We study spectral properties of transfer operators for diffeomorphisms $T:X\to X$ on a Riemannian manifold $X$. Suppose that $\Omega$ is an isolated hyperbolic subset for $T$, with a compact isolating neighborhood $V\subset X$. We first introduce Banach spaces of distributions supported on $V$, which are anisotropic versions of the usual space of $C^p$ functions $C^p\left(V\right)$ and of the generalized Sobolev spaces $W^{p,t}\left(V\right)$, respectively. We then show that the transfer operators associated to $T$ and a smooth weight $g$ extend boundedly to these...

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\<p\<\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\<q\<\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups...

For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related $L^p$ boundedness properties of these operators and show in particular that, although they are not bounded on $L^p$ in general, they are always bounded on suitable weighted $L^p$ spaces.

Let $A,B$ denote generic binary forms, and let ${𝔲}_r={(A,B)}_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\left\{{𝔲}_r\right\}$. As a consequence, we show that each of the higher transvectants $\{{𝔲}_r:r\ge 2\}$ is redundant in the sense that it can be completely recovered from ${𝔲}_0$ and ${𝔲}_1$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S{L}_2$-representations,...

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...

The main object of this work is to describe such weight functions w(t) that for all elements $f\in L_{p,\Omega }$ the estimate $\parallel wf{\parallel }_p\ge K\left(\Omega \right)\parallel f{\parallel }_p$ is valid with a constant K(Ω), which does not depend on f and it grows to infinity when the domain Ω shrinks, i.e. deforms into a lower dimensional convex set ${\Omega }_{\infty }$. In one-dimensional case means that $K\left(\sigma \right):=K\left({\Omega }_{\sigma }\right)\to \infty$ as σ → 0. It should be noted that in the framework of the signal transmission problem such estimates describe a signal’s behavior under the influence of detection and amplification....