First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending this result to non-compact manifolds, with modifications based on the ends of the manifold. These results can be thought of as generalizations of the original work which established genericity for the specific property of ergodicity (Oxtoby and Ulam, 1941) and subsequent work for other properties such as weak mixing (Katok and Stepin, 1970). The techniques used to obtain the titled theorem are also applied to related areas, such as fixed point theorems and chaos theory, and some new results are obtained.

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that:

1. U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);

2. f\in U_1 if and only if f is Anosov and transitive;

3. f\in U_2 if and only if f is partially hyperbolic; and

4. f\in U_3 if and only if f has a stable completely elliptic periodic point.

For any stationary process X=X_1,X_2,\dotsc with shift map \sigma, and T an invertible, measure-preserving flow on a probability space, consider the random walk on a random scenery \hat{T}_X(x,\omega)=(\sigma(x),T_{x_0}(\omega)). We prove that if X satisfies a certain property and T has positive entropy, then the number of names in the Vershik metric v_n increases superpolynomially in n. This fact allows us to prove that a class of smooth maps are not standard in the sense of Vershik.

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

We show that, for any compact surface, there is a residual (dense G_{\delta}) set of C^{1} area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mañé, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous SL(2,\mathbb{R})-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

Let G be a connected Lie group and \mathcal{C}(G) be the space of all closed subgroups of G, equipped with the Fell topology. We consider actions on \mathcal{C}(G), of G (by conjugation) and of groups of Lie automorphisms of G (by set operation). We describe conditions under which the orbit of a closed subgroup with finitely many connected components is locally closed (open in its closure) in \mathcal{C}(G). The results are applied to conclude, in particular, that for any ergodic action of G on a standard Borel space the connected components of the identity in almost all the stabilizers are isomorphic to each other, and that if all the stabilizers are assumed to have only finitely many connected components then almost all of them are isomorphic to each other. Various conditions are also described under which almost all the stabilizers are conjugate to each other.

Using the orbital approach to the entropy theory we extend from \mathbb{Z}-actions to general countable amenable group actions T (or provide new short proofs for) the following results: (1) the relative and absolute Krieger Theorem about finite generating partitions (and its infinite Rokhlin counterpart in the case h(T)=\infty); (2) the relative and absolute Sinai Theorem about Bernoullian factors; (3) the Thouvenot Theorem that every intermediate factor of a relatively Bernoullian action is also relatively Bernoullian; (4) the Thouvenot Theorem that a factor of T with the strong Pinsker property enjoys this property; (5) the Smorodinsky–Thouvenot Theorem that T can be spanned by three Bernoullian factors; (6) the Ornstein–Weiss isomorphism theory for Bernoullian actions of the same entropy (provided that they possess generating partitions with at least three elements); and (7) there are uncountably many non-equivalent completely positive entropy extensions of T of the same relative entropy.

In proving these theorems, we were able to bypass the machinery of Orstein and Weiss except for the Rokhlin lemma. It is shown that the language of orbit equivalence relations and their cocycles (unlike the standard dynamical one) is well suited for the inducing operation needed to settle (1), (5) and (6).

The entropy for actions of a finitely-generated nilpotent group G is investigated. The Pinsker algebra of such actions is described explicitly. The systems with completely positive entropy are shown to have a sort of ‘asymptotic independence property’, just as in the case of \mathbb{Z}^d-actions. The invariant partitions are used to prove that the property of completely positive entropy is equivalent to the property of the K-system (the property of the existence of special ‘good’ partitions). A complete spectral characterization of K-systems is given. A construction of examples of completely positive non-Bernoullian actions of general countable nilpotent groups and a class of solvable groups is presented.

Let T:J \to J be an expanding rational map of the Riemann sphere acting on its Julia set J and f:J\to \mathbb{R} denote a Hölder continuous function satisfying f(x) > \log|T^\prime(x)| for all x in J. Then for any point z_0 in J define the set D_{z_0}(f) of ‘well-approximable’ points to be the set of points in J which lie in the Euclidean ball

B\bigg(y,\exp\bigg(-\sum_{i=0}^{n-1} f(T^iy)\bigg)\bigg)

for infinitely many pairs (y,n) satisfying T^n(y)=z_0. In our 1997 paper, we calculated the Hausdorff dimension of D_{z_0} (f). In the present paper, we shall show that the Hausdorff measure \mathcal{H}^s of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.

We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.

Not every Cantor set can arise as the minimal set for a C^1 diffeomorphism of the circle. For example, McDuff has shown that the usual ‘middle thirds’ Cantor set cannot arise in this way. In this paper we exclude a suitable neighborhood of the class of affine Cantor sets.

We prove a relative version of Krieger's Theorem constructing relative generating partitions with cardinality estimated via the relative entropy. We introduce relative asymptotically entropy expansive random transformations and construct for them relative topological generators with cardinality estimated via the relative topological entropy, which enables us to represent such systems by means of random symbolic subshifts.

In this paper, we consider random dynamical systems (abbreviated as RDSs) generated by compositions of one-sided stationary random endomorphisms of class C^2 of a compact manifold. We will first introduce the notions of entropy and Lyapunov exponents for such RDSs, and then prove that the entropy formula of Pesin type holds if the sample measures of an invariant measure are absolutely continuous with respect to the Lebesgue measure on the manifold. Our result covers those obtained by Ledrappier and Young [11], and Liu [12] for i.i.d. random diffeomorphisms or (non-invertible) endomorphisms, and that obtained by [12] for two-sided stationary random endomorphisms. If the phase spaces are compact and finite-dimensional manifolds without boundary, this result may be considered as the almost-final form of Pesin entropy formula for RDSs with absolutely continuous invariant or sample measures.

We show that Anosov systems in manifolds with trivial tangent bundles and with the property that the derivatives of the return maps at periodic orbits are multiples of the identity in the stable and unstable bundles are locally rigid. That is, any other smooth map, in a C^1 neighborhood such that it has the same Jordan normal form at corresponding periodic orbits is smoothly conjugate to it. This generalizes results of Castro and Moriyón (1997). We present several arguments for the main results. In particular, we use quasi-conformal regularity theory. We also extend the examples of an earlier paper of the author (1992) to show that some of the hypotheses we make in this paper are indeed necessary.

Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.

We construct a Teichmüller space for the C^{1+}-conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C^{1+}-conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C^{1+}-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space.

We establish a version of the local product structure (weak local product structure) for ergodic measures \overline{\mu} which are the invertible extension of ergodic weak Gibbs measures \mu invariant under piecewise C^0-invertible (infinite to one) Markov maps T. As a special case, \overline{\mu} possesses asymptotically ‘almost’ local product structure in the sense of Barreira, Pesin and Schmeling. Under piecewise conformality of T and the existence of a piecewise smooth representation of the dual map of T, the weak local product structure allows one to show that the pointwise dimension of \overline{\mu} exists almost everywhere and is the sum of the pointwise dimension of \mu and the pointwise dimension of the dual of \mu. Our results can be applicable to a natural extension of piecewise conformal two-dimensional Markov map which is related to a complex continued fraction and admits indifferent periodic points.