We study the Perron-Frobenius operator P of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue ρ of P guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than − log ρ. The open systems with slow escape rates are easily iden...

We construct a universal tangle invariant on a quantum algebra. We show that the invariant maps tangle to commutants of the algebra; every (1, 1)-tangle is mapped to a Casimir operator of the algebra; the eigenvalue of the Casimir operator in an irreducible representation of the algebra is a link polynomial for the closure of the tangle. This result is applied to a discussion of the Alexander–C...

In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. Our construction...

The absolute separability problem asks for a characterization of the quantum states ρ ∈ Mm ⊗Mn with the property that UρU† is separable for all unitary matrices U. We provide evidence that ρ is absolutely separable if and only if UρU† has positive partial transpose for all unitary matrices U. In particular, we show that many well-known separability criteria are unable to detect entanglement in ...

We establish a nonlinear lower bound for halfplane range searching over a group. Specifically, we show that summing up the weights of n (weighted) points within n halfplanes requires Ω(n logn) additions and subtractions. This is the first nontrivial lower bound for range searching over a group. By contrast, range searching over a semigroup (which forbids subtractions) is almost completely under...

In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. Our construction...

In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. Our construction...

In this paper we study one-parameter families of signed transfer operators P± q associated to the Farey map, as well as two-parameter families of operators for the Gauss map, obtained from the Farey map by inducing. We characterise all the analytic eigenfunctions of P± q with eigenvalue λ not embedded in the continuous spectrum, which for λ = 1 are the period functions for Maass forms studied b...

We consider a Gaussian rotationally invariant ensemble of random real totally symmetric tensors with independent normally distributed entries, and estimate the largest eigenvalue typical tensor in this by examining rate growth initial vector under successive applications nonlinear map defined tensor. In limit large number dimensions, we observe that simple form melonic dominance holds, quantity...

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges i...