Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel's classification of zero entropy maps of S 2 for non-wandering homeomorphisms; we show that if f is a Hamiltonian homeomorphism of the annulus, then the rotation set of f is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks-Misiurewicz Conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms. F. A. Tal was partially supported by CAPES, FAPESP and CNPq-Brasil. 1 arXiv:1503.09127v3 [math.DS] 8 Nov 2017The study of non-contractible periodic orbits for Hamiltonian maps of sympletic manifolds has been receiving increased attention (see for instance [GG]). A natural question in the area, posed by V. Ginzburg, is to determine if the existence of non-contractible periodic points is generic for smooth Hamiltonians. A consequence of Corollary I is an affirmative answer for the case of the torus:Proposition J. Let Ham ∞ (T 2 ) be the set of Hamiltonian C ∞ diffeomorphisms of T 2 endowed with the Whitney C ∞ -topology. There exists a residual subset A of Ham ∞ (T 2 ) such that every f in A has non-contractible periodic points.Let us explain now the results related to the entropy. For example we can give a short proof of the following improvement of a result due to Handel [H1].Theorem K. Let f : S 2 → S 2 be an orientation preserving homeomorphism such that the complement of the fixed point set is not an annulus. If f is topologically transitive then the number of periodic points of period n for some iterate of f grows exponentially in n. Moreover, the entropy of f is positive.Another entropy result we obtain is related to the existence and continuous variation of rotation numbers for homeomorphisms of the open annulus. A stronger version of this result for diffeomorphisms was already proved in an unpublished paper of Handel [H2]. Given a homeomorphism of T 1 × R and a liftf to R 2 , we say that a point z ∈ T 1 × R has a rotation number rot(z) if the ω-limit of its orbit is not empty, and if for any compact set K ⊂ T 1 × R and every increasing sequence of integers n k such that f n k (z) ∈ K and anyž ∈ π −1 (z), lim k→∞ 1 n k π 1 (f n k (ž) − π 1 (ž) = rot(z),where π is the covering projection from R 2 to T 1 × R and π 1 : R 2 → R is the projection on the first coordinate.

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus T 2 which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points Ine(f ) is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points Ess(f ) is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by Jäger. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in Ess(f ) is as rich as in T 2 from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.

Abstract. We construct a C ∞ area-preserving diffeomorphism of the twodimensional torus which is Bernoulli (in particular, ergodic) with respect to Lebesgue measure, homotopic to the identity, and has a lift to the universal covering whose rotation set is {(0, 0)}, which in addition has the property that almost every orbit by the lifted dynamics is unbounded and accumulates in every direction of the circle at infinity.

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a liftf to the infinite stripà which is transitive. We show that, if the rotation numbers of both boundary components of A are strictly positive, then there exists a closed nonempty unbounded setif p 1 is the projection on the first coordinate ofÃ, then there exists d > 0 such that, for anỹIn particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior.