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On the computability of rotation sets and their entropies
- Source :
- Ergodic Theory and Dynamical Systems. 40:367-401
- Publication Year :
- 2018
- Publisher :
- Cambridge University Press (CUP), 2018.
-
Abstract
- Let$f:X\rightarrow X$be a continuous dynamical system on a compact metric space$X$and let$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$be an$m$-dimensional continuous potential. The (generalized) rotation set$\text{Rot}(\unicode[STIX]{x1D6F7})$is defined as the set of all$\unicode[STIX]{x1D707}$-integrals of$\unicode[STIX]{x1D6F7}$, where$\unicode[STIX]{x1D707}$runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy$\unicode[STIX]{x210B}(w)$to each$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where$f$is a subshift of finite type. We prove that$\text{Rot}(\unicode[STIX]{x1D6F7})$is computable and that$\unicode[STIX]{x210B}(w)$is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general,$\unicode[STIX]{x210B}$is not continuous on the boundary of the rotation set when considered as a function of$\unicode[STIX]{x1D6F7}$and$w$. In particular,$\unicode[STIX]{x210B}$is, in general, not computable at the boundary of$\text{Rot}(\unicode[STIX]{x1D6F7})$.
- Subjects :
- Applied Mathematics
General Mathematics
010102 general mathematics
Boundary (topology)
Function (mathematics)
Topological entropy
Subshift of finite type
01 natural sciences
Combinatorics
Compact space
0103 physical sciences
010307 mathematical physics
0101 mathematics
Invariant (mathematics)
Dynamical system (definition)
Mathematics
Probability measure
Subjects
Details
- ISSN :
- 14694417 and 01433857
- Volume :
- 40
- Database :
- OpenAIRE
- Journal :
- Ergodic Theory and Dynamical Systems
- Accession number :
- edsair.doi...........b7997b1e33c7d323211b8275ac16e574