Smooth Ergodic Theory Research Articles

Smooth ergodic theory of $ \mathbb {Z}^d $-actions

Smooth ergodic theory of $ \mathbb {Z}^d $-actions

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n, \mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha\colon \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha\colon \Gamma \rightarrow\mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Time Weighted Pressure

Schmeling introduced the concept of time weighted entropy in 2000 as a generalization of Pesin’s topological entropy. Since the potential function $$\phi $$ in topological pressure plays an important role in multifractal analysis and smooth ergodic theory, by analogy, we propose time weighted pressures. It is proved that time weighted pressures are generally different on non-invariant sets, and their properties concerning on the time complexity (with respect to the potential function $$\phi $$) of systems are explored. Further, it is shown that for a $$C^{1+\alpha }$$-diffeomorphism with a finite Markov partition, the corresponding coding map preserves time weighted pressures.

Entropy, Lyapunov exponents, and rigidity of group actions

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled Workshop for young researchers: Groups acting on manifolds held in Teresopolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory-particularly Lya-punov exponents and metric entropy-as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [114] and recent the work of the author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [31, 36]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures. i

Equidistribution for Nonuniformly Expanding Dynamical Systems, and Application to the Almost Sure Invariance Principle

Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $\mu$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, \mu)$. In smooth ergodic theory there are various natural choices of $\mu$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We show that in a large class of measures, it is possible to couple (redefine on a new probability space) every two processes so that they are almost surely close to each other, with explicit estimates of "closeness". The purpose of this work is to close a gap in the proof of the almost sure invariance principle for nonuniformly hyperbolic transformations by Melbourne and Nicol.

Mixed Order Fractional Differential Equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

What are Lyapunov exponents, and why are they interesting?

Taking as inspiration the Fields Medal work of Artur Avila, I’d like to introduce you to Lyapunov exponents. My plan is to show how Lyapunov exponents play a key role in three areas in which Avila’s results lie: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schrodinger operators. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats. The barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices. Here’s what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide:

Smooth approximation of stochastic differential equations

Consider an It\^{o} process $X$ satisfying the stochastic differential equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a multidimensional Brownian motion. Suppose that $W_n$ has smooth sample paths and that $W_n$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_n$ to the ordinary differential equation $dX_n=a(X_n)\,dt+b(X_n)\,dW_n$. The classical Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)\,dW$ depends sensitively on how the smooth approximation $W_n$ is chosen. In applications, a natural class of smooth approximations arise by setting $W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$ where $\phi_t$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on $\phi_t$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)\,dW$. Our theory applies to Anosov or Axiom A flows $\phi_t$, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on $\phi_t$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

Preface Issue 2-2015

Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the second half of the twentieth century. This applies not only to his mathematical achievements –mainly in topology, geometry and number theory– and his huge school of at least 52 PhD students. He also made enormous efforts and had great success in reshaping mathematics in Germany after the Second World War. The Sonderforschungsbereich Theoretische Mathematik, the Arbeitstagung and the Max-Planck-Institut fur Mathematik played an important role in making Germany once again a place which mathematicians from all over the world would like to visit. The Deutsche Mathematiker-Vereinigung (DMV) owes a particularly deep gratitude to Friedrich Hirzebruch who served twice as its chairman, in 1962 and 1990. In 1962, one year after the wall was built, the Gesellschaft fur Mathematik der DDR was founded and urged to branch off from the DMV. One purpose of his second election in 1990 was that he should chair the reunification of both societies and to celebrate at the same time the 100th anniversary of the DMV. In the 28 years in between he ensured that the ties to the East were not severed. One of his best known former PhD-students, Don Zagier, gives a fairly detailed picture of all these and many more facets of Friedrich Hirzebruch’s life within mathematics. Often, pictures are available only in a corrupted form and one has to try to reconstruct the original ones. One may think for instance of a real valued function describing the gray-value of the picture at the actual position. If one assumes that the corruption mechanism (the corruption mapping) is known up to some “noise” one has to seek a preimage of the corrupted picture under this mapping. In principle one could apply the corruption mapping to all possible pictures and search for those whose images minimise a suitable distance functional from the given corrupted picture. Due to the “bad” properties of the corruption mapping this problem is in this form not well posed. In order to resolve this issue, one adds a regularisation term containing first and second order derivatives of the unknown preimage to the distance functional. Gabriele Steidl explains in her survey article on “Combined first and second order variational approaches for image processing” the recent development of such techniques, and illustrates them by two rather different applications. Functions of bounded variation and related function spaces are fundamental for Gabriele Steidl’s survey article and are also the topic of a recent book by Jurgen Appell, Jozef Banas, and Nelson Jose Merentes Diaz, which is reviewed by Michel Willem. The second review concerns a book by Luis Barreira and Yakov Pesin on smooth ergodic theory and is provided by Gerhard Knieper.

Nonuniform Hyperbolicity and Admissibility

Abstract For a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓp, ℓq), with p and q not necessarily equal. This includes the notion of a nonuniform exponential dichotomy as a very special case. Moreover, we consider a general noninvertible dynamics and a strong nonuniform exponential behavior. The latter is the typical situation from the point of view of smooth ergodic theory.

