Topological Pressure Research Articles

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.

Bowen’s formula for meromorphic functions

AbstractLet f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

Given a continuous dynamical system ( X , T ) with the specification property, and a sequence of asymptotically additive continuous functions, we consider the irregular set for it and show that this set is either empty or carries full asymptotically additive topological pressure.

Hausdorff dimension of the limit set of conformal iterated function systems with overlaps

We give a new approach to the study of conformal iterated function systems with arbitrary overlaps. We provide lower and upper estimates for the Hausdorff dimension of the limit sets of such systems; these are expressed in terms of the topological pressure and the function d d , counting overlaps. In the case when the function d d is constant, we get an exact formula for the Hausdorff dimension. We also prove that in certain cases this formula holds if and only if the function d d is constant.

Topological pressure and topological entropy of a semigroup of maps

By using the Carathéodory-Pesin structure(C-P structure), withrespect to arbitrary subset, the topological pressure andtopological entropy, introduced for a single continuous map, isgeneralized to the cases of semigroup of continuous maps. Several oftheir basic properties are provided.

Exponential stabilization without geometric control

We present examples of exponential stabilization for the damped wave equation on a compact manifold in situations where the geometric control condition is not satisfied. This follows from a dynamical argument involving a topological pressure on a suitable uncontrolled set.

Random graph directed Markov systems

We introduce and explore random conformal graph directed Mar-kovsystems governed by measure-preserving ergodic dynamicalsystems. We first develop the symbolic thermodynamicformalism for random finitely primitive subshifts of finite typewith a countable alphabet (by establishing tightness ina narrow topology). We then construct fibrewise conformal andinvariant measures along with fibrewise topological pressure.This enables us to define the expected topological pressure$\mathcal EP(t)$ and to prove a variant of Bowen's formulawhich identifies the Hausdorff dimension of almost everylimit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is theunique zero of the expected pressure if the alphabet is finite orthe system is regular. We introduce the class of essentiallyrandom systems and we show that in the realm of systems withfinite alphabet their limit set fibers are never homeomorphic in abi-Lipschitz fashion to the limit sets of deterministic systems;they thus make up a drastically new world. We also provide alarge variety of examples, with exact computations of Hausdorffdimensions, and we study in detail the small random perturbationsof an arbitrary elliptic function.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Energy Decay for the Damped Wave Equation Under a Pressure Condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.

(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions.

An asymptotic analysis in thermodynamic formalism

We consider an asymptotic behaviour of the topological pressure, the Gibbs measure and the measure-theoretic entropy concerning a potential defined on a subshift. Our results are obtained by considering asymptotic perturbation of transfer operators and by using a method that avoids resolvent’s perturbation. In application, we investigate an asymptotic behaviour of the Hausdorff dimension of a perturbed cookie-cutter set.

Bowen’s equation in the non-uniform setting

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

The local variational principle of topological pressure for sub-additive potentials

Let ( X , T ) be a topological dynamical system and F = { f n } n = 1 ∞ be a sub-additive potential on C ( X , R ) . Let U be an open cover of X . Then for any T -invariant measure μ , let F ∗ ( μ ) = lim n → ∞ 1 n ∫ f n d μ . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle: P ( T , F , U ) = sup μ { h μ ( T , U ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } where h μ ( T , U ) denotes the measure-theoretic entropy of μ relative to U and the supremum can be attained by a T -invariant ergodic measure. The main purpose of this paper is to generalize a result of Huang and Yi (2007) [17]. In the paper [17], they proved the local variational principle of pressure for additive potentials. Furthermore, we prove the result P ( T , F ) = lim d i a m ( U ) → 0 P ( T , F ; U ) . Moreover, we obtained P ( T , F ) = sup μ { h μ ( T ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } , which gives another proof of the topological pressure variational principle for sub-additive potentials from Cao et al. (2008) [14].

Thermodynamic formalism for subadditive sequence of discontinuous functions

The topological pressure for subadditive sequence of discontinuous functions is defined on any invariant subset having a nested family of subsets in the compact metric space. Two subadditive variational principles associated with two different relatively weak conditions are developed for the defined topological pressure. As an application, we give an example on systems with nonzero Lyapunov exponents.

Phenomenology of polymorphism and topological pressure–temperature diagrams

Although polymorphism of drug molecules is often studied with extensive and excellent experimental data, pressure appears to be a forgotten variable. In this article, an analysis is provided of the phase behavior of Atovaquone using available literature data. A pressure- temperature diagram is constructed topologically by way of the Clapeyron equation. The method leads to the conclu- sion that Atovaquone phase I and III behave enantiotropi- cally, like a- and b-sulfur do in their paradigmatic P-T diagram, and that phase I is stable at room temperature and under ''ordinary'' pressure.

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

A relative local variational principle for topological pressure

We define the relative local topological pressure for any given factor map and open cover, and prove the relative local variational principle of this pressure. More precisely, for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover U of X, a continuous, real-valued function f on X and an S-invariant measure ν on Y, we show that the corresponding relative local pressure P(T, f, U, y) satisfies $$ \mathop {\sup }\limits_{\mu \in \mathcal{M}(X,T)} \left\{ {h_\mu (T,\mathcal{U}|Y) + \int_X {f(x)d\mu (x):\pi \mu = \nu } } \right\} = \int_Y {P(T,f,\mathcal{U},y)d\nu (y)} , $$ where \( \mathcal{M} \)(X, T) denotes the family of all T-invariant measures on X. Moreover, the supremum can be attained by a T-invariant measure.

A thermodynamic definition of topological pressure for non-compact sets

AbstractWe give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

Lyapunov Spectrum of Asymptotically Sub-additive Potentials

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map that is not upper-semi continuous.

Which hole is leaking the most: a topological approach to study open systems

We study the process of escape of orbits through a hole in the phase space of a dynamical system generated by a chaotic map of an interval. If this hole is an element of Markov partition we are able to use the machinery of symbolic dynamics and estimate the probability of an orbit to escape at the instant of time n in terms of the topological pressure over corresponding symbolic dynamical systems. We obtain exact formulae allowing us to compare different holes according to their ability to support escaping flow of orbits. These results are applicable to the classification of vertices of dynamical networks in terms of the loads on nodes and edges of a network.