The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

On Meta-Thin Association Schemes

An association scheme is a combinatorial object derived from the orbitals of a transitive permutation group. Let G be a transitive permutation group acting on a finite set X. Then _{x X}G_x is a normal subgroup of G where G_x:={g G x^g=x}. A meta-thin association scheme can be considered as a generalization of the situation where _{x X}G_x normalizes G_x. In this paper, we consider the automorphism group of a meta-thin association scheme, and obtain a sufficient condition for a meta-thin association scheme to have a transitive automorphism group. This enables us to conclude that every meta-thin association scheme with its thin residue isomorphic to the cyclic group of order pq, where p and q are primes, has a transitive automorphism group.     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

Almost everywhere convergence of the spherical partial sums for radial functions

LetfL ^p( ⁿ ),n2, be a radial function and letS _Rf be the spherical partial sums operator. We prove that if thenS _Rf(x)f(x) a.e. asR. The result is false for and \frac{{2n}}{{n + 1}}$$ " align="middle" border="0"> .Partially supported by M.P.I.     

Generalization of two theorems of M. I. Kadets concerning the indefinite integral of abstract almost periodic functions

Conditions are obtained for the almost periodicity (or almost automorphy) of an abstract functionf (t) on a group G satisfying the difference equationsf (t)–f(t)=g(t), where, for each G, the function (t) is almost periodic (or almost automorphic) (the difference problem). The investigation of the almost periodicity of the integral of an almost periodic function (t) on the real line R is reduced to a study of the difference problem.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 311–321, March, 1971.In conclusion I wish to thank V. V. Zhikov for suggesting this problem, and B. M. Levitan and E. A. Gorin for their discussion of the work.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Surfaces and the second homology of a group

LetG be a group andK(G, 1) an Eilenberg—MacLane space, i.e. ₁(K(G,1))G, _i (K(G,1))=0,i1. We give a purely algebraic proof that the second homology groupH ₂(G)=H ₂(G,)H ₂(K(G,1)) is isomorphic to the group of stable equivalence classes of continuous mapsFK(G,1) inducing surjections on fundamental groups (resp. surjections, whereF{F _g=closed orientable surface of genusg,g}. As a corollary we obtain an algebraic proof of the well-known isomorphismH ₂(G)₂(K(G,1)) (2-dimensional bordism group).     

Sugli insiemi di permutazioni strettamente k-transitivi (k ≥ 3)

Summary Let G be a sharply 3-transitive permutation set on a finite set E of even cardinality and let 1 be in G. The following theorems are proved. G is one of the known examples if and only if there exists a non-identity normal subgroup N of G and an element of E such that NG G.G is a group if and only if G for every G and for every G and for every G .By using the classification of finite single groups a result concerning sharply k-transitive permutation sets k>3 is also proved.

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Dedicato a Guido Zappa in occasione del suo 70° compleanno

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Lavoro eseguito nell'ambito dei progetti finanziati dal Ministero della Pubblica Istruzione.     

On Groups with Finite Involution and Locally Finite 2-Isolated Subgroup of Even Period

A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >C _G(g) H 1 and 2(C_G(g)) imply that C_G(g)H. An involution i in a group G is said to be finite if |ii ^g| < (for any g G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:1) all 2-elements of the group G belong to H;2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;3) G=FFC_G(i) is a locally finite Frobenius group with Abelian kernel F;4) H=V D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O₂(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set =U ^g g G.     

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

Asymptotic Behavior of Entire Functions with Exceptional Values in the Borel Relation

Let M _f(r) and _f (r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let l(r) be a continuously differentiable function convex with respect to ln r. We establish that, in order that ln M _f(r) ln _f (r), r +, for every entire function f such that _f (r) l(r), r +, it is necessary and sufficient that ln (rl(r)) = o(l(r)), r +.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ²ⁱ )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH ₀=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a _i (), 0) at 0 ^k , whereH (z)–z=a _i ()z ⁱ . Then one can find a parameter solutionS (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS (z)=z+b _i ()(z–z ₀) ⁱ with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 ² such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé