Topological Pressure Research Articles

A note of topological pressure for non-compact sets of a factor map

Using the notion of topological pressure for non-compact sets, we prove a relation for two topological pressures with a factor map. We also provide an application in symbolic dynamics and conformal repellers. These results are generalized to the cases of BS-dimensions.

Sub-Additive and Asymptotically Sub-Additive Topological Pressure for $$\mathbb{Z }^d$$ -Actions

We introduce the topological pressure for any sub-additive and asymptotically sub-additive potentials of $$\mathbb{Z }^d$$ -actions, and establish the variational principle for them.

Variation of topological pressure and dimension: from polynomials to complex Hénon maps

AbstractWe study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

Variational principles for relative local pressure with subadditive potentials

We prove two relative local variational principles of topological pressure functions \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U},y)$\end{document}P(T,F,U,y) and \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U}|Y)$\end{document}P(T,F,U|Y) for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover \documentclass[12pt]{minimal}\begin{document}$\mathcal {U}$\end{document}U of X and a subadditive potential \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F in \documentclass[12pt]{minimal}\begin{document}$C(X,\mathbb {R})$\end{document}C(X,R). By proving the upper semi-continuity and affinity of the entropy maps \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ(T,U∣Y) and \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h^+_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ+(T,U∣Y) on the space of all invariant Borel probability measures, we show that the relative local topological pressure with subadditive potentials determines the local measure-theoretic conditional entropies.

On non-regular g-measures

We prove that g-functions whose set of discontinuity points has strictly negative topological pressure and which satisfy an assumption that is weaker than non-nullness, have at least one stationary g-measure. We also obtain uniqueness by adding conditions on the set of continuity points.

Equidistribution results for geodesic flows

AbstractUsing the works of Mañé [On the topological entropy of the geodesic flows.J. Differential Geom.45(1989), 74–93] and Paternain [Topological pressure for geodesic flows.Ann. Sci. Éc. Norm. Supér.(4)33(2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a$\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

Topological pressure of simultaneous level sets

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets.Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace.This technique allows us to extend a number of previous multifractal results from the C1+ε case to the C1 case. We consider ergodic ratios Snφ/Snψ where the function ψ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.

Delocalization of Slowly Damped Eigenmodes on Anosov Manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.

Bicalutamide polymorphs I and II

For a complete picture of the phase behavior of a compound exhibiting dimorphism, not only the temperature but also the pressure, the second variable of the Gibbs energy, should be taken into account. Because volume reflects the dependence of the Gibbs energy on pressure, investigating the specific volumes of a compound provides information on its phase behavior under pressure. This can be quantified using the Clapeyron equation, which leads to a so-called topological pressure–temperature phase diagram. Bicalutamide is used as a sample case and with literature data its topological phase diagram has been constructed. Even though the phase relationship between bicalutamide’s two known solid phases is monotropic at ordinary pressure, it becomes enantiotropic at higher pressure. The steep slope of the solid–solid equilibrium in the P–T diagram, 22.5 MPa K−1, indicates that pressure has virtually no influence on it and that it is mainly entropy driven.

Thermodynamic phase transitions for Pomeau-Manneville maps

We study phase transitions in the thermodynamic description of Pomeau-Manneville intermittent maps from the point of view of infinite ergodic theory, which deals with diverging measure dynamical systems. For such systems, we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase. In particular, topological pressure and Rényi entropy are calculated exactly for such systems. Finally, we show the connection of the distributional limit theorem with non-Gaussian fluctuations of the algorithmic complexity proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA 85, 4591 (1988)].

Phenomenology of polymorphism: The topological pressure–temperature phase relationships of the dimorphism of finasteride

Knowledge of the phase behavior in the solid state of active pharmaceutical ingredients is important for the development of stable drug formulations. The topological method for the construction of pressure–temperature phase diagrams has been applied to study the phase behavior of finasteride. It is demonstrated that with basic calorimetric measurements and X-ray diffraction sufficient data can be obtained to construct a complete topological pressure–temperature phase diagram. The dimorphism observed for finasteride gives rise to a phase diagram similar to the paradigmatic diagram of sulfur. The solid–solid phase relationship is enantiotropic at ordinary pressure and becomes monotropic at elevated pressure, where solid I is the only stable phase.

Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle–Perron–Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

Topological pressure and the variational principle for actions of sofic groups

AbstractWe introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Approximating entropy for a class of ℤ2Markov random fields and pressure for a class of functions on ℤ2shifts of finite type

AbstractFor a class of ℤ2Markov Random Fields (MRFs)μ, we show that the sequence of successive differences of entropies of induced MRFs on strips of heightnconverges exponentially fast (inn) to the entropy ofμ. These strip entropies can be computed explicitly whenμis a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor ℤ2shift of finite typeX. We state this result in terms of approximations to the (topological) pressures of certain functions on such anX, and we show that these pressures are computable if the values taken on by the functions are computable. Finally, we show that our results apply to the hard core model and Ising model for certain parameter values of the corresponding interactions, as well as to the topological entropy of certain nearest-neighbor ℤ2shifts of finite type, generalizing a result in [R. Pavlov. Approximating the hard square entropy constant with probabilistic methods.Ann. Probab.to appear].

Variational Principle for Conditional Pressure with Subadditive Potential

The purpose of this paper is to define and study conditional pressure for subadditive potentials which is an extension of conditional entropy. This study reveals a variational principle of topological conditional pressure for subadditive potentials. Besides, we discuss the topological conditional pressure for asymptotically subadditive potentials. The main idea of the proof is quite similar to that of Cao, Feng and Huang's approximations.

Topological pressure of historic set for [formula omitted]-actions

We show that, if a continuous Z d - or Z + d -action of a compact metric space has the almost product property, the topological pressure of historic set can be equal to that of the whole space.

Rich phase transitions in step skew products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated with the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behaviour of different types. However, in some sense the expanding behaviour is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

Topological pressure for sub-additive potentials of amenable group actions

We introduce the topological pressure for any sub-additive potentials of a countable discrete amenable group action and any given open cover, and establish a local variational principle for it.

Thermodynamic formalism for suspension flows over countable Markov shifts

We introduce a new notion of topological pressure for suspension flows over countable Markov shifts. We show that our definition, which is a natural analogue of Gurevich pressure for a Markov shift, is equivalent to previous notions of topological pressure which were defined on a restricted class of flows. We also widen the domain of definition of these previous notions, making them suitable for application to some important examples.