Topological Pressure Research Articles

Topological pressure for an iterated function system

In this paper, we introduce a notion of topological pressure, which is different from the LMW's and ML's for an iterated function system. We find out the properties of the topological pressure, which are more similar to the properties of the classical topological pressure than LMW's and ML's. For an iterated function system, we obtain a partial variational principle on topological pressure, which improves the LMW's related result. Finally, we give a lower bound estimation of the topological pressure for a Ruelle-expanding iterated function system. In particular, we point out the exponential growth rate of fixed points is a lower bound of WLLZ's topological entropy for a Ruelle-expanding iterated function system.

Topological Pressure of Free Semigroup Actions for Non-Compact Sets and Bowen’s Equation, I

The applicability of Bowen’s equation to an arbitrary subset Z of a compact metric space has been well-studied by Climenhaga (Ergod Theory Dyn Syst 31(4):1163–1182, 2011). This paper aims to generalize the main results obtained by Climenhaga to free semigroup actions. To this end, we adopt the Caratheodory–Pesin structure (C–P structure) and introduce the notions of the topological pressure and lower and upper capacity topological pressures of a free semigroup action for an arbitrary subset. Some properties of these notions follow and we obtain the following three main results. First, by Bowen’s equation, we characterize the Hausdorff dimension of an arbitrary subset, where the points of the subset have the positive lower Lyapunov exponents and satisfy a tempered contraction condition. Second, we give an estimation of the topological pressure of a free semigroup action on an arbitrary subset. Finally, we analyze the relationship between the upper capacity topological pressure of a skew-product transformation and a free semigroup action in an arbitrary subset.

Topological pressures of a factor map for free semigroup actions

In this paper, we mainly study the topological pressures of finitely generated free semigroup actions on a compact metric space. By means of a factor map, we give an inequality relation for two topological pressures with a factor map of free semigroup actions. We establish a formula for two topological pressures with a factor map of free semigroup actions.

On the joint continuity of topological pressures for sub-additive singular-valued potentials

Given a non-conformal repeller [Formula: see text] of a [Formula: see text] map [Formula: see text], we give a variational principle of a dimensional upper bound of the non-conformal repellers. If [Formula: see text] is [Formula: see text] then we prove the joint continuity of topological pressure for sub-additive singular-valued potentials on maps and parameters.

自由半群作用的拓扑r压和拓扑压

自由半群作用的拓扑r压和拓扑压

Quantitative recurrence properties in the historic set for symbolic systems

Let be a topologically mixing subshift of finite type on an alphabet consisting of m symbols. Let be a continuous function. For a positive function and a continuous positive function , define the sets of recurrence points and In this paper, we give quantitative estimates of the sets of recurrence points, that is, can be expressed by ψ and is the solution of the Bowen equation of topological pressure.

The approximation of uniform hyperbolicity for C1 diffeomorphisms with hyperbolic measures

Let f be a C1-diffeomorphism which preserves an ergodic hyperbolic measure μ with positive measure-theoretic entropy. Assume that the Oseledec splittingTxM=E1(x)⊕⋯⊕Es(x)⊕Es+1(x)⊕⋯⊕El(x) is dominated on the Oseledec basin Γ. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes. Based on this, we also establish the continuity of the sub-additive topological pressure of the singular value potential.

On the rotation sets of generic homeomorphisms on the torus

AbstractWe study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

On local metric pressure of dynamical systems

In this paper, a local version of the topological pressure of dynamical systems is presented. It is a function defined on the product space which does not depend on any measure. It is shown that, for any invariant measure, integration of the introduced function with respect to its corresponding diagonal measure results in the metric pressure of the dynamical system.

Additive, Almost Additive and Asymptotically Additive Potential Sequences Are Equivalent

Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually equivalent to a standard (additive) potential in the sense that there exists a continuous potential with the same topological pressure, equilibrium states, variational principle, weak Gibbs measures, level sets (and irregular set) for the Lyapunov exponent and large deviations properties. In this sense, our result shows that almost and asymptotically additive potential sequences do not extend the scope of the theory compared to standard potentials, and that many results in the literature about such sequences can be recovered as immediate consequences of their counterpart in the additive case. A corollary of our main result is that all quasi-Bernoulli measures are weak Gibbs.

