Tail Entropy Research Articles

Entropies for factor maps of amenable group actions

In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles (Zhu 2021 Nonlinearity 34 5163–85, theorems 2.12 and 3.9) proved by Zhu under additional assumptions. Then along the line of Misiurewicz (1976 Studia Math. 55 175–200), we introduce a new invariant called relative topological tail entropy and prove a Ledrappier’s type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.

Complex cellular structures

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex cell $\mathcal{C}$ we define its holomorphic extension $\mathcal{C}\subset\mathcal{C}^\delta$, which is again a complex cell. The hyperbolic geometry of $\mathcal{C}$ within $\mathcal{C}^\delta$ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved. Using this theory, we refine the Yomdin-Gromov algebraic lemma on $C^r$-smooth parametrizations of semialgebraic sets: we show that the number of $C^r$ charts can be taken to be polynomial in the smoothness order $r$ and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of $C^\infty$ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin. The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

Critical role of histone tail entropy in nucleosome unwinding.

The nucleosome is the fundamental packaging unit for the genome. It must remain tightly wound to ensure genome stability while simultaneously being flexible enough to keep the DNA molecule accessible for genome function. The set of physicochemical interactions responsible for the delicate balance between these naturally opposed processes have not been determined due to challenges in resolving partially unwound nucleosome configurations at atomic resolution. Using a near atomistic protein-DNA model and advanced sampling techniques, we calculate the free energy cost of nucleosome DNA unwinding. Our simulations identify a large energetic barrier that decouples the outer and the inner DNA unwinding into two separate processes, occurring on different time scales. This dynamical decoupling allows the exposure of outer DNA at a modest cost to ensure accessibility while keeping the inner DNA and the histone core intact to maintain stability. We also reveal that this energetic barrier arises from a delayed loss of contacts between disordered histone tails and the DNA and is, surprisingly, largely offset by an entropic contribution from these tails. Implications of this enthalpy entropy compensation for the regulation of nucleosome stability and genome function are discussed.

Upper bounds on measure-theoretic tail entropy for dominated splittings

For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.

Local unstable entropies of partially hyperbolic diffeomorphisms

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

Tail entropy and hyperbolicity of measures

We study the relationship between the tail entropy and the hyperbolicity of invariant measures. An upper bound of the tail entropy is given in terms of the hyperbolic index.

Sub-additive topological and measure-theoretic tail pressures

In this paper, we give some definitions of the topological tail pressures for sub-additive potentials and prove that they are equivalent if the potentials are continuous. Under some assumptions, we get a variational principle which exhibits the relationship between topological tail pressure and measure-theoretic tail entropy. Finally, we define a new measure-theoretic tail pressure for sub-additive potentials and some interesting properties of it are obtained.

Relative tail entropy for random bundle transformations

We introduce the relative tail entropy to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail entropy is conserved by the principal extension.

2D lattice model of a lipid bilayer: Microscopic derivation and thermodynamic exploration.

Based on all-atom Molecular Dynamics (MD) simulations of a lipid bilayer we present a systematic mapping on a 2D lattice model. Keeping the lipid type and the chain order parameter as key variables we derive a free energy functional, containing the enthalpic interaction of adjacent lipids as well as the tail entropy. The functional form of both functions is explicitly determined for saturated and polyunsaturated lipids. By studying the lattice model via Monte Carlo simulations it is possible to reproduce the temperature dependence of the distribution of order parameters of the pure lipids, including the prediction of the gel transition. Furthermore, application to a mixture of saturated and polyunsaturated lipids yields the correct phase separation behavior at lower temperatures with a simulation time reduced by approximately 7 orders of magnitude as compared to the corresponding MD simulations. Even the time-dependence of the de-mixing is reproduced on a semi-quantitative level. Due to the generality of the approach we envisage a large number of further applications, ranging from modeling larger sets of lipids, sterols, and solvent proteins to predicting nucleation barriers for the melting of lipids. Particularly, from the properties of the 2D lattice model one can directly read off the enthalpy and entropy change of the 1,2-dipalmitoyl-sn-glycero-3-phosphocholine gel-to-liquid transition in excellent agreement with experimental and MD results.

