Pesin's Entropy Research Articles

Entropy formula for systems with inducing schemes

We obtain entropy formulas for Sinai–Ruelle–Bowen (SRB) measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures are given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold.

A comparison of topological entropies for nonautonomous dynamical systems

We study the distance entropy, Bowen topological entropy, and the classical topological entropy of nonautonomous dynamical systems. We show, in particular, that the distance entropy, Bowen entropy, Pesin entropy and the classical entropy are equivalent when the system is weakly mixing. Furthermore, we investigate the relationship between distance entropy and Hausdorff dimension on subsets from several aspects in detail.

Entropies of commuting transformations on Hilbert spaces

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $C^{2}$ $\mathbb{N}^2$-actions.

Pesin’s entropy formula for $$C^1$$ C 1 non-uniformly expanding maps

We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential $\psi=-\log|\det Df|$. In particular, this holds for any $C^1$-expanding map and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak$^*$-closed convex hull of the ergodic weak-SRB-like measures.

Topological entropy on points without physical-like behaviour

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f \cap \Gamma_f \subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f \cap \Gamma_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M \setminus I_f) \cap \Gamma_f$ of regular points without physical-like behaviour, has full topological entropy.

Ruelle's inequality in negative curvature

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

On Pesin's entropy formula for dominated splittings without mixed behavior

For C1 diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture for diffeomorphisms having such kind of splittings.

The Pesin entropy formula for diffeomorphisms with dominated splitting

AbstractFor any$C^1$diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.

Electrical Current in Sinai Billiards Under General Small Forces

The Lorentz gas of $\mathbb{Z}^2$-periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on an ionized medal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.

The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula

Consider a random cocycle $\Phi$ on a separableinfinite-dimensional Hilbert space preserving a probability measure$\mu$, which is supported on a random compact set $K$. We show thatif $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrableconditions of the differentials, then Pesin's entropy formula holdsif $\mu$ has absolutely continuous conditionalmeasures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.

Dominated splitting and Pesin's entropy formula

Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a dominated splitting $T_{orb(x)}M=E\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\mu(f)$ satisfies $$h_{\mu}(f)\geq\int \chi(x)d\mu,$$ where $\chi(x)=\sum_{i=1}^{dim\,F(x)}\lambda_i(x)$ and $\lambda_1(x)\geq\lambda_2(x)\geq...\geq\lambda_{dim\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\mu.$ Consequently, by using a dichotomy for generic volume-preserving diffeomorphism we show that Pesin's entropy formula holds for generic volume-preserving diffeomorphisms, which generalizes a result of Tahzibi in dimension 2.

Current in Periodic Lorentz Gases with Twists

We study electrical current in two-dimensional periodic Lorentz gas in the presence of a twist force on the scatterers. In this deterministic system, billiard orbits are still geodesics between collisions, but do not reflect elastically when reaching the boundary. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the strength of the twist force measured by e. We also prove the existence of a unique SRB measure, for which the Pesin entropy formula and Young’s expression for the fractal dimension are valid.

Stochastic stability at the boundary of expanding maps

We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastic stability. We also characterize the zero-noise limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter. The physical measures are obtained as zero-noise limits which are shown to satisfy Pesin's entropy formula.

Formula omitted]-generic Pesin's entropy formula

The metric entropy of a C 2-diffeomorphism with respect to an invariant smooth measure μ is equal to the average of the sum of the positive Lyapunov exponents of μ. This is the celebrated Pesin's entropy formula, h μ ( f)=∫ M ∑ λ i >0 λ i . The C 2 regularity (or C 1+ α ) of diffeomorphism is essential to the proof of this equality. We show that at least in the two dimensional case this equality is satisfied for a C 1-generic diffeomorphism and in particular we obtain a set of volume preserving diffeomorphisms strictly larger than those which are C 1+ α where Pesin's formula holds. To cite this article: A. Tahzibi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1057–1062.

PARTITION-BASED ENTROPIES OF DETERMINISTIC AND STOCHASTIC MAPS

In this paper we explore the relationship between the Kolmogorov–Sinai entropy, the sum of positive Lyapunov exponents, denoted here as the Pesin entropy and a new measure, the T-entropy, for nonlinear maps. We demonstrate that threshold-crossing partitions are effective in deriving representative symbolic realisations for the real-valued time series. We describe the recently developed entropy measure for finite strings and compare with values derived from the application of Shannon's theory. These techniques are further applied to a simple stochastic system, appearing to further confirm recent theoretical results on Lyapunov exponents.

Characterization of Measures Satisfying the Pesin Entropy Formula for Random Dynamical Systems

In this paper we first give a formulation of SRB (Sinai–Ruelle–Bowen) property for invariant measures of “stationary” random dynamical systems and then prove that this property is sufficient and necessary for a formula of Pesin's type relating entropy and Lyapunov exponents of such dynamical systems. This result is a random version of the main result in Part I of Ledrappier and Young's celebrated paper [11].

Pesin's entropy formula for stochastic flows of diffeomorphisms

Pesin's entropy formula relating entropy and Lyapunov exponents within the context of random dynamical systems generated by (discrete or continuous) stochastic flows of diffeomorphisms (including solution flows of stochastic differential equations on manifolds) is proved.

Ergodic theory for axiom A endomorphisms

AbstractIn this paper the Pesin's entropy formula and ‘large ergodic theorem’ are constructed for Axiom A endomorphisms.

Steady-state electrical conduction in the periodic Lorentz gas

We study nonequilibrium steady states in the Lorentz gas of periodic scatterers when an external field is applied and the particle kinetic energy is held fixed by a ``thermostat'' constructed according to Gauss' principle of least constraint (a model problem previously studied numerically by Moran and Hoover). The resulting dynamics is reversible and deterministic, but does not preserve Liouville measure. For a sufficiently small field, we prove the following results: (1) existence of a unique stationary, ergodic measure obtained by forward evolution of initial absolutely continuous distributions, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid; (2) exact identity of the steady-state thermodyamic entropy production, the asymptotic decay of the Gibbs entropy for the time-evolved distribution, and minus the sum of the Lyapunov exponents; (3) an explicit expression for the full nonlinear current response (Kawasaki formula); and (4) validity of linear response theory and Ohm's transport law, including the Einstein relation between conductivity and diffusion matrices. Results (2) and (4) yield also a direct relation between Lyapunov exponents and zero-field transport (=diffusion) coefficients. Although we restrict ourselves here to dimension $d=2,$ the results carry over to higher dimensions and to some other physical situations: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical systems and the method of Markov sieves, an approximation of Markov partitions. In our context we discuss also the van Kampen objection to linear response theory, which, we point out, overlooks the ``structural stability'' of strongly hyperbolic flows.

The Metric Entropy of Diffeomorphisms: Part I: Characterization of Measures Satisfying Pesin's Entropy Formula

Soit M une variete de Riemann compacte, soit f:M→M un diffeomorphisme, et soit m une mesure de probabilite de Borel f-invariante sur M. On identifie les mesures pour lesquelles l'inegalite de Margulis et Ruelle qui relie l'entropie aux exposants de Lyapunov atteint l'egalite. On demontre une formule valable pour toutes les mesures invariantes