Nonadditive topological pressure for flows**Partially supported by FCT/Portugal through UID/MAT/04459/2013. C H was supported by FCT/Portugal through the Grant PD/BD/135523/2018.

We introduce a version of the nonadditive topological pressure for flows and we describe some of its main properties. In particular, we discuss how the nonadditive topological pressure varies with the data and we establish a variational principle in terms of the Kolmogorov–Sinai entropy. We also consider corresponding capacity topological pressures. In the particular case of subadditive families of functions we give a simpler characterization of these pressures. To the possible extent we follow corresponding arguments for maps, although various proofs require nontrivial modifications.

The phase relationship between the pyrazinamide polymorphs α and γ.

Pyrazinamide is an active pharmaceutical compound for the treatment of tuberculosis. It possesses at least four crystalline polymorphs. Polymorphism may cause solubility problems as the case of ritonavir has clearly demonstrated; however, polymorphs also provide opportunities to improve pharmaceutical formulations, in particular if the stable form is not very soluble. The four polymorphs of pyrazinamide constitute a rich system to investigate the usefulness of metastable forms and their stabilization. However, despite the existence of a number of papers on the polymorphism of pyrazinamide, well-defined equilibrium conditions between the polymorphs appear to be lacking. The main objectives of this paper are to establish the temperature and pressure equilibrium conditions between the so-called α and γ polymorphs of pyrazinamide, its liquid phase, and vapor phase and to determine the phase-change inequalities, such as enthalpies, entropies, and volume differences. The equilibrium temperature between α and γ was experimentally found at 392(1) K. Moreover, vapor pressures and solubilities of both phases have been determined, clearly indicating that form α is the more stable form at room temperature. High-pressure thermal analysis and the topological pressure-temperature phase diagram demonstrate that the γ form is stabilized by pressure and becomes stable at room temperature under a pressure of 260MPa.

Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings

We study infinite iterated function systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focussing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

Weak Gibbs measures and equilibrium states

ABSTRACT We present a simple proof of the fairly general fact that an invariant right-weak Gibbs probability measure is an equilibrium state, using a generalization of the formula of Brin and Katok for the local metric entropy. This new formula also allows us to clarify the role of the topological pressure on the standard definition of the Gibbs property.

Pressure and exponential rate over periodic orbits

For a measure μ preserved by a C1+α0 (α0>0) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑eSnϕ(x) over the orbits of some periodic points and the free energy hμ(f)+∫ϕdμ (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ˜ such that for every f-invariant measure supported on Λ˜, the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ˜ such that for every f-invariant measure supported on Δ˜, the exponential growth rate equals to the topological pressure.

A variational principle of topological pressure on subsets for amenable group actions

<p style='text-indent:20px;'>In this paper, we establish a variational principle for topological pressure on compact subsets in the context of amenable group actions. To be precise, for a countable amenable group action on a compact metric space, say <inline-formula><tex-math id="M1">\begin{document}$ G\curvearrowright X $\end{document}</tex-math></inline-formula>, for any potential <inline-formula><tex-math id="M2">\begin{document}$ f\in C(X) $\end{document}</tex-math></inline-formula>, we define and study topological pressure on an arbitrary subset and measure theoretic pressure for any Borel probability measure on <inline-formula><tex-math id="M3">\begin{document}$ X $\end{document}</tex-math></inline-formula> (not necessarily invariant); moreover, we prove a variational principle for this topological pressure on a given nonempty compact subset <inline-formula><tex-math id="M4">\begin{document}$ K\subseteq X $\end{document}</tex-math></inline-formula>.

Double variational principle for mean dimension with potential

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.

Hidden Gibbs measures on shift spaces over countable alphabets

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation on irreducible countable sofic shifts. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. Applications are given to some dimension problems and study of factors of (generalized) Gibbs measures on certain subshifts over countable alphabets.

Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems

Let \begin{document}$ \mathcal{F} $\end{document} be a random partially hyperbolic dynamical system generated by random compositions of a set of \begin{document}$ C^2 $\end{document} -diffeomorphisms. For the unstable foliation, the corresponding local unstable measure-theoretic entropy, local unstable topological entropy and local unstable pressure via the dynamics of \begin{document}$ \mathcal{F} $\end{document} along the unstable foliation are introduced and investigated. And variational principles for local unstable entropy and local unstable pressure are obtained respectively.