Measure-preserving Research Articles

Relative weak mixing is generic

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos’ result to the collection of ergodic extensions of a fixed, but arbitrary, aperiodic transformation T0. We then use a result of Ornstein and Weiss to extend this relative theorem to the general (countable) amenable group.

A Brief Review of Generalized Entropies.

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

A double return times theorem

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem.

Generic behavior of a measure-preserving transformation

Del Junco–Lemańczyk [Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115 (3) (1992)] showed that a generic measure-preserving transformation satisfies certain orthogonality conditions. More precisely, there is a dense $G_{\unicode[STIX]{x1D6FF}}$ subset of measure preserving transformations such that, for every $T\in G$ and $k(1),k(2),\ldots ,k(l)\in \mathbb{Z}^{+}$, $k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime })\in \mathbb{Z}^{+}$, the convolutions $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{T^{k(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k(l)}}\quad \text{and}\quad \unicode[STIX]{x1D70E}_{T^{k^{\prime }(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k^{\prime }(l^{\prime })}},\end{eqnarray}$$ where $\unicode[STIX]{x1D70E}_{T^{k}}$ is the maximal spectral type of $T^{k}$, are mutually singular, provided that $(k(1),k(2),\ldots ,k(l))$ is not a rearrangement of $(k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime }))$. We will introduce analogous orthogonality conditions for continuous unitary representations of the group of all measurable functions with values in the circle, $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$, which we denote by the DL-condition. We connect the DL-condition with a result of Solecki [Unitary representations of the groups of measurable and continuous functions with values in the circle. J. Funct. Anal., 267 (2014), pp. 3105–3124] which identifies continuous unitary representations of $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$ with a collection of measures $\{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}\}$, where $\unicode[STIX]{x1D705}$ runs over all increasing finite sequence of non-zero integers. In particular, we show that the ‘probabilistic’ DL-condition translates to ‘deterministic’ orthogonality conditions on the measures $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}$. As a corollary, we show that the same orthogonality conditions as in the result by Del Junco–Lemańczyk hold for a generic unitary operator on a separable infinite-dimensional Hilbert space.

Partially bounded transformations have trivial centralizers

We prove that for infinite rank-one transformations satisfying a property called “partial boundedness,” the only commuting transformations are powers of the original transformation. This shows that a large class of infinite measure-preserving rank-one transformations with bounded cuts have trivial centralizers. We also characterize when partially bounded transformations are isomorphic to their inverse.

Independence and Alpern multitowers

ABSTRACTLet T be any invertible, ergodic, aperiodic measure-preserving transformation of a Lebesgue probability space , and P any finite measurable partition of X. We show that a (finite) Alpern multitower may always be constructed whose base is independent of P.

Isometry Invariant Shape Descriptors for Abnormality Detection on Brain Surfaces Affected by Alzheimer's Disease.

Alzheimer's disease (AD), a progressive brain disorder, is the most common neurodegenerative disease in older adults. There is a need for brain structural magnetic resonance imaging (MRI) biomarkers to help assess AD progression and intervention effects. Prior research showed that surface based brain imaging features hold great promise as efficient AD biomarkers. However, the complex geometry of cortical surfaces poses a major challenge to defining such a feature that is sensitive in qualification, robust in analysis, and intuitive in visualization. Here we propose a novel isometry invariant shape descriptor for brain morphometry analysis. First, we calculate a global area-preserving mapping from cortical surface to the unit sphere. Based on the mapping, the Beltrami coefficient shape descriptor is calculated. An analysis of average shape descriptors reveals that our detected features are consistent with some previous AD studies where medial temporal lobe volume was identified as an important AD imaging biomarker. We further apply a novel patch-based spherical sparse coding scheme for feature dimension reduction. Later, a support vector machine (SVM) classifier is applied to discriminate 135 amyloid-beta positive persons with the clinical diagnosis of Mild Cognitive Impairment (MCI) from 248 amyloid-beta-negative normal control subjects. The 5-folder cross-validation accuracy is about 81.82\% on the dataset, outperforming some traditional, Freesurfer based, brain surface features. The results show that our shape descriptor is effective in distinguishing dementia due to AD from age-matched normal aging individuals. Our isometry invariant shape descriptors may provide a unique and intuitive way to inspect cortical surface and its morphometry changes.

