acceptor substituent at C<Subscript>(5)</Subscript> in reactions with nucleophilic reagents

Depending on the substituent at position 5 of the heterocycle, reaction of 2-isoxazolines with K-selectride, molybdenum hexacarbonyl, dimsyl sodium, and butyl lithium gave 1,3-cyclo-decomposition, aromatization of the isoxazoline ring, or reduction of the functional groups in the substituted isoxazolines. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No.3, 447–455, March, 2007.