In the present paper we introduce positive flows and processes, which generalize ordinary dynamical systems and stochastic processes. The theory of positive flows and processes is based on the methods of positive operator theory and operator algebras theory. Its basic concepts are the phase and positive algebras, the spectral potential, dual entropy, equilibrium measures, action functional, sensitive states, and empirical measures. Within the frames of the theory we prove the law of large numbers with respect to the sensitive states and calculate asymptotics for the probabilities of large deviations in terms of the action functional.

We establish the analogue for maps on trees of the result established by Bobok (Studia Math.152 (2002), 249–261 and Studia Math.166 (2005), 11–27) for interval maps, that a continuous self-map for which all but countably many points have at least $m$ preimages (and none have less than two) has topological entropy bounded below by $\log m$.

By definition, an $\omega$-residually free tower is positive-genus if all surfaces used in its construction are of positive-genus. We prove that every limit group is virtually a subgroup of a positive-genus, $\omega$-residually free tower. By combining this construction with results of Gaboriau, we prove that elementarily free groups are measure-equivalent to free groups.

The classical KAM theorem establishes the persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. A long-standing question is to what extent the perturbed tori are unique. Using the fact that at the level of tori there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine torus union of full measure.

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

For polynomials $f$ on the complex plane with a dendrite Julia set we study invariant probability measures, obtained from a reference measure. To do this we follow Keller [K1] in constructing canonical Markov extensions. We discuss ‘liftability’ of measures (both $f$-invariant and non-invariant) to the Markov extension, showing that invariant measures are liftable if and only if they have a positive Lyapunov exponent. We also show that $\delta$-conformal measure is liftable if and only if the set of points with positive Lyapunov exponent has positive measure.

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede–Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ‘fundamental homoclinic point’ and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $\tau$. We prove that, if $\Gamma$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimes\Gamma$ also has the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed product von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup property. Moreover, if the actions have property $(\tau)$, then the latter factors are full.

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb P\mathit{Curr}(F_k)$ that is either $\mathit{Out}(F_k)$-equivariant or $\mathit{Out}(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the ‘length function’ boundary of Culler–Vogtmann's Outer space $CV(F_k)$, and $\mathbb P\mathit{Curr}(F_k)$ is the space of projectivized geodesic currents for $F_{k}$. We also prove that, if $k\ge 3$, for the action of $\mathit{Out}(F_k)$ on $\mathbb P\mathit{Curr}(F_{k})$ and for the diagonal action of $\mathit{Out}(F_k)$ on the product space $\partial CV(F_k)\times \mathbb P\mathit{Curr}(F_k)$, there exist unique non-empty minimal closed $\mathit{Out}(F_k)$-invariant sets. Our results imply that for $k\ge 3$ any continuous $\mathit{Out}(F_k)$-equivariant embedding of $CV(F_k)$ into $\mathbb P\mathit{Curr}(F_k)$ (such as the Patterson–Sullivan embedding) produces a new compactification of Outer space, different from the usual ‘length function” compactification $\overline{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.

Given a compact surface $F$ with non-empty boundary and a homotopy class of self-maps on $F$, we consider the problem of finding a representative which carries the minimal number of fixed points possible for the class. When $F$ is either the pants surface or the once punctured Möbius band this problem is solved by a graph map, where the graph is homotopy equivalent to $F$. A partial result is presented for the remaining two surfaces whose fundamental group is the free group on two generators.

We generalize the classical result of J. Mather stating the existence of a drift orbit inside a region of instability of an exact-symplectic positive twist map, to the case of a finite family ${\cal F}$ of such maps. A special case is the case where the maps $F\in{\cal F}$ have no common invariant continuous graph. We prove the existence of a sequence $(z_i)_{i\in\mathbb{Z}}$ in $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$ and of a sequence $(F_i)_{i\in\mathbb{Z}}$ in ${\cal F}$ such that

\[ z_{i+1}=F_i(z_i),\quad \lim_{i\to -\infty} p_2(z_i)=-\infty,\quad \lim_{i\to +\infty} p_2(z_i)=+\infty, \]

where $p_{2} :(x,y)\mapsto y$ is the second projection.

In a previous paper (Presentations of subshifts and their topological conjugacy invariants. Doc. Math.4 (1999), 285–340), the notion of $\lambda$-graph system has been introduced. The $\lambda$-graph systems are generalizations of finite directed labeled graphs. In this paper, we study periodic points of $\lambda$-graph systems. We introduce some invariants for a $\lambda$-graph system $\mathfrak L$ to count the cardinal number of $p$-periodic points of $\mathfrak L$. They are invariant under strong shift equivalence of $\lambda$-graph systems. We then consider the zeta functions of $\lambda$-graph systems, which are also invariant under strong shift equivalence of $\lambda$-graph systems. Some examples are also presented.

In the 1970s, Nekhorochev proved that, for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians that are non-steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors. We generalize this notion for an arbitrary integrable Hamiltonian and prove Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents. Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonians which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.

Let $(X,d,T)$ be a dynamical system, where $(X,d)$ is a compact metric space and $T:X\rightarrow X$ a continuous map. We introduce two conditions for the set of orbits, called respectively the $\texttt{g}$-almost product property and the uniform separation property. The $\texttt{g}$-almost product property holds for dynamical systems with the specification property, but also for many others. For example, all $\beta$-shifts have the $\texttt{g}$-almost product property. The uniform separation property is true for expansive and more generally asymptotically $h$-expansive maps. Under these two conditions we compute the topological entropy of saturated sets. If the uniform separation condition does not hold, then we can compute the topological entropy of the set of generic points, and show that for any invariant probability measure $\mu$, the (metric) entropy of $\mu$ is equal to the topological entropy of generic points of $\mu$. We give an application of these results to multi-fractal analysis and compare our results with those of Takens and Verbitskiy (Ergod. Th. & Dynam. Sys.23 (2003), 317–348).

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

In this paper we will show that, for any strong orbit equivalence class for Cantor minimal systems, the class contains (minimal) subshifts which attain arbitrary finite topological entropy.

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

We define and study a new invariant called pre-image pressure and its relationship with invariant measures. More precisely, for a given dynamical system $(X,f)$ (where $X$ is a compact metric space and $f$ is a continuous map from $X$ to itself) and $\varphi\in C(X,R)$ (the space of real-valued continuous functions on $X$), we prove a variational principle for pre-image pressure $P_{\rm pre}(f,\varphi),\ P_{\rm pre}(f,\varphi) = \sup_{\mu \in \mathcal{M}(f)} \{ h_{{\rm pre},\mu}(f) + \int \varphi \,d\mu\}$, where $h_{{\rm pre},\mu}(f)$ is the pre-image entropy (W.-C. Cheng and S. Newhouse. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113) and $\mathcal{M}(f)$ is the set of invariant measures of $f$. Moreover, we also prove that pre-image pressure determines the invariant measures and give some applications of pre-image pressure to equilibrium states.