Normal Forms for Three-parameter Families of Area-preserving Maps near an Elliptic Fixed Point

We study dynamics of area-preserving maps in a neighbourhood of an elliptic fixed point. We describe simplified normal forms for a fixed point of co-dimension 3. We also construct normal forms for a generic three-parameter family which contains such degeneracy and use the normal form theory to describe generic bifurcations of periodic orbits in these families.

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $${\mu_\phi}$$ for each expanding Thurston map $${f : S^2\rightarrow S^2}$$ together with a real-valued Holder continuous potential $${\phi}$$ . Here the sphere S2 is equipped with a natural metric induced by f, called a visual metric. We also prove that identical equilibrium states correspond to potentials that are co-homologous up to a constant, and that the measure-preserving transformation f of the probability space $${(S^2,\mu_\phi)}$$ is exact, and in particular, mixing and ergodic. Moreover, we establish versions of equidistribution of preimages under iterates of f, and a version of equidistribution of a random backward orbit, with respect to the equilibrium state. As a consequence, we recover various results in the literature for a postcritically-finite rational map with no periodic critical points on the Riemann sphere equipped with the chordal metric.

A New Chaos-Based Color Image Encryption Scheme with an Efficient Substitution Keystream Generation Strategy

This paper suggests a new chaos-based color image cipher with an efficient substitution keystream generation strategy. The hyperchaotic Lü system and logistic map are employed to generate the permutation and substitution keystream sequences for image data scrambling and mixing. In the permutation stage, the positions of colored subpixels in the input image are scrambled using a pixel-swapping mechanism, which avoids two main problems encountered when using the discretized version of area-preserving chaotic maps. In the substitution stage, we introduce an efficient keystream generation method that can extract three keystream elements from the current state of the iterative logistic map. Compared with conventional method, the total number of iterations is reduced by 3 times. To ensure the robustness of the proposed scheme against chosen-plaintext attack, the current state of the logistic map is perturbed during each iteration and the disturbance value is determined by plain-pixel values. The mechanism of associating the keystream sequence with plain-image also helps accelerate the diffusion process and increase the degree of randomness of the keystream sequence. Experimental results demonstrate that the proposed scheme has a satisfactory level of security and outperforms the conventional schemes in terms of computational efficiency.

Ruelle's inequality in negative curvature

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case

Henon map is a well-studied classical example of area-contracting maps, modelling dissipative dynamics. The rich phenomena of coexistence of stable islands and their separatrices is typical of area-preserving maps, modelling conservative dynamics. In this paper we use the Henon map to ascertain that coexistence of sinks is greater and greater approaching the conservative case, and that part of it can be organized following a renormalization argument. Using a numerical continuation that we devised, and called method [ 5 ], one can follow bifurcation paths from the area-preserving case into the dissipative one, organizing families of coexisting attractive periodic orbits with diverging period. When the dissipation parameter goes to zero, we will give numerical evidence of the increasing coexistence of such periodic orbits, in the coordinate and parameter space values. Vanishing dissipation and diverging period constitute a double limit that we study as such, giving evidence of a singularity in the limit. The families we study all appear as homoclinic bifurcation, and the fixed point causing the homoclinic onset also structures the renormalization scheme. One of the goals of this paper is to improve the results obtained by looking to higher periods, and to approach dissipation down to an area-contraction factor of \begin{document}$1- 10^{-8}$\end{document} . Using the same dribbling method, as further promising application, we also deal with the dissipative Standard map.

