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Given discrete groups $Gamma subset Delta$ we characterize $(Gamma,sigma)$-invariant spaces that are also invariant under $Delta$. This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal $(Gamma,sigma)$-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the analogous in the context of shift-invariant spaces in terms of the Fourier transform, which is very well-known. As a consequence of our results, we give a solution for the problem of finding the $(Gamma,sigma)$-invariant space nearest - in the sense of least squares - to a given set of data.
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Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components. We prove that the map $pi_alpha(P)$ is Fredholm if, and only if, $P$ is {em transversally $alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
Let $A$ be a normal operator in a Hilbert space $mathcal{H}$, and let $mathcal{G} subset mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $mathcal{G}$ , and $L$ that makes the system of iterations ${A^ng: gin mathcal{G},;0leq n< L(g)}$ complete, Bessel, a basis, or a frame for $mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.
In this paper we study chaotic behavior of actions of a countable discrete group acting on a compact metric space by self-homeomorphisms. For actions of a countable discrete group G, we introduce local weak mixing and Li-Yorke chaos; and prove that local weak mixing implies Li-Yorke chaos if G is infinite, and positive topological entropy implies local weak mixing if G is an infinite countable discrete amenable group. Moreover, when considering a shift of finite type for actions of an infinite countable amenable group G, if the action has positive topological entropy then its homoclinic equivalence relation is non-trivial, and the converse holds true if additionally G is residually finite and the action contains a dense set of periodic points.
Let $H_3(Bbb R)$ denote the 3-dimensional real Heisenberg group. Given a family of lattices $Gamma_1supsetGamma_2supsetcdots$ in it, let $T$ stand for the associated uniquely ergodic $H_3(Bbb R)$-{it odometer}, i.e. the inverse limit of the $H_3(Bbb R)$-actions by rotations on the homogeneous spaces $H_3(Bbb R)/Gamma_j$, $jinBbb N$. The decomposition of the underlying Koopman unitary representation of $H_3(Bbb R)$ into a countable direct sum of irreducible components is explicitly described. The ergodic 2-fold self-joinings of $T$ are found. It is shown that in general, the $H_3(Bbb R)$-odometers are neither isospectral nor spectrally determined.
We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme $f : X rightarrow S$ endowed with an action of the additive group scheme $mathbb{G}_{ a,S}$ over a base scheme or algebraic space $S$, which we call the (relative) Rees algebra of the $mathbb{G}_{ a,S}$-action. We illustrate these properties on several examples which played important roles in the development of the algebraic theory of locally nilpotent derivations and give some applications to the construction of families of affine threefolds with Ga-actions.
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