Ergodic Theory of Generic Continuous Maps
- First Online:
- 324 Downloads
We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps.
To further explore the mysterious behaviour of C0 generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.
Unable to display preview. Download preview PDF.
- AHK.Akin, E., Hurley, M., Kennedy, J.A.: Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783), viii+130, Providence RI: Amer. Math. Soc., 2003Google Scholar
- AP.Prasad, S., Alpern, V.S.: Typical dynamics of volume preserving homeomorphisms. Volume 139 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2000Google Scholar
- Bow.Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Volume 470 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, revised edition, 2008, with a preface by David Ruelle, Edited by Jean-René ChazottesGoogle Scholar
- Con.Conley, C.: Isolated invariant sets and the Morse index, Volume 38 of CBMS Regional Conference Series in Mathematics. Providence, RI: Amer. Math. Soc., 1978Google Scholar
- DGS.Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, Vol. 527. Berlin: Springer-Verlag, 1976Google Scholar
- Mis.Misiurewicz, M.: Ergodic natural measures. In: Algebraic and topological dynamics, Volume 385 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2005, pp. 1–6Google Scholar
- Moi.Moise, E.E.: Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics 47, New York: Springer-Verlag, 1977Google Scholar
- Pal.Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque, (261):xiii–xiv, 335–347, 2000Google Scholar
- Rue2.Ruelle, D.: Historical behaviour in smooth dynamical systems. In: Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 63–66Google Scholar