For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (φ1, ..., φm) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f -invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of R m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in R m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits RotP t(Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = RotP t(Φ). Finally, we study the entropy function w → H(w), w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (subshifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w → H(w) is real-analytic in the interior of the rotation set.

It is shown that on certain Banach spaces, including C[0, 1] and L 1 [0, 1], there is no strongly continuous semigroup (T t ) 0

We study sectorial operators with a special type of functional calculus, which we term an absolute functional calculus. A typical example of such an operator is an invertible operator A (defined on a Banach space X ) considered on the real interpolation space (Dom(A), X ) θ, p where 0 < θ < 1 and 1 < p < ∞. In general the absolute functional calculus can be characterized in terms of real interpolation spaces. We show that operators of this type have a strong form of the H ∞ -calculus and behave very well with respect to the joint functional calculus. We give applications of these results to recent work of Arendt, Batty and Bu on the existence of Hölder-continuous solutions for the abstract Cauchy problem.

Given a continuous dynamical system f on a compact metric space X and a continuous potential Φ : X → R m , the generalized rotation set is the subset of R m consisting of all integrals of Φ with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.

We consider a continuous dynamical system f : X → X on a compact metric space X equipped with an m-dimensional continuous potential Φ = (φ1, · · · , φm) : X → R m . We study the set of ground states GS(α) of the potential α · Φ as a function of the direction vector α ∈ S m−1 . We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of Φ. In particular, for each α the set of rotation vectors of GS(α) forms a non-empty, compact and connected subset of a face Fα(Φ) of the rotation set associated with α. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in Fα(Φ). We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any m ∈ N examples with an exposed boundary point (i.e. Fα(Φ) being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face Fα(Φ) with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of GS(α) is a non-trivial line segment.