Measure-theoretic Point Research Articles

Variational principles for pointwise preimage entropies

Based on the preimage structure of the system , Hurley introduced the notion of pointwise topological preimage entropies and . Furthermore, from the measure-theoretic point of view, Wu and Zhu introduced a notion of pointwise metric preimage entropy for a T-invariant measure µ on X, and obtained the variational principle between and under the condition of uniform separation of preimages. A natural question is whether a variational principle for and without any additional assumptions. In this paper, we define a new version of topological preimage entropy relative to a T-invariant measure µ, and show that the inequality holds for every T-invariant probability measure µ. As a consequence, we obtain that there is a topological dynamical system such that the following strict inequality holds: where denote the set of all T-invariant probability measures.

Recovering the Homology of Immersed Manifolds

Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold. It consists in estimating its tangent bundle -- seen as subset of another Euclidean space -- in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology. The construction we propose is consistent and stable, and does not involve the knowledge of the dimension of the manifold. In order to obtain quantitative results, we introduce the normal reach, which is a notion of reach suitable for an immersed manifold.

Entropy via preimage structure

To measure the complexity of a “non-invertible” continuous map f on a compact metric space via the preimage structure, Hurley introduced the notion of pointwise topological preimage entropies hm(f) and hp(f) in 1995. A natural question was: Can one introduce the counterpart of hm(f) or hp(f) from the measure-theoretic point of view for an f-invariant measure μ and obtain properties analogous to that for the classical entropies? Cheng and Newhouse made the first step on this topic and introduced the notions of preimage entropies hpre(f) and hpre,μ(f) and investigated their properties in 2005. Recently, Wu and Zhu proposed a notion of pointwise metric preimage entropy hm,μ(f) and obtained some properties for f with uniform separation of preimages.In this paper, we give a complete answer to the above question for general continuous maps. The Shannon-McMillan-Breiman theorem for hm,μ(f) and the variational principle relating hm,μ(f) and hm(f) are obtained. It is shown that hm(f)=hpre(f) and the three types of measure-theoretic entropy-like invariants hpre,μ(f), hm,μ(f) and the folding entropy Fμ(f) (introduced by Ruelle) via the preimage structure coincide with each other. We also show that the preimage entropy variational principle can be realized as a variant of the conditional variational principle. These results answer several open questions asked by Cheng-Newhouse and Downarowicz. The upper semi-continuity of hm,μ(f) with respect to invariant measures with the same separation rate is obtained. Moreover, the preimage pressure and preimage equilibrium states are investigated.

"Statistics 103" for Multitarget Tracking.

The finite-set statistics (FISST) foundational approach to multitarget tracking and information fusion was introduced in the mid-1990s and extended in 2001. FISST was devised to be as “engineering-friendly” as possible by avoiding avoidable mathematical abstraction and complexity—and, especially, by avoiding measure theory and measure-theoretic point process (p.p.) theory. Recently, however, an allegedly more general theoretical foundation for multitarget tracking has been proposed. In it, the constituent components of FISST have been systematically replaced by mathematically more complicated concepts—and, especially, by the very measure theory and measure-theoretic p.p.’s that FISST eschews. It is shown that this proposed alternative is actually a mathematical paraphrase of part of FISST that does not correctly address the technical idiosyncrasies of the multitarget tracking application.

A Subnormal Completion Problem for Weighted Shifts on Directed Trees

Given a directed tree and a collection of weights on a subtree, the subnormal completion problem is to determine whether the weights may be completed to the weights of an injective, bounded, subnormal weighted shift on the Hilbert space arising from the full tree. We study this problem (which generalizes significantly the classical subnormal completion problem for weighted shifts) both from a measure-theoretic point of view and in terms of initial data, for various classes of trees with a single branching point. We give several characterizations of when such a completion is possible. Considered also are connections with Stieltjes moment sequences, flatness of a completion, completions in which the resulting measures may be taken to be finitely atomic, and we provide a result showing that in certain circumstances the present completion problem is equivalent to a related classical completion problem.

Is direct measurement of time possible?

Is direct measurement of time possible? The answer to this question may depend upon how one understands time. Is time an essential constituent of physical reality? Or is what scientists are talking about when they use the symbol ‘t’ or the word ‘time’ an human cultural construct, as the Chief of the USA NIST Divisions of Time and Frequency and of Quantum Physics has suggested. Few aspects of physics do not reference activity to time, but many discussions within either view of time seem to use one same, largely traditional, language of time. Briefly considering the question of measurement, including from a formal measure-theoretic point of view, clarifies the situation.

