Measure Of Maximal Entropy Research Articles

Regularity of dynamical Green’s functions

For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green’s functions give rise to invariant currents which intersect to yield measures of maximal entropy. ‘Nice enough’ is often a condition on the regularity of the Green’s function. In this paper we look at a variety of regularity properties that have been considered for dynamical Green’s functions. We simplify and extend some known results and prove several others which are new. We also give some examples indicating the limits of what one can hope to achieve in complex dynamics by relying solely on the regularity of a dynamical Green’s function.

A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

AbstractWe consider the mapTα,β(x):=βx+α mod 1, which admits a unique probability measureμα,βof maximal entropy. Forx∈[0,1], we show that the orbit ofxisμα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit ofx=0 isμα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalizedβ-transformations (in particular, the tent map). We show that the critical orbitx=1 is normal with respect to the measure of maximal entropy for almost allβ.

Bernoulli coding map and almost sure invariance principle for endomorphisms of $${\mathbb P^k}$$

Let f be an holomorphic endomorphism of $${\mathbb{P}^k}$$ and μ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems $${(\mathbb{P}^k,f,\mu)}$$ . Our class $${\mathcal {U}}$$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $${\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)}$$ . We obtain the invariance principle for an observable ψ on $${(\mathbb{P}^k,f,\mu)}$$ by applying Philipp–Stout’s theorem for $${\chi = \psi \circ \omega}$$ on (Σ, s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class $${\mathcal {U}}$$ . As an application, we give a direct proof of the absolute continuity of the measure μ when it satisfies Pesin’s formula. This approach relies on the central limit theorem for the unbounded observable log $${{\tt Jac}\, f \in \mathcal{U}}$$ .

Requirements for total uncertainty measures in Dempster–Shafer theory of evidence

Recently, an alternative measure of total uncertainty in Dempster–Shafer theory of evidence (DST) has been proposed in place of the maximum entropy measure. It is based on the pignistic probability of a basic probability assignment and it is proved that this measure verifies a set of needed properties for such a type of measure. The proposed measure is motivated by the problems that maximum (upper) entropy has. In this paper, we analyse the requirements, presented in the literature, for total uncertainty measures in DST and the shortcomings found on them. We extend the set of requirements, which we consider as a set of requirements of properties, and we use the set of shortcomings found on them to define a set of requirements of the behaviour for total uncertainty measures in DST. We present the differences of the principal total uncertainty measures presented in DST taking into account their properties and behaviour. Also, an experimental comparative study of the performance of total uncertainty measures in DST on a special type of belief decision trees is presented.

Basic dynamical stability results for

AbstractWe study stability questions for dynamics on $\mathbb {P}^n$ using holomorphic families, which we require to be locally simply parameterized. Given such a holomorphic family $f\colon \mathbb {P}^n\times M\to \mathbb {P}^n$, we define the notion of a ‘Fatou section’, which is intuitively a choice of one point in $\mathbb {P}^n$ for each map in the family such that: (1) the chosen point depends holomorphically on M, and (2) the points chosen for two different maps behave dynamically ‘comparably’ under iteration of those respective maps. We prove a weak version of the λ-lemma for dynamical systems on $\mathbb {P}^n$ by showing that the set of all Fatou sections is a compact space. We introduce the notion of a postcritically bounded holomorphic family and show that if a (locally simply parameterized) holomorphic family is postcritically bounded by some open subset of $\mathbb {P}^n$ then: (1) repelling periodic points in the Julia set for one member of the family cannot bifurcate, nor can they leave the Julia set in other members of the family, (2) there is a Fatou section which never leaves the Julia set through each point of the Julia set of any member of the family, and (3) any intersection between the Julia set and the critical set is persistent, in the sense that there is a Fatou section through the point of intersection which never leaves either the critical set or the Julia set. Here, for conciseness, we are using the term ‘Julia set’ to mean the support of the unique measure of maximal entropy, although we recognize that how this term should be used in higher dimensions is still undecided. We also provide a brief summary of the development of the field in our introduction.

Fractal analysis for sets of non-differentiability of Minkowski's question mark function

In this paper we study various fractal geometric aspects of the Minkowski question mark function Q. We show that the unit interval can be written as the union of the three sets Λ 0 : = { x : Q ′ ( x ) = 0 } , Λ ∞ : = { x : Q ′ ( x ) = ∞ } , and Λ ∼ : = { x : Q ′ ( x ) does not exist and Q ′ ( x ) ≠ ∞ } . The main result is that the Hausdorff dimensions of these sets are related in the following way: dim H ( ν F ) < dim H ( Λ ∼ ) = dim H ( Λ ∞ ) = dim H ( L ( h top ) ) < dim H ( Λ 0 ) = 1 . Here, L ( h top ) refers to the level set of the Stern–Brocot multifractal decomposition at the topological entropy h top = log 2 of the Farey map F, and dim H ( ν F ) denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with F. The proofs rely partially on the multifractal formalism for Stern–Brocot intervals and give non-trivial applications of this formalism.

Equilibrium measures for maps with inducing schemes

We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions \phi on I which admit unique equilibrium measures \mu_\phi minimizing the free energy for a certain class of measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the Central Limit Theorem. Our results apply in particular to some one-dimensional unimodal and multimodal maps as well as to multidimensional nonuniformly hyperbolic maps admitting Young's tower. Examples of potential functions to which our theory applies include \phi_t=-t\log|df| with t\in(t_0, t_1) for some t_0<1<t_1. In the particular case of S-unimodal maps we show that one can choose t_0<0 and that the class of measures under consideration comprises all invariant Borel probability measures. Thus our results establish existence and uniqueness of both the measure of maximal entropy (by a different method than Hofbauer) and the absolutely continuous invariant measure extending results by Bruin and Keller for the parameters under consideration.