Pseudo-rotations with sufficiently Liouvillean rotation number are $$C^0$$ C 0 -rigid

It is an open question in smooth ergodic theory whether there exists a Hamiltonian disk map with zero topological entropy and (strong) mixing dynamics. Weak mixing has been known since Anosov and Katok first constructed examples in 1970. Currently all known examples with weak mixing are irrational pseudo-rotations with Liouvillean rotation number on the boundary. Our main result however implies that for a dense subset of Liouville numbers (strong) mixing cannot occur. Our approach involves approximating the flow of a suspension of the given disk map by pseudoholomorphic curves. Ellipticity of the Cauchy–Riemann equation allows quantitative \(L^2\)-estimates to be converted into \(C^0\)-estimates between the pseudoholomorphic curves and the trajectories of the flow on growing time scales. Arithmetic properties of the rotation number enter through these estimates.

A new entropy formula of Ledrappier–Young type for linear toral dynamics

AbstractA detailed Ledrappier–Young theory is presented for linear toral dynamics. First, a proof is given for a simplified definition for local entropies which holds in more general settings besides the current linear dynamics. Then it is shown that the transverse dimensions can be defined directly via the Smale structure of the linear dynamical system. A new entropy formula of Ledrappier–Young type is obtained. The conjecture of Ledrappier and Xie [Vanishing transverse entropy in smooth ergodic theory. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1229–1235] is also discussed for such linear dynamics.

Vanishing transverse entropy in smooth ergodic theory

AbstractFor a measure-preserving transformation, the entropy being zero means that there is no increasingσ-algebra. In this note, we prove that a similar phenomenon occurs forC2diffeomorphisms when considering the increment between the partial entropies associated with different exponents.

On the work of Dolgopyat on partial and nonuniform hyperbolicity

This paper is a nontechnical survey and aims to illustrate Dolgopyat'sprofound contributions to smooth ergodic theory. I will discuss some ofDolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity withemphasis on the interaction between the two-the class of dynamical systemswith mixed hyperbolicity. On one hand, this includes uniformlypartially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in thecenter direction. The study of their ergodic properties has provided analternative approach to the Pugh-Shub stable ergodicity theory for bothconservative and dissipative systems. On the other hand, ideas of mixedhyperbolicity have been used in constructing volume-preservingdiffeomorphisms with nonzero Lyapunov exponents on any manifold.

Preface

This special issue of Discrete and Continuous Dynamical Systems is dedicated to Yakov Pesin on the occasion of his sixtieth birthday, which took place in December of 2006. Pesin is one of the world leaders in the field of dynamical systems. His work exerted a deep and lasting influence in the area. Subjects of his landmark works include nonuniform hyperbolicity, smooth ergodic theory, partial hyperbolicity, thermodynamic formalism, and dimension theory in dynamics. <br><br> For more information please click the “Full Text” above.

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Henon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Henon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism.

Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics

The existence of stable manifolds for nonuniformly hyperbolic trajectories is well know in the case of $C^{1+\alpha}$ dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a $C^1$ diffeomorphism that is not of class $C^{1+\alpha}$ for any $\alpha$ and for which there exists no stable manifold. The $C^{1+\alpha}$ hypothesis appears to be crucial in some parts of smooth ergodic theory, such as for the absolute continuity property and thus in the study of the ergodic properties of the dynamics. Nevertheless, we establish the existence of invariant stable manifolds for nonuniformly hyperbolic trajectories of a large family of maps of class at most $C^1$, by providing a condition which is weaker than the $C^{1+\alpha}$ hypothesis but which is still sufficient to establish a stable manifold theorem. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We note that our proof of the stable manifold theorem is new even in the case of $C^{1+\alpha}$ nonuniformly hyperbolic dynamics. In particular, the optimal $C^1$ smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.

On symmetric logarithm and some old examples in smooth ergodic theory

On symmetric logarithm and some old examples in smooth ergodic theory

Dynamics of random transformations: smooth ergodic theory

A selection of basic results in smooth ergodic theory and in the thermodynamic formalism of dynamical systems generated by compositions of random maps is reviewed in this article.

The Morse spectrum of linear flows on vector bundles

For a linear flow Φ \Phi on a vector bundle π : E → S \pi : E \rightarrow S a spectrum can be defined in the following way: For a chain recurrent component M \mathcal {M} on the projective bundle P E \mathbb {P} E consider the exponential growth rates associated with (finite time) ( ε , T ) (\varepsilon ,T) -chains in M \mathcal {M} , and define the Morse spectrum Σ M o ( M , Φ ) \Sigma _{Mo}(\mathcal {M},\Phi ) over M \mathcal {M} as the limits of these growth rates as ε → 0 \varepsilon \rightarrow 0 and T → ∞ T \rightarrow \infty . The Morse spectrum Σ M o ( Φ ) \Sigma _{Mo}(\Phi ) of Φ \Phi is then the union over all components M ⊂ P E \mathcal {M}\subset \mathbb {P}E . This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec̆ spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that Σ M o ( Φ ) \Sigma _{Mo}(\Phi ) contains all Lyapunov exponents of Φ \Phi for arbitrary initial values, and the Σ M o ( M , Φ ) \Sigma _{Mo}(\mathcal {M},\Phi ) are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that Φ \Phi is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of E E may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in P E \mathbb {P} E ) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledeč, the Lyapunov, and the Morse spectrum. We present an example (flows induced by C ∞ C^{\infty } vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen’s Shadowing Lemma.