Usc/fibred entropy structure and applications

ABSTRACTFor a given metrizable space X, we study continuity properties of the entropy as function not only of the measure but also of the dynamical system on X. We introduce the notion of robust tail entropy, which implies upper semicontinuity of the topological entropy but also stability of measures of maximal entropy (when the topological entropy is continuous). This gives, inter alia, simple proofs of results of Misiurewicz and Raith for multimodal interval maps. We also consider fibred entropy structures which allow us to investigate the symbolic extensions of (smooth) skew-product systems.

On the tail pressure

In this paper, we give two equivalent definitions of tail pressure involving open covers and establish a variational principle which exhibits the relationship between tail pressure and measure-theoretic tail entropy.

Tail variational principle for a countable discrete amenable group action

In this paper, we define the measure-theoretic tail entropy for a countable discrete amenable group action and prove the tail variational principle when there exists a refining sequence of essential partitions. Additionally, some properties of local entropy are investigated.

Asymptotich-expansiveness rate ofC∞maps

We study the rate of convergence to zero of the tail entropy of C ∞ maps. We give an upper bound of this rate in terms of the growth in k of the derivative of order k and give examples showing the optimality of the established rate of convergence. We also consider the case of multimodal maps of the interval. Finally, we prove that homoclinic tangencies give rise to C r ( r ⩾ 2 ) robustly non- h-expansive dynamical systems.

Uniform tail entropy for real analytic maps

Let M be a compact real analytic manifold of finite dimension. There is a function a:(0,+∞)→[0,+∞) with limt→0⁡a(t)=0 such that the tail entropy h⁎(f,ε) of any real analytic map f on M is uniformly bounded above by the scale a(ε).

Weak expansiveness for actions of sofic groups

In this paper, we shall introduce h-expansiveness and asymptotical h-expansiveness for actions of sofic groups. By definition, each h-expansive action of a sofic group is asymptotically h-expansive. We show that each expansive action of a sofic group is h-expansive, and, for any given asymptotically h-expansive action of a sofic group, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy.Observe that asymptotically h-expansive property was first introduced and studied by Misiurewicz for Z-actions using the language of tail entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz's ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting.

EXPONENTIAL CONVERGENCE RATES OF SECOND QUANTIZATION SEMIGROUPS AND APPLICATIONS

Exponential convergence rates in the L 2 -tail norm and entropy are characterized for the second quantization semigroups by using the corresponding base Dirichlet form. This supplements the well known result on the L 2 -exponential convergence rate of second quantization semigroups. As applications, birth-death type processes

Generalized beta-generated distributions

This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also has tractable properties: we present various expansions for moments, generating function and quantiles. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.

Effect of Temperature on the Structure of Charged Membranes

Interactions between charged and neutral self-assembled phospholipid membranes are well understood and take into account temperature dependence. Yet, the manner in which the structure of the membrane is affected by temperature was hardly studied. Here we study the effect of temperature on the thickness, area per lipid, and volume per lipid of charged membranes. Two types of membranes were studied: membranes composed of charged lipids and dipolar (neutral) membranes that adsorbed divalent cations and became charged. Small-angle X-ray scattering data demonstrate that the thickness of charged membranes decreases with temperature. Wide-angle X-ray scattering data show that the area per headgroup increases with temperature. Intrinsically charged membranes linearly thin with temperature, whereas neutral membranes that adsorb divalent ions and become charged show an exponential decrease of their thickness. The data indicate that, on average, the tails shorten as the temperature rises. We attribute this behavior to higher lipid tail entropy and to the weaker electrostatic screening of the charged headgroups, by their counterions, at elevated temperatures. The latter effect leads to stronger electrostatic repulsion between the charged headgroups that increases the area per headgroup and decreases the bilayer thickness.

Tail pressure and the tail entropy function

AbstractBurguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.

Directional entropy and tail uncertainty, with applications to financial hazard

“Mine is a long and sad tale”, said the Mouse, turning to Alice and sighing. “It is a long tail certainly,” said Alice, looking down with wonder at the Mouse's tail; “but why do you call it sad?” And she kept on puzzling about it while the mouse was speaking … Financial risk management metrics such as value at risk (VaR) can be illuminated by means of a regime-specific concept of directional entropy. This enables a change of measure via a rescaling function to an equivalent logistic distribution, one that has the same total and directional entropies at the chosen critical point. VaR rescaling adjusts the critical probability to capture the long tail entropy. The scaling function can be used as a comparative metric for tail length, or equivalent conditional value at risk, even where moments do not exist. Directional entropy can also be used to identify regions of maximal exposure to new information, which can actually increase entropy rather than collapse it.