Generic points in systems of specification and Banach valued Birkhoff ergodic average

We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.

REMARKS ON LARGE DEVIATION FOR RATIONAL MAPS ON THE RIEMANN SPHERE

We consider rational maps of degree d ≥ 2 on the Riemann sphere and obtain large deviation results for Hölder observables under the measure of maximal entropy.

Families of ergodic and exact one-dimensional maps

We give a parametrized family of rational interval maps of degree two, each ergodic, exact and preserving a measure equivalent to a Lebesgue measure. The family includes the unique quadratic Chebyshev polynomial as its only polynomial map. We extend the family to other settings on the circle and real line. We also give numerical approximations to the entropy of the equivalent invariant measure and the Hausdorff dimension of the singular measure of maximal entropy.

On thermodynamics of rational maps on the Riemann sphere

We show that there is a spectral gap for the transfer operator associated with a rational map $f$ on the Riemann sphere. Using this and the method of perturbed operators, we establish the (local) central limit theorem for the measure of maximal entropy of $f$, with an estimate on the speed of convergence.

Continuity of Lyapunov exponents for polynomial automorphisms of $\mathbb{C}^2$

We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of . The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a one-dimensional map.

Visible measures of maximal entropy in dimension one

A real one-dimensional analogue of Zdunik’s dichotomy is proven giving dynamical conditions for a multimodal map to have a measure of maximal entropy of dimension one.

Invariant densities for random $\beta$-expansions

Let \beta &gt;1 be a non-integer. We consider expansions of the form \sum_{i=1}^{\infty} \frac{d_i}{\beta^i} , where the digits (d_i)_{i \geq 1} are generated by means of a Borel map K_{\beta} defined on \{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right] . We show existence and uniqueness of an absolutely continuous K_{\beta} -invariant probability measure w.r.t. m_p \otimes \lambda , where m_p is the Bernoulli measure on \{0,1\}^{\N} with parameter p ( 0 &lt; p &lt; 1) and \lambda is the normalized Lebesgue measure on [0 ,\lfloor \beta \rfloor /(\beta -1)] . Furthermore, this measure is of the form m_p \otimes \mu_{\beta,p} , where \mu_{\beta,p} is equivalent with \lambda . We establish the fact that the measure of maximal entropy and m_p \otimes \lambda are mutually singular. In case 1 has a finite greedy expansion with positive coefficients, the measure m_p \otimes \mu_{\beta,p} is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].

Lyapunov spectra for KMS states on Cuntz-Krieger algebras

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.

Dynamics of a two parameter family of plane birational maps: Maximal entropy

We mix combinatorial with complex methods to study the dynamics of a real two parameter family of plane birational maps. Specifically, we consider the action of the maps on the Picard group of an appropriate compactification of the complex plane, on the homology groups of a forward invariant real subset of this compactification, and on a Markov partition of the real plane determined by the critical set. For the range of parameters considered, the three actions are equivalent. This allows us to construct a measure of maximal entropy on the real nonwandering set, and it allows us to show that all wandering points are attracted to infinity in a well-defined fashion.

Uniqueness of maximal entropy measure on essential spanning forests

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for which ⋃Gn=G. Pemantle’s arguments imply that the uniform measures on spanning trees of Gn converge weakly to an Aut (G)-invariant measure μG on essential spanning forests of G. We show that if G is a connected, amenable graph and Γ⊂Aut (G) acts quasitransitively on G, then μG is the unique Γ-invariant measure on essential spanning forests of G for which the specific entropy is maximal. This result originated with Burton and Pemantle, who gave a short but incorrect proof in the case Γ≅ℤd. Lyons discovered the error and asked about the more general statement that we prove.

A subjective assessment of approximate probabilities with a portfolio application

This paper extends the assessment of approximate probabilities in two important directions. The first is to investigate some mathematical relations between the probability ranges and derives the most unbiased probability for the case when the limits are subjectively defined. The second is to suggest a simple method to determine the optimal solution which represents the optimal portfolio proportions of securities that possess the minimum risk measured by the maximum entropy measure. The paper considers the derivation of portfolio modeling under a fuzzy situation using probability theory, and provides various other (non-probabilistic) scenarios with their utility in risk modeling. A simple method for identification of mean-entropic frontier is proposed. Then, a comparison of mean-variance procedure with the discrete mean-entropic method is implemented by an example.

The Julia Set of Hénon Maps

In this paper we investigate the support of the unique measure of maximal entropy of complex Henon maps, J ∗ . The main question is whether this set is the same as the analogue of the Julia set J.

Image Enhancement by Using DirectionalWavelet Transform

The purpose of this paper is to introduce a novel image enhancement technique by using directional wavelet transform. Directional wavelet transform decomposes an image into four-dimensional space which augments the image by the scale and directional information. We show that the directional information significantly improves image enhancement in noisy images in comparison with the classical techniques. Image enhancement is based on the multiscale singularity detection with an adaptive threshold whose value is calculated via maximum entropy measure. The proposed technique was tested on synthetic images at different signal-to-noise ratios and clinical images.