On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials

ABSTRACT

We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behaviour and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described as a graph of the solution of the Bellman equation.     

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.     

Mutation-periodic quivers, integrable maps and associated Poisson algebras

We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville's equation and the map plays the role of a B?cklund transformation.     

The organization of biological sequences into constrained and unconstrained parts determines fundamental properties of genotype–phenotype maps

Biological information is stored in DNA, RNA and protein sequences, which can be understood as genotypes that are translated into phenotypes. The properties of genotype–phenotype (GP) maps have been studied in great detail for RNA secondary structure. These include a highly biased distribution of genotypes per phenotype, negative correlation of genotypic robustness and evolvability, positive correlation of phenotypic robustness and evolvability, shape-space covering, and a roughly logarithmic scaling of phenotypic robustness with phenotypic frequency. More recently similar properties have been discovered in other GP maps, suggesting that they may be fundamental to biological GP maps, in general, rather than specific to the RNA secondary structure map. Here we propose that the above properties arise from the fundamental organization of biological information into ‘constrained'' and ‘unconstrained'' sequences, in the broadest possible sense. As ‘constrained'' we describe sequences that affect the phenotype more immediately, and are therefore more sensitive to mutations, such as, e.g. protein-coding DNA or the stems in RNA secondary structure. ‘Unconstrained'' sequences, on the other hand, can mutate more freely without affecting the phenotype, such as, e.g. intronic or intergenic DNA or the loops in RNA secondary structure. To test our hypothesis we consider a highly simplified GP map that has genotypes with ‘coding'' and ‘non-coding'' parts. We term this the Fibonacci GP map, as it is equivalent to the Fibonacci code in information theory. Despite its simplicity the Fibonacci GP map exhibits all the above properties of much more complex and biologically realistic GP maps. These properties are therefore likely to be fundamental to many biological GP maps.     

Piecewise linear interval maps both expanding and contracting

The dynamics of piecewise linear interval maps is studied with two branches, one expanding and one contracting. It is proved that such a map either has a periodic attractor or it is eventually expanding. In the latter case there exists an absolutely continuous invariant measure.     

Renormalization in a class of interval translation maps of d branches

Bruin and Troubetzkoy's 2003 results are generalized to a class of interval translation maps with arbitrarily many pieces. It is shown that there is an uncountable set of parameters leading to type ∞ interval translation maps (ITMs), but that the Lebesgue measure of these parameters is 0. Furthermore, conditions are given that imply that the ITMs have multiple ergodic invariant measures.     

Dense distribution of chaotic maps in continuous map spaces

This article is concerned with distribution of several kinds of chaotic maps in a continuous map space, in which the maps are defined in a closed bounded set of a Banach space. It is shown that the map space contains a dense set of maps that are strictly coupled-expanding, have nondegenerate and regular snap-back repellers, have nondegenerate and regular homoclinic orbits to repellers, and consequently that are chaotic in the sense of Devaney as well as in the original sense of Li–Yorke, and have the topological entropy larger than any given positive constant. Further, in the finite-dimensional case, there exists a dense residual set of the map space such that every map?f in the set is strictly coupled-expanding in k pairwise disjoint compact sets for any given integer k?≥?2, is chaotic in the sense of Li–Yorke and has the infinite topological entropy and a nontrivial invariant measure.     

An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator

The Lyapunov exponents of the linearization x = - x + 2betax + rho zeta t x of a noisy Duffing-van der Pol oscillator are key quantities in the investigation of the stochastic Hopf bifurcation of this system. Considering the white noise case we derive a simple equation exhibiting them explicitly as functions of the fourth moment of the invariant measure of an associated diffusion with drift given by a potential function and additive noise, and, consequently, in terms of hypergeometric functions. This representation leads to diff erent kinds of complete and explicit asymptotic expansions, as well as a rather complete account of global properties of the Lyapunov exponents as functions of beta and rho.     

A moment invariant for evaluating the chirality of three-dimensional objects

Chirality is an important feature of three-dimensional objects and a key concept in chemistry, biology and many other disciplines. However, it has been difficult to quantify, largely owing to computational complications. Here we present a general chirality measure, called the chiral invariant (CI), which is applicable to any three-dimensional object containing a large amount of data. The CI distinguishes the hand of the object and quantifies the degree of its handedness. It is invariant to the translation, rotation and scale of the object, and tolerant to a modest amount of noise in the experimental data. The invariant is expressed in terms of moments and can be computed in almost no time. Because of its universality and computational efficiency, the CI is suitable for a wide range of pattern-recognition problems. We demonstrate its applicability to molecular atomic models and their electron density maps. We show that the occurrence of the conformations of the macromolecular polypeptide backbone is related to the value of the CI of the constituting peptide fragments. We also illustrate how the CI can be used to assess the quality of a crystallographic electron density map.     

