For S-unimodal maps $f$, we study equilibrium states maximizing the free energies $F_t(\mu) := h(\mu) - t\int \log|f'|\,d\mu$ and the pressure function $P(t):=\sup_\mu F_t(\mu)$. It is shown that if $f$ is uniformly hyperbolic on periodic orbits, then $P(t)$ is analytic for $t\approx 1$. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for $t=1$ and of observable measure do not coincide.

Basic results in the rigorous theory of weighted dynamical zeta functions or dynamically defined generalized Fredholm determinants are presented. Analytic properties of the zeta functions or determinants are related to statistical properties of the dynamics via spectral properties of dynamical transfer operators, acting on Banach spaces of observables.

We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

We show that the entropy of a finitely generated pseudogroup (respectively, of a foliation on a compact Riemannian manifold) can be calculated by suitable counting separated pseudo-orbits (respectively, pseudoleaves).

Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories never enter holes under the past iterations of the map is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

Consider a Hamiltonian system with $d$ degrees of freedom whose motions are all linear on tori of some fixed dimension $n\led$ is such a system necessarily completely (or else non-commutatively) integrable? We show that the answer is affirmative under quite broad conditions, but not always, and we provide counterexamples

In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.

In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.

The free shift $\alpha$ of the reduced free product $C^*$-algebra $A$ is studied from both the analytic and non-commutative ergodic theoretic viewpoints. For an automorphism $\beta$ of $B$, we show that the entropy of $\mathop{\rm Ad}\nolimits u(\alpha \otimes \beta)$ is equal to the entropy of $\mathop{\rm Ad}\nolimits u(\beta)$. We also show that if $B$ is unital, nuclear, and simple, and if the crossed product $B \rtimes_\beta {\Bbb Z}$ is simple and purely infinite, then $(O_\infty \otimes B)\rtimes_{\alpha \otimes \beta} {\Bbb Z}$ is isomorphic to $B \rtimes_\beta {\Bbb Z}$.

We give an asymptotic estimate for the number of periodic orbits of an Anosov flow which are subject to multi-dimensional constraints. We also study their spatial distribution. For instance, we describe the distribution of periodic orbits with respect to homology classes, for both homologically full Anosov flows and suspensions of Anosov transformations.

Recently, Goldman [2] proved that the mapping class group of a compact surface $S$, ${\it MCG}(S)$, acts ergodically on each symplectic stratum of the Poisson moduli space of flat $ S(2)$-bundles over $S$, $X(S, S(2))$. We show that this property does not extend to that of cyclic subgroups of ${\it MCG}(S)$, for $S$ a punctured torus. The symplectic leaves of $X(T^2-pt., SU(2))$ are topologically copies of the 2-sphere $S^2$, and we view mapping class actions as a continuous family of discrete Hamiltonian dynamical systems on $S^2$. These deformations limit to finite rotations on the degenerate leaf corresponding to $-{\rm Id}$. boundary holonomy. Standard KAM techniques establish that the action is not ergodic on the leaves in a neighborhood of this degenerate leaf.

We survey some recent progress in the theory of dynamical zeta functions and explain its implications for counting problems.

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 paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, Section I.5]. The main theorem is that if $X$ is a complete, simply connected Riemannian manifold without conjugate points, and $M=X\times R$ is given the Riemannian product metric $g$, then any metric without conjugate points on $M$ which agrees with $g$ outside a compact set is isometric to $g$.

We describe the dynamical behaviour of the entire transcendental non-critically finite function $f_\lambda (z) = \lambda(e^z - 1)/z$, $\lambda > 0$. Our main result is to obtain a computationally useful characterization of the Julia set of $f_\lambda (z)$ as the closure of the set of points with orbits escaping to infinity under iteration, which in turn is applied to the generation of the pictures of the Julia set of $f_\lambda (z)$. Such a characterization was hitherto known only for critically finite entire transcendental functions [11]. We find that bifurcation in the dynamics of $f_\lambda (z)$ occurs at $\lambda = \lambda^{*}$ ($\approx 0.64761$) where $\lambda^\ast = {(x^{*})}^{2} /({e}^{x^{*}} -1)$ and $x^{*}$ is the unique positive real root of the equation $e^{x}(2 -x ) -2 = 0$.

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

We summarize the results of several recent papers, together with a few new results, which rely on a connection between semi-dispersing billiards and non-regular Riemannian geometry. We use this connection to solve several open problems about the existence of uniform estimates on the number of collisions, topological entropy and periodic trajectories of such billiards.

We prove that if $F$ is an orientation-preserving homeomorphism of the plane that leaves invariant an irreducible plane separating continuum $\Delta$, then, with the possible exception of three numbers, if $p/q$ is a reduced rational in the interior of the convex hull of the rotation set of $F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$. This result also applies to certain nonseparating invariant continua.

This paper investigates the dynamical properties of a class of urn processes and recursive stochastic algorithms with constant gain which arise frequently in control, pattern recognition, learning theory, and elsewhere.

It is shown that, under suitable conditions, invariant measures of the process tend to concentrate on the Birkhoff center of irreducible (i.e. chain transitive) attractors of some vector field $F: {\Bbb R}^d \rightarrow {\Bbb R}^d$ obtained by averaging. Applications are given to simple situations including the cases where $F$ is Axiom A or Morse–Smale, $F$ is gradient-like, $F$ is a planar vector field, $F$ has finitely many alpha and omega limit sets.

The generic unfolding of codimension $n$ flips (one eigenvalue $-1$ and the others with norm different from 1) embedded in a Morse–Smale diffeomorphism is analyzed. Local and global bifurcations are described.