R. Mañé has given a proof of the C1 Stability Conjecture and conjectured that every element of ℱ1(M) satisfies Axiom A. Here we prove that this conjecture is true.

It is proved that for the map f: ℂ → ℂ given by f(z) = ez, the family is not normal at any point. In particular, f is topologically transitive.

For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.

We consider examples of Finsler metrics symmetric or not) on Sn, Pnℂ, Pnℍ, and P2Ca with only finitely many closed geodesies or with only few short closed geodesies. The number of closed geodesies in these examples and properties of the closed geodesies are considered.

A complete set of computable invariants is given for deciding whether two irreducible subshifts of finite type have topologically equivalent suspension flows.

A minimal subshift $(X,T)$ is linearly recurrent (LR) if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word, $u$, the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a LR subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

It is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:

where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.

In a topological dynamical system $(X,T)$ the complexity function of a cover ${\cal C}$ is the minimal cardinality of a sub-cover of $\bigvee_{i=0}^n T^{-i}{\cal C}$. It is shown that equicontinuous transformations are exactly those such that any open cover has bounded complexity. Call scattering a system such that any finite cover by non-dense open sets has unbounded complexity, and call 2-scattering a system such that any such 2-set cover has unbounded complexity: then all weakly mixing systems are scattering and all 2-scattering systems are totally transitive. Conversely, any system that is not 2-scattering has covers with complexity at most $n+1$. Scattering systems are characterized topologically as those such that their cartesian product with any minimal system is transitive; they are consequently disjoint from all minimal distal systems. Finally, defining $(x,y)$, $x\ne y$, to be a complexity pair if any cover by two non-trivial closed sets separating $x$ from $y$ has unbounded complexity, we prove that 2-scattering systems are disjoint from minimal isometries; that in the invertible case the complexity relation is contained in the regionally proximal relation and, when further assuming minimality, coincides with it up to the diagonal.

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

We providenecessary and sufficient conditions for a suspension flow, over a subshift of finite type, to mix faster than any power of time. Then we show that these conditions are satisfied if the flow has two periodic orbits such that the ratio of the periods cannot be well approximated by rationals.

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

We study the effects of small random errors on the asymptotic distribution of points in the basin of a hyperbolic attractor.

We find asymptotics for the number of cylinders and saddle connections on flat surfaces. These results extend previous results of Veech.

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X-X is locally finite. Such sets are characterized by their patch-counting function N_X (T) of radius T being finite for all T. We formulate conjectures relating slow growth of the patch-counting function N_X (T) to the set X having a non-trivial translation symmetry.

A Delone set X of finite type is repetitive if there is a function M_X(T) such that every closed ball of radius M_X(T) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with \mathbb{R}^n-action. There is a lower bound for M_X(T) in terms of N_X(T), namely M_X(T) \ge c( N_X (T))^{1/n} for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for M_X(T) purely in terms of N_X(T). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M_X (T). For example, the function M_X (T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M_X (T) = O(T) as T \to \infty and is densely repetitive if M_X (T) = O(N_X(T))^{1/n} as T \to \infty. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure. In the reverse direction, we construct a repetitive Delone set X in \mathbb{R}^n which has M_X (T) = O(T (\log T)^{2/n} (\log\log\log T )^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of ‘perfectly ordered quasicrystals’.

We prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

We investigate a variant of the baker's transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.

Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).

Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.