Topological Pressure Research Articles

Ergodic and Bernoulli properties of analytic maps of complex projective space

We examine the measurable ergodic theory of analytic maps F of complex projective space. We focus on two different classes of maps, Ueda maps of P n , and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy (h μ (F) = h top (F) = log(deg F)). We find analytic maps of P 1 and P 2 which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer d > 1, there exists a rational map of the sphere which is one-sided Bernoulli of entropy log d with respect to a smooth measure.

A note on the Ruelle pressure for a dilute disordered Sinai billiard

The topological pressure is evaluated for a dilute random Lorentz gas, in the approximation that takes into account only uncorrelated collisions between the moving particle and fixed, hard sphere scatterers. The pressure is obtained analytically as a function of the temperature-like parameter, β, and of the density of scatterers, n. The effects of correlated collisions on the topological pressure can be described qualitatively, at least, and they significantly modify the results obtained by considering only uncorrelated collision sequences. As a consequence, for large systems, the range of β-values over which our expressions for the topological pressure are valid becomes very small, approaching zero, in most cases, as the inverse of the logarithm of system size.

Gibbs states on the symbolic space over an infinite alphabet

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Conformal measures for multidimensional piecewise invertible maps

Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow ]0,\infty[ , a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g} d\nu with a constant \lambda>0 . Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

Phase Transitions for Countable Markov Shifts

We study the analyticity of the topological pressure for some one-parameter families of potentials on countable Markov shifts. We relate the non-analyticity of the pressure to changes in the recurrence properties of the system. We give sufficient conditions for such changes to exist and not to exist. We apply these results to the Manneville–Pomeau map, and use them to construct examples with different critical behavior.

Fractality of the hydrodynamic modes of diffusion.

Transport by normal diffusion can be decomposed into hydrodynamic modes which relax exponentially toward the equilibrium state. In chaotic systems with 2 degrees of freedom, the fine scale structures of these modes are singular and fractal, characterized by a Hausdorff dimension given in terms of Ruelle's topological pressure. For long-wavelength modes, we relate the Hausdorff dimension to the diffusion coefficient and the Lyapunov exponent. This relationship is tested numerically on two Lorentz gases, one with hard repulsive forces, the other with attractive, Yukawa forces. The agreement with theory is excellent.

Large deviation for weak Gibbs measures and multifractalspectra

We introduce the class of `medium varying functions' and correspondingweak Gibbs measures both defined on a symbolic shift space. We provethat the Helmholtz free energy of the stochastic process of a randomlystopped Birkhoff sum measured by a weak Gibbs measure can be expressedin terms of the topological pressure. This leads to the notion of themultifractal entropy function which provides large deviation bounds.Weshow that the multifractal entropy functions coincide (up toconstants)with multifractal spectra when Gibbs or g-measures are involved.

Discontinuities of the pressure for piecewise monotonic interval maps

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.

Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$

In this paper, we review several notions from thermodynamic formalism,like topological pressure and entropy and show how they can be employed, in orderto obtain information about stable and unstable sets of holomorphic endomorphismsof $\mathbb P^2$ with Axiom A.In particular, we will consider the non-wandering set of such a mapping and its'saddle' part $S_1$, i.e the subset of points with both stable and unstable directions.Under a derivative condition, the stable manifolds of points in S1 will have a very'thin' intersection with $S_1$, from the point of view of Hausdorff dimension. Whilefor diffeomorphisms there is in fact an equality between $HD(W^s_\varepsilon(x)\cap S_1)$ and theunique zero of $P(t \cdot \phi^s)$ (Verjovsky-Wu [VW]) in the case of endomorphisms this willnot be true anymore; counterexamples in this direction will be provided. We alsoprove that the unstable manifolds of an endomorphism depend Hölder continuouslyon the corresponding prehistory of their base point and employ this in the end togive an estimate of the Hausdorff dimension of the global unstable set of $S_1$. Thisset could be à priori very large, since, unlike in the case of H´enon maps, there is anuncountable collection of local unstable manifolds passing through each point of $S_1$.

Parabolic iterated function systems

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

On Invariant Line Fields

It is shown that a rational function of degree [ges ] 2 admits an invariant line field with respect to some measure μ, which is an equilibrium state of a Holder continuous potential whose topological pressure is greater than its supremum, only in very special cases when the Julia set is either a geometric circle or an interval, or totally disconnected and contained in a real-analytic curve.

On an example with a non-analytic topological pressure

Abstract We construct a function φ = φ ( x 0 , x 1 ) on a topologically mixing countable Markov shift such that the set of values β >0 for which the pressure function P top ( βφ ) is not analytic in a neighbourhood of β has positive Lebesgue measure. The construction is based on renewal theoretic ideas.

Topological pressure for geodesic flows

We give a Riemannian formula for the topological pressure of the geodesic flow of a closed Riemannian manifold. As a consequence, we derive an asymptotic formula for the stable norm in cohomology in terms of geodesic arcs for manifolds with topological entropy h top=0. We introduce a Poincaré series also in terms of geodesic arcs and we show that it defines a holomorphic function on the half plane given by those complex numbers with real part > h top.

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC 1-mapX→ℝ for which {c∈X: T′c=0} is finite. Set $$T\left( T \right) = \cap _{j = 1}^\infty \overline {T^{ - j} X} $$ . Fix ann ∈ ℕ. For e>0, theC 1-map $$\tilde T:X \to \mathbb{R}$$ is called an e-perturbation ofT if $$\tilde T$$ is a piecewise monotonic map with at mostn intervals of monotonicity and $$\tilde T$$ is e-close toT in theC 1-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation $$\tilde T$$ ofT has a unique measure $$\tilde \mu $$ of maximal entropy, and the map $$\tilde T \mapsto \tilde \mu $$ is continuous atT in the weak star-topology.

Thermodynamic formalism for countable Markov shifts

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

Abnormal escape rates from nonuniformly hyperbolic sets

Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log |J^u f|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.

The entropy formula for SRB-measures of lattice dynamical systems

We give a detailed proof of the entropy formula for SRB-measures of coupled hyperbolic attractors over integer lattices. We show that the topological pressure for the potential function of the SRB-measure is zero.

An integral formula for topological entropy of $C^{\infty}$ maps

In this paper we consider a smooth dynamical system $f$ and give estimates of the growth rates of vector fields and differential forms in the $L_p$ norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy $$h_{\rm top}=\lim_{n\to\infty} \frac 1n \log \int \Vert Df_x^n\,^{\wedge}\Vert \,dx,$$ where $Df_x^n\,^{\wedge}$ is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.

Box Dimensions and Topological Pressure for some Expanding Maps

We consider an expanding map defined on an open region of the plane and study the box dimensions of its invariant sets. Under the condition that the map leaves invariant a “strong unstable foliation”\(\), we prove that the box dimension of an invariant set is given by δF+δT, where δT is its dimension transverse to \(\) and δF is the root of a certain function involving topological pressure.

Invariant of topological pressure under some semi-conjugates

In this paper, the topological pressure is preserved under some semi-conjugates, and a formula of computing topological pressure by use of periodic points for positively expansive continuous map with specification is given.