Markov Extensions Research Articles

Markov extensions and conditionally invariant measures for certain logistic maps with small holes

We study the family of quadratic maps fa(x) = 1 − ax 2 on the interval (−1,1) with 0 ≤ a ≤ 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a density which is bounded away from zero and is analogous to the density for the corresponding closed system. These results establish the exponential escape rate of Lebesgue measure from the system, despite the contraction in a neighborhood of the critical point of the map. We also prove convergence of the conditionally invariant measure to the SRB measure for fa as the size of the hole goes to zero.

Limiting negations in bounded-depth circuits: An extension of Markov's theorem

A theorem of Markov shows the necessary and sufficient number of negations for computing an arbitrary collection of Boolean functions. In this paper, we consider a class of bounded-depth circuits, in which each gate computes an arbitrary monotone Boolean function or its negation. We show tight bounds on the number of negations for computing an arbitrary collection of Boolean functions when such a class of circuits is under consideration.

On Certain Extensions of a Rotation Invariant Markov Process

Let (Xt,Px) be a rotation invariant (RI) Feller process on Rd∖{0}, d≥2. We study certain type of strong Markov extensions of (Xt,Px) to Rd. It turns out that that the unique (RI) extension always exists and is of that type. We give a kind of quasi-ergodicy condition (E) under which the (RI) extension is the only extension of that type. A class of processes fulfilling (E) is also characterized.

Frobenius Extensions and Weak Hopf Algebras

We study a symmetric Markov extension of k-algebras N↪M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N↪M↪M1↪M2 can be obtained by taking relative tensor products with centralizers A=CM1(N) and B=CM2(M). Under this condition, we prove that N↪M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N=MA. The endomorphism algebra M1=EndNM is shown to be isomorphic to the smash product algebra M#A. We also extend results of W. Szymański (1994, Proc. Amer. Math. Soc. 120, 519–528), D. Nikshych and L. Vainerman (2000, Funct. Anal.171, 278–307), and L. Kadison and D. Nikschych (Comm. Algebra, 29(9) 2001).

Walsh's Brownian Motion-Type of Extensions

Let (X t ) be a rotation invariant Feller process on the state space F ∈ R2{} consisting of finite number of rays, meeting at 0. We study a certain class of possible strong Markov extensions of (X t ) to F ∪ ,{} given the corresponding radial extension to [0, ∞). A well-known example is the class of “Walsh's Brownian motions,” in the case where (X t ) is the Brownian motion on F. It turns out that while the symmetric extension of “Walsh's Brownian motion-type” always exists, the non-symmetric extension exists iff (X t ), roughly speaking, does not jump from one ray to another before hitting 0.

Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

Markov extensions for multi-dimensional dynamical systems

By a result of F. Hofbauer [11], piecewise monotonic maps of the interval can be identified with topological Markov chains with respect to measures with large entropy. We generalize this to arbitrary piecewise invertible dynamical systems under the following assumption: the total entropy of the system should be greater than the topological entropy of the boundary of some reasonable partition separating almost all orbits. We get a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy. We illustrate our results by quoting an application to a class of multi-dimensional, non-linear, non-expansive smooth dynamical systems.

Multi-state models: a review.

Multi-state models are models for a process, for example describing a life history of an individual, which at any time occupies one of a few possible states. This can describe several possible events for a single individual, or the dependence between several individuals. The events are the transitions between the states. This class of models allows for an extremely flexible approach that can model almost any kind of longitudinal failure time data. This is particularly relevant for modeling different events, which have an event-related dependence, like occurrence of disease changing the risk of death. It can also model paired data. It is useful for recurrent events, but has limitations. The Markov models stand out as much simpler than other models from a probability point of view, and this simplifies the likelihood evaluation. However, in many cases, the Markov models do not fit satisfactorily, and happily, it is reasonably simple to study non-Markov models, in particular the Markov extension models. This also makes it possible to consider, whether the dependence is of short-term or long-term nature. Applications include the effect of heart transplantation on the mortality and the mortality among Danish twins.

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy.

Entropy, Reversible Diffusion Processes, and Markov Uniqueness

Consider a symmetric bilinear form Eϕdefined on C∞c(Rd) by[formula]In this paper we study the stochastic process associated with the smallest closed markovian extension of (Eϕ, C∞c), and give a new proof of Markov uniqueness (i.e. the uniqueness of a closed markovian extension) based on purely probabilistic arguments. We also give another purely analytic one. As a consequence, we show that all invariant measures are reversible, provided they are of finite energy. The problem of uniqueness of such measures is also partially solved.

Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains

In this work we introduce a new parameter,s≥1, in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) inequalities and show their validity (with an appropriates) for any compact subanalytic domain. The classical form of these SGN inequalities (s=1 in our formulation) fails for domains with outward pointing cusps. Our parameters measures the degree of cuspidality of the domain. For regular domainss=1. We also introduce an extension, depending on a parameter σ≥1, to several variables of a local form of the classical Markov inequality on the derivatives of a polynomial in terms of its own values, and show the equivalence of Markov and SGN inequalities with the same value of parameters, σ=s. Our extension of Markov's inequality admits, in the case of supremum norms, a geometric characterization. We also establish several other characterizations: the existence of a bounded (linear) extension ofC∞ functions with a homogeneous loss of differentiability, and the validity of a global Markov inequality. Our methods may broadly be classified as follows: 1. Desingularization and anLp-version of Glaeser-type estimates. In fact we obtain a bounds<-2d+1, whered is the maximal order of vanishing of the jacobian of the desingularization map of the domain. 2. Interpolation type inequalities for norms of functions and Bernstein-Markov type inequalities for multivariate polynomials (classical analysis). 3. Geometric criteria for the validity of local Markov inequalities (local analysis of the singularities of domains). 4. Multivariate Approximation Theory.

Induced maps, Markov extensions and invariant measures in one-dimensional dynamics

A way to study ergodic and measure theoretic aspects of interval maps is by means of the Markov extension. This tool, which ties interval maps to the theory of Markov chains, was introduced by Hofbauer and Keller. More generally known are induced maps, i.e. maps that, restricted to an element of an interval partition, coincide with an iterate of the original map.We will discuss the relation between the Markov extension and induced maps. The main idea is that an induced map of an interval map often appears as a first return map in the Markov extension. For S-unimodal maps, we derive a necessary condition for the existence of invariant probability measures which are absolutely continuous with respect to Lebesgue measure. Two corollaries are given.

Upper bounds for survival probability of the contact process

A precise description of the nontrivial upper invariant measure for λ>λc is still an open problem for the basic contact process, which is a self-dual, attractive, but nonreversible Markov process of an interacting particle system. By its self-duality, to identify the invariant measure is equivalent to determining the initial-state dependence of the survival probability of the process. A procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by Harris. Two new bounds are given, improving the simple branching-process bound. In the one-dimensional case, the present procedure can be viewed as a trial to make approximate measures by generalized Markov extensions.

Three-Point Markov Extension and an Improved Upper Bound for Survival Probability of the One-Dimensional Contact Process

The survival probability σ( A ) is studied for the one-dimensional contact process. An approximate function h (3) ( A ) for σ( A ) is obtained; a partial stationary condition is used to determine h (3) ( C i ) for small sets C i 's and then it is extended to all finite subsets A of Z 1 by the Markov extension. It is proved that h (3) ( A ) gives an upper bound for σ( A ) in the parameter region where the infection rate λ is greater than Griffeath's lower bound for λ c , i.e. \(\lambda>\frac{1}{6}\) (\(1+\sqrt{37}\)). The present result improves previous upper bounds.

Regular isomorphism of Markov chains is almost topological

AbstractTwo Markov chains are regularly isomorphic if and only if they have a common Markov extension by right closing block codes of degree one. A certain ideal class in the integral group ring of the ratio group associated to a Markov chain is a new invariant of regular isomorphism and some other coding relations.

Markov Extensions, Zeta Functions, and Fredholm Theory for Piecewise Invertible Dynamical Systems

Transfer operators and zeta functions of piecewise monotonic and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.

Lifting measures to Markov extensions

Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove: (1) IfT is anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set. (2) IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ≪m iffhμ (T)=Σlogf dμ.

Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems

Transfer operators and zeta functions of piecewise monotonic and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.

Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant

If p p is a polynomial of degree at most n n such that | p ( x ) | ⩽ 1 − x 2 |p(x)| \leqslant \sqrt {1 - {x^2}} for − 1 ⩽ x ⩽ 1 - 1 \leqslant x \leqslant 1 , then for each k k , max | p ( k ) ( x ) | \max |{p^{(k)}}(x)| on [ − 1 , 1 ] [ - 1,\,1] is maximized by the polynomial ( x 2 − 1 ) U n − 2 ( x ) ({x^2} - 1){U_{n - 2}}(x) where U m {U_m} is the m m th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume | p ( x ) | ⩽ 1 − x 2 |p(x)| \leqslant \sqrt {1 - {x^2}} at some appropriately chosen set of n + 1 n + 1 points in [ − 1 , 1 ] [ - 1,\,1] . The problem is inspired by a well-known extension of Markov’s inequality due to Duffin and Schaeffer.

On sofic systems II

The results ofOn sofic systems I on topological Markov chains extending sofic systems are completed. To homomorphisms of sofic systems are canonically associated homomorphisms of Markov extensions. Also considered is a class of finitary codes for sofic systems.