Spark-level sparsity and the ℓ1 tail minimization

Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A , and then the unique sparsest solution exists, and is recoverable by ℓ 1 -minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements A x = b . Specifically, suppose A is of full spark with m rows, and suppose m 2 < s < m . Then the solution to A x = b is unique for x with ‖ x ‖ 0 ≤ s up to a set of measure 0 in every s -sparse plane. This phenomenon is observed and confirmed by an ℓ 1 -tail minimization procedure, which recovers sparse signals uniquely with s > m 2 in thousands and thousands of random tests. We further show instead that the mere ℓ 1 -minimization would actually fail if s > m 2 even from the same measure theoretical point of view.

On small sets from the measure-theoretical point of view

For nonzero invariant (quasi-invariant) σ-finite measures on an uncountable group (G,⋅), the behaviour of small sets with respect to the group operation in G is studied.

Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier–Stokes Equations

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray–Hopf weak solution of the three-dimensional incompressible Navier–Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also shown that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray–Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.

Most linear flows on [formula omitted] are Benford

A necessary and sufficient condition (“exponential nonresonance”) is established for every signal obtained from a linear flow on Rd by means of a linear observable to either vanish identically or else exhibit a strong form of Benford's Law (logarithmic distribution of significant digits). The result extends and unifies all previously known (sufficient) conditions. Exponential nonresonance is shown to be typical for linear flows, both from a topological and a measure-theoretical point of view.

Generalized conformal densities for higher products of rank one Hadamard spaces

Let $$X$$ be a product of locally compact rank one Hadamard spaces and $$\Gamma $$ a discrete group of isometries which contains two elements projecting to a pair of independent rank one isometries in each factor. In Link (Asymptotic geometry in higher products of rank one Hadamard spaces. arXiv:1308.5584 , 2013) we gave a precise description of the structure of the geometric limit set $$L_\Gamma $$ of $$\Gamma $$ ; our aim in this paper is to describe this set from a measure theoretical point of view, using as a basic tool the properties of the exponent of growth of $$\Gamma $$ established in the aforementioned article. We first show that the conformal density obtained from the classical Patterson–Sullivan construction is supported in a unique $$\Gamma $$ -invariant subset of the geometric limit set; generalizing this classical construction we then obtain measures supported in each $$\Gamma $$ -invariant subset of the regular limit set and investigate their properties. We remark that apart from Kac–Moody groups over finite fields acting on the Davis complex of their associated twin building, the probably most interesting examples to which our results apply are isometry groups of reducible CAT(0)-cube complexes without Euclidean factors.

On growth of random groups of intermediate growth

We study the growth of typical groups from the family of p -groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by e^{n^\alpha} for some \alpha &lt; 1.

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T; that there exist frequently hypercyclic operators on the sequence space c0 admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.

Convergence of a typical martingale (A remark on the Doob theorem)

We study convergence behavior of discrete martingales with values in the interval [0,1] from a measure theoretical point of view as well as from a topological one. We show that almost all martingales converge to 0 or 1 almost everywhere. On the other hand, a typical martingale diverges on a comeager set.

Reciprocal processes. A measure-theoretical point of view

The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process. The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes. Abstract results are illustrated by several examples and counter-examples.

On spectral methods for variance based sensitivity analysis

Consider a mathematical model with a finite number of random parameters. Variance based sensitivity analysis provides a framework to characterize the contribution of the individual parameters to the total variance of the model response. We consider the spectral methods for variance based sensitivity analysis which utilize representations of square integrable random variables in a generalized polynomial chaos basis. Taking a measure theoretic point of view, we provide a rigorous and at the same time intuitive perspective on the spectral methods for variance based sensitivity analysis. Moreover, we discuss approximation errors incurred by fixing inessential random parameters, when approximating functions with generalized polynomial chaos expansions.

ON DYNAMICS OF ξS QUADRATIC STOCHASTIC OPERATORS

In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2D-simplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.

Physical measures for infinite-modal maps

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover we show that both the densities of these measures and their entropy vary continuously with the parameter. In addition we obtain exponential rate of mixing for these measures and also that they satisfy the Central Limit Theorem.

An Averaging Theorem for Hamiltonian Dynamical Systems in the Thermodynamic Limit

It is shown how to perform some steps of perturbation theory if one assumes a measure-theoretic point of view, i.e. if one renounces to control the evolution of the single trajectories, and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase–space. For a system of coupled rotators, estimates uniform in N for finite specific energy can be obtained in quite a direct way. This is achieved by making reference not to the sup norm, but rather, following Koopman and von Neumann, to the much weaker L 2 norm.

Statistical stability of saddle-node arcs

We study the dynamics of generic unfoldings of saddle-node circle local diffeomorphisms from the measure theoretical point of view, obtaining statistical and stochastic stability results for deterministic and random perturbations in this kind of one-parameter families.