Sub-actions for weakly hyperbolic one-dimensional systems

Let f be an expanding map of degree 2 in the unitary interval [0,1], with an indifferent fixed point at x = 0 (i. e. (0) = 1), increasing, surjective and C 1 in each injective branch [0, c ] and ( c ,1], with inf{ f ' ( x ) | x ] [ g , 1)\{ c }} > 1 for all g > 0. Let A : [0,1] M be a function f -Hölder in each injective branch, monotone in a small neighbourhood of the origin, with A (0)< m and A (1)< m, where m is the maximum value of Z A d w over all invariant probability measures of f. Our main result is to prove that there exists a f -Hölder function S : [0,1] M , which satisfies the subcohomology equation Using S we prove that, if A admits a unique measure which maximizes Z , then this measure is uniquely ergodic. If A =log f ' , we are analysing the measure of maximal Lyapunov exponent. We also use the function f to define a bidimensional bijective function B in the unitary square [0,1) 2 [0,1), which is a modification of the Baker's Map, having an indifferent fixed point at the origin. If A : [0,1) 2     

A numerical study of infinitely renormalizable area-preserving maps

It has been shown in Gaidashev and Johnson [D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: stable sets, J. Mod. Dyn. 3(4) (2009), pp. 555–587.] and Gaidashev et al. [D. Gaidashev, T. Johnson, and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, in preparation.] that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Hölder continuous of exponent α?>?0. In this article we investigate numerically the specific value of α. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real.     

Random and deterministic perturbation of a class of skew-product systems

Abstract. This paper is concerned with the stability properties of skew-products T (x,y) = (f(x), g(x,y)) in which (f,X,mu) is an ergodic map of a compact metric space X and g: Xx Rn Rn is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. We study the orbit stability and stability of mixing of T(x,y) = (f(x), g(x,y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance of each other except for a fraction of the time. Furthermore, we show that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which 'contract on average'.     

Mapping the backbone of science

Summary This paper presents a new map representing the structure of all of science, based on journal articles, including both the natural and social sciences. Similar to cartographic maps of our world, the map of science provides a bird’s eye view of today’s scientific landscape. It can be used to visually identify major areas of science, their size, similarity, and interconnectedness. In order to be useful, the map needs to be accurate on a local and on a global scale. While our recent work has focused on the former aspect,1 this paper summarizes results on how to achieve structural accuracy. Eight alternative measures of journal similarity were applied to a data set of 7,121 journals covering over 1 million documents in the combined Science Citation and Social Science Citation Indexes. For each journal similarity measure we generated two-dimensional spatial layouts using the force-directed graph layout tool, VxOrd. Next, mutual information values were calculated for each graph at different clustering levels to give a measure of structural accuracy for each map. The best co-citation and inter-citation maps according to local and structural accuracy were selected and are presented and characterized. These two maps are compared to establish robustness. The inter-citation map is then used to examine linkages between disciplines. Biochemistry appears as the most interdisciplinary discipline in science.     

Almost complete Lyapunov spectrum in step skew-products

We study the spectrum of Lyapunov exponents of a family of partially hyperbolic and topologically transitive local diffeomorphisms that are step skew-products over a horseshoe map, continuing previous investigations. These maps are genuinely non-hyperbolic and the central Lyapunov spectrum contains negative and positive values. We show that, besides one gap, this spectrum is complete. We also investigate how Lyapunov regular points with corresponding (central) exponents are distributed in phase space. The principal ingredients of our proofs are minimality of the underlying iterated function system and shadowing-like arguments.     

Integration of colour and affine invariant feature for multi-view depth video estimation

Although many depth map estimation methods have been developed, depth map estimation still has some difficulties in low-texture regions and occlusion. This paper presents a novel approach for multi-view depth video estimation which adopts an adaptive matching scheme for high-texture regions and low-texture regions, respectively. For low-texture regions, the proposed method integrates affine invariant features with colour to make the matching more robust. Furthermore, according to the smoothness assumption of depth maps, a refinement is then performed on the estimated depth maps. Experimental results show that the proposed method can achieve better or comparable performances than the state-of-the-art method in the category of local methods even with the less running time.     

On discrete multivariate distributions symmetric in frequencies

In this paper we describe a new class of discrete multivariate distributions which verify that their probability mass function is invariant when their univariate variables are permuted. These distributions may be generated by a multivariate extension of the Gauss function₂ F ₁ with matrix argument. A methodology that permits the fit of these distributions to real data is developed. A fit of a distribution for bivariate real data is shown and is compared with fits obtained by means of other usual bivariate distributions generated by extensions of the Gauss function.     

On the asymptotic properties of piecewise contracting maps

We are interested in the phenomenology of the asymptotic dynamics of piecewise contracting maps. We consider a wide class of such maps and we give sufficient conditions to ensure some general basic properties, such as the periodicity, the total disconnectedness or the zero Lebesgue measure of the attractor. These conditions show in particular that a non-periodic attractor necessarily contains discontinuities of the map. Under this hypothesis, we obtain numerous examples of attractors, ranging from finite to connected and chaotic, contrasting with the (quasi-)periodic asymptotic behaviours observed so far.