Synchronization, phase slips, and coherent structures in area-preserving maps

The problem of synchronization of coupled Hamiltonian systems exhibits interesting features due to the non-uniform or mixed nature (regular and chaotic) of the phase space. We study these features by investigating the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Carroll method. We find that coupled standard maps show complete synchronization for values of the nonlinearity parameter at which regular structures are still present in phase space. The distribution of synchronization times has a power law tail indicating long synchronization times for at least some of the synchronizing trajectories. With the introduction of coherent structures using parameter perturbation in the system, this distribution crosses over to exponential behavior, indicating shorter synchronization times, and the number of initial conditions which synchronize increases significantly, indicating an enhancement in the basin of synchronization. On the other hand, coupled blinking vortex maps display both phase synchronization and phase slips, depending on the location of the initial conditions. We discuss the implication of our results.

On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

Since its inception in the 1970s at the hands of Feigenbaum and, independently, Coullet and Tresser the study of renormalization operators in dynamics has been very successful at explaining universality phenomena observed in certain families of dynamical systems. The first proof of existence of a hyperbolic fixed point for renormalization of area-preserving maps was given by Eckmann et al. (Mem Am Math Soc 47(289):vi+122, 1984). However, there are still many things that are unknown in this setting, in particular regarding the invariant Cantor sets of infinitely renormalizable maps. In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann–Koch–Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques.

Chaotic magnetic field lines and fractal structures in a tokamak with magnetic limiter

Open hamiltonian systems have typically fractal structures underlying chaotic dynamics with a number of physical consequences on transport. We consider such fractal structures related to the formation of a chaotic magnetic field line region near the tokamak wall and the corresponding field line dynamics, described by a two-dimensional area-preserving map. We focus on the exit basins, which are sets of points which originate orbits escaping through some exit and are typically fractal, also exhibiting the so-called Wada property. We show qualitative as well as quantitative evidences of these properties.

Dynamical intricacy and average sample complexity

ABSTRACTWe propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define intricacy and average sample complexity for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.

Universal exponent for transport in mixed Hamiltonian dynamics.

We compute universal distributions for the transition probabilities of a Markov model for transport in the mixed phase space of area-preserving maps and verify that the survival probability distribution for trajectories near an infinite island-around-island hierarchy exhibits, on average, a power-law decay with exponent γ=1.57. This exponent agrees with that found from simulations of the Hénon and Chirikov-Taylor maps. This provides evidence that the Meiss-Ott Markov tree model describes the transport for mixed systems.

On the limits of probabilistic forecasting in nonlinear time series analysis II: Differential entropy.

In a previous paper, the authors studied the limits of probabilistic prediction in nonlinear time series analysis in a perfect model scenario, i.e., in the ideal case that the uncertainty of an otherwise deterministic model is due to only the finite precision of the observations. The model consisted of the symbolic dynamics of a measure-preserving transformation with respect to a finite partition of the state space, and the quality of the predictions was measured by the so-called ignorance score, which is a conditional entropy. In practice, though, partitions are dispensed with by considering numerical and experimental data to be continuous, which prompts us to trade off in this paper the Shannon entropy for the differential entropy. Despite technical differences, we show that the core of the previous results also hold in this extended scenario for sufficiently high precision. The corresponding imperfect model scenario will be revisited too because it is relevant for the applications. The theoretical part and its application to probabilistic forecasting are illustrated with numerical simulations and a new prediction algorithm.

Limiting Curves for Polynomial Adic Systems

We prove the existence of and describe limiting curves resulting from deviations in the partial sums in the ergodic theorem for cylinder functions and polynomial adic systems. For a general ergodic measure-preserving transformation and a summable function, we give a necessary condition for a limiting curve to exist. Our work generalizes results by E. Janvresse, T. de la Rue, and Y. Velenik and answers several questions from their work.

Quantitative Pesin theory for Anosov diffeomorphisms and flows

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.