TY - JOUR
T1 - C 0-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces
AU - Brandenbursky, Michael
AU - Khanevsky, Michael
N1 - Publisher Copyright:
© 2022, The Hebrew University of Jerusalem.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - Let Σ be a surface equipped with an area form. There is a long-standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the C0-closure of the set of integrable diffeomorphisms. A slightly weaker version of this question asks: “Does every entropy-zero Hamiltonian diffeomorphism of a surface lie in the C0-closure of the set of autonomous diffeomorphisms?” In this paper we answer in the negative the latter question. In particular, we show that on a surface Σ the set of autonomous Hamiltonian diffeomorphisms is not C0-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.
AB - Let Σ be a surface equipped with an area form. There is a long-standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the C0-closure of the set of integrable diffeomorphisms. A slightly weaker version of this question asks: “Does every entropy-zero Hamiltonian diffeomorphism of a surface lie in the C0-closure of the set of autonomous diffeomorphisms?” In this paper we answer in the negative the latter question. In particular, we show that on a surface Σ the set of autonomous Hamiltonian diffeomorphisms is not C0-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.
UR - http://www.scopus.com/inward/record.url?scp=85143654439&partnerID=8YFLogxK
U2 - 10.1007/s11856-022-2418-z
DO - 10.1007/s11856-022-2418-z
M3 - Article
AN - SCOPUS:85143654439
SN - 0021-2172
VL - 255
SP - 311
EP - 324
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -