Geometry and entropy of generalized rotation sets

Abstract

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.

The thermodynamic approach to multifractal analysis

“…Nevertheless, many of the same techniques can be used: the thermodynamic approach was applied in [14] to show that differentiability of the appropriate cross-section of the pressure function still leads to a conditional variational principle (see also [4,32]). A related notion of 'rotation sets' is also studied; see [75], for example.…”

Entropy and rotation sets: A toy model approach

“…, φ n dµ the rotation vector of µ with respect to Φ. Following [15,12,20] we call the set of rotation vectors of all f -invariant probability measures the (generalized) rotation set of f with respect to the m-dimensional potential Φ (see Sec. 2.1 for details and further references).…”

“…Let now Φ be a general m-dimensional continuous potential and let w be a point in the rotation set of Φ. Following [15,20], we call h m (w) = sup{h µ (f ) : rv Φ (µ) = w} (1) the localized entropy at w with respect to Φ (see Sec. 2.2 for details and references).…”

“…In [20] we study the relationship between the pointwise, ergodic and generalized rotation sets of Φ. In particular, we construct explicit examples showing that both of the inclusions in (7) can be strict.…”

“…For example, these sets coincide for topologically mixing subshifts of finite type, expansive homeomorphisms with specification, axiom A basic sets. On the other hand, the inclusion Rot E (Φ) ⊂ Rot Pt (Φ) maybe strict even for shift maps (see [20]). …”