Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]).     

The Adjoint Action of an Expansive Algebraic -Action

 Let , and let α be an expansive -action by continuous automorphisms of a compact abelian group X with completely positive entropy. Then the group of homoclinic points of α is countable and dense in X, and the restriction of α to the α-invariant subgroup is a -action by automorphisms of . By duality, there exists a -action by automorphisms of the compact abelian group : this action is called the adjoint action of α. We prove that is again expansive and has completely positive entropy, and that α and are weakly algebraically equivalent, i.e. algebraic factors of each other. A -action α by automorphisms of a compact abelian group X is reflexive if the -action on the compact abelian group adjoint to is algebraically conjugate to α. We give an example of a non-reflexive expansive -action α with completely positive entropy, but prove that the third adjoint is always algebraically conjugate to . Furthermore, every expansive and ergodic -action α is reflexive. The last section contains a brief discussion of adjoints of certain expansive algebraic -actions with zero entropy. Received 11 June 2001; in revised form 29 November 2001     

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shiftV _β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV _β and certainα-invariant probability measures onX. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras. Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic.     

The cup product of Hilbert schemes for K3 surfaces

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A ^[n] so that there is canonical isomorphism of rings (H ^*(X;ℚ)[2]) ^[n] ≅H ^*(X ^[n] ;ℚ)[2n] for the Hilbert scheme X ^[n] of generalised n-tuples of any smooth projective surface X with numerically trivial canonical bundle. Oblatum 25-I-2001 & 18-IX-2002?Published online: 24 February 2003     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

On automorphisms of infinite graphs with forbidden subgraphs

LexX be anm-connected infinite graph without subgraphs homeomorphic toKm, n, for somen, and let α be an automorphism ofX with at least one cycle of infinite length. We characterize the structure of α and use this characterization to extend a known result about orientation-preserving automorphisms of finite plane graphs to infinite plane graphs. In the last section we investigate the action of α on the ends ofX and show that α fixes at most two ends (Theorem 3.2).     

Osservazioni su alcuni operatori ibridi iperbolico-ipoellittici

Riassunto Viene dimostrata l'equivalenza di alcune condizioni sul polinomio a coefficienti costantiP(ζ). La prima (si vedaH?rmander [3]), garantisce l'esistenza di una soluzione fondamentale diP(D) con supporto singolare in un cono convesso; l'ultima esprime il carattere ibrido diP(ζ). Viene infine costruita una soluzione fondamentale ultradistribuzione per un operatore 1/β-iperbolico-1/α-ipoellittico estendendo un precedente risultato [5] dell'autore.

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Summary The equivalence of some conditions on the polynomialP(ζ) with constant coefficients is proved. The first one (seeH?rmander [3]), implies the existence of a fundamental solution forP(D) with singular support contained in a convex cone; the last one shows thatP(ζ) is an hyperbolic-hypoelliptic polynomial. Finally an ultradistribution fundamental solution for a 1/β-hyperbolic-1/α-hypoelliptic operator is constructed, so extending a previous result [5] of the author.

     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Analytic Semigroups on C [0,1] Generated by Some Classes of Second Order Differential Operators

C [0,1], α > 0 in (0,1) and α(1), we consider the second order differential operator on C[0,1] defined by Au: = αu″ + βu′, where D(A) may include Wentzell boundary conditions. Under integrability conditions involving √α and β/√α, we prove the analyticity of the semigroup generated by (A,D(A)) on Co[0,1], Cπ[0,1] and on C[0,1], where Co[0,1]: {u∈ C[0,1]|u (1)} and Cπ[0,1]: = {u∈ C[0,1]| u (0) = u (1)}. We also prove different characterizations of D(A) related to some results in [1], where β≡ 0, exhibiting peculiarities of Wentzell boundary conditions. Applications can be derived for the case αx = x ^k (1 - x )^kγ(x )(k≥j/2, x∈ [0,1], γ∈ C{0,1}).     

Homology Groups of Conormal Varieties

Let G be a real Lie group acting on real analytic manifold with finitely many orbits. We prove that the characteristic cycle map is a surjective homomorphism from the K-group of G-equivariant sheaves on X to the top homology group of the conormal variety of the G-action on X. We also show that the top homology group of the G-action on X is a free -module of rank equal to the number of G-orbits. This work was completed with the support of the Ministry of Science, Education and Sport of Croatia, and Infodesign d.o.o., Varaždin.     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Markov Chains with Positive Transitions Are Not Determined by Any p-Marginals

 We prove that if is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number p, there exists a continuum of finite valued non Markovian processes which have the p-marginal distributions of X and with positive entropy, whereas for an irrational rotation and essentially bounded real measurable function f with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain. (Received 22 July 1999; in revised form 24 February 2000)     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

Extension of stationary stochastic processes

Summary LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX ^{[0, 1]}, equipped with the Borel -field of the product space. We prove the existence of a stationary extension of this process to ₀ ⁺ . Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX ^{[0, 1]} is equipped with the product of the Borel -fields.Corporate Research and Development, SIEMENS AG, D-81730 Munich, Germany