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

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.

Let $A\in SL(2,{\Bbb Z})$ be given and $n\in{\Bbb N}$. $A$ induces a map $A_n$ on ${\Bbb Z}^2_n$ by $(x,y)\mapsto(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$ mod $n$. $A_n$ can be thought of as a discrete approximation of the similarly defined map on the unit square. It is known that the $A_n$ have a surprisingly small period: there always is an $r\le3n$ such that $A^r_n$ is the identity on ${\Bbb Z}^2_n$.

One can visualize the properties of the iterates of $A_n$ by considering the sets $P,A_nP,A^2_nP,\ldots$ for a fixed subset $P$ of ${\Bbb Z}^2_n$. (Here the cat comes into play: $P$ has often been chosen to be the digitalized picture of a cat, in particular when treating the special case of the Anosov map $A={2\ 1\atopwithdelims() 1\ 1}$.) It turns out that in the sequence $(A^k_nP)_k$ mostly ‘chaotic’ pictures occur as one would expect. However, some kind of regular behaviour can also be observed for certain powers $A^k_n$ before the end of the period.

It is the aim of this paper to describe these phenomena and to explain why and how they occur. In order to quantify the chaotic behaviour of the maps on ${\Bbb Z}^2_n$ under consideration, we introduce suitable functions.

Our methods are elementary. Only basic results from the geometry of numbers will be used.

Integer $m\times m$ matrices $A$ with determinant $1$ define diffeomorphisms of the $m$-dimensional torus $T^m=({\Bbb R}/{\Bbb Z})^m$ into itself. Likewise, they define bijective self-maps of the discretized tori $({\Bbb Z}/n{\Bbb Z})^m=({\Bbb z}_n)^m$. We present estimates of the surprisingly low order (or period) $\Per_A(n)$ of the iteration $A^r, r=1,2,3,\ldots,$ on the discretized torus $({\Bbb z}_n)^m$. We obtain $\Per_A(n)\leq 3n$ for dimension $m=2$. In the special case of the Anosov map $A={2\ 1 \atopwithdelims() 1\ 1}$, this result is due to Dyson and Talk [DT92]. For arbitrary dimensions $m>2$ we obtain $$\Per_A(n)\leq {\rm constant}\cdot n^{m-1},$$ provided $n$ is a power of a prime number. For general $n$, number theoretic problems arise.

Let $F:X\rightarrow X$ be a $C^k(X)$, $k=[0, \infty]$, map on a topological space (smooth manifold) $X$, $A:X\rightarrow \End(C^m)$ and let $\{U_\alpha\}$ be an $F\mbox{-invariant}$ covering of $X$. We introduce spaces of cohomologies associated with $\{U_\alpha\}$ and an operator $T=I - R$, where $(R\phi)(x)=A(x)\phi(F(x))$ is a weighted substitution operator in $C^k(X)$. This yields a correspondence between $\Im T$ and $\Im T|U_\alpha$ and the description of $\Im T$ in cohomological terms. In particular, it is proven that for any structurally stable diffeomorphism on a circle and for large enough $k$, the operator $T$ is semi-Fredholm, and a similar result holds for the substitution operators generated by simple multidimensional maps. On the other hand, we show that, in general, the closures of $\Im T$ and $\Im T|U_\alpha$ are independent.

For each irreducible hyperbolic automorphism $A$ of the $n$-torus we construct a sofic system $(\Sigma,\sigma)$ and a bounded-to-one continuous semiconjugacy from $(\Sigma,\sigma)$ to $({\Bbb T}^n,A)$. This construction is natural in the sense that it depends only on the characteristic polynomial of $A$ and, furthermore, it has an arithmetic interpretation.

Let $F$ be a nonquasi-unipotent one-parameter (cyclic) subgroup of a unimodular Lie group $G$, $\Gamma$ a discrete subgroup of $G$. We prove that for certain classes of subsets $Z$ of the homogeneous space $G/\Gamma$, the set of points in $G/\Gamma$ with $F$-orbits staying away from $Z$ has full Hausdorff dimension. From this we derive applications to geodesic flows on manifolds of constant negative curvature.

We use the concept of enveloping Ellis semigroups in order to classify the behaviour of dynamical systems on intervals. Several notions of chaos and non-chaos are expressed by means of algebraic properties of the corresponding Ellis semigroups.

In this paper we consider a smooth dynamical system $f$ and give estimates of the growth rates of vector fields and differential forms in the $L_p$ norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy $$h_{\rm top}=\lim_{n\to\infty} \frac 1n \log \int \Vert Df_x^n\,^{\wedge}\Vert \,dx,$$ where $Df_x^n\,^{\wedge}$ is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.

In this paper we derive some properties of a variety of entropy that measures rotational complexity of annulus homeomorphisms, called asymptotic or rotational entropy. In previous work [KS] the authors showed that positive asymptotic entropy implies the existence of infinitely many periodic orbits corresponding to an interval of rotation numbers. In our main result, we show that a Hölder $C^1$ diffeomorphism with nonvanishing asymptotic entropy is isotopic rel a finite set to a pseudo-Anosov map. We also prove that the closure of the set of recurrent points supports positive asymptotic entropy for a ($C^0$) homeomorphism with nonzero asymptotic entropy.

It is shown that systems with hyperbolic structure have the Bernoulli property. Some new results on smooth cross-sections of hyperbolic Bernoulli flows are also derived. The proofs involve an abstract version of our original methods for showing that the geodesic flow on surfaces of negative curvature are Bernoulli.

We study the local connectedness of the Julia set of the family $f_b(z)=z^m(z^n-b)$. Assuming that the Julia set is connected and depending on whether $m=1$ or $m>1$, a sufficient condition for the local connectivity of the Julia set is obtained. This complements the known results of Yoccoz–Hubbard–Branner for $m=n=1$.

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.

We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.

In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.

Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

We show in the setting of measure-theoretical dynamical systems that certain singular integral operators and Toeplitz operators with the same coefficients but represented on different $L^2$-spaces are $C^\ast $-algebraically equivalent.

The proofs of Theorems 5.1 and 7.1 of [2] contain a gap. We will show below how to close it under some suitable additional assumptions in these theorems and their corollaries. We will assume the notation of [2] throughout. In particular, $\mu$ is a measure invariant and ergodic under an $R^k$-action $\alpha$. Let us first explain the gap. Both theorems are proved by establishing a dichotomy for the conditional measures of $\mu$ along the intersection of suitable stable manifolds. They were either atomic or invariant under suitable translation or unipotent subgroups $U$. Atomicity eventually led to zero entropy. Invariance of the conditional measures showed invariance of $\mu$ under $U$. We then claimed that $\mu$ was algebraic using, respectively, unique ergodicity of the translation subgroup on a rational subtorus or Ratner's theorem (cf. [2, Lemma 5.7]). This conclusion, however, only holds for the $U$-ergodic components of $\mu$ which may not equal $\mu$. In fact, in the toral case, the $R^k$-action may have a zero-entropy factor such that the conditional measures along the fibers are Haar measures along a foliation by rational subtori. Since invariant measures with zero entropy have not been classified, we cannot conclude algebraicity of the total measure $\mu$ at this time. In the toral case, the existence of zero entropy factors turns out to be precisely the obstruction to our methods. The case of Weyl chamber flows is somewhat different as the ‘Haar’ direction of the measure may not be integrable. In this case, we need to use additional information coming from the semisimplicity of the ambient Lie group to arrive at the versions of Theorem 7.1 presented below.