Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Giuseppe Della Sala [a]

Prolongation of diffeomorphisms and smoothness of invariant submanifolds

Pages: 99-122
Received: 14 January 2019
Accepted: 25 June 2019
Mathematics Subject Classification (2010): 37F99, 32V40.
Keywords: Local diffeomorphism, invariant submanifolds, smoothness and analyticity.
Authors address:
[a]: Department of Mathematics, American University of Beirut, Riad El Sohl, Beirut, Lebanon

Full Text (PDF)

Abstract: We study various questions related to the smoothness of a real submanifold \(M\) which is invariant under a family of real-analytic or holomorphic diffeomorphisms. We show that in various situations it is possible to conclude that \(M\) is necessarily real-analytic (or the same smoothness of the diffeomorphisms involved if these are not analytic). The prolongation method we use also allows to recover some known results by employing relatively simple tools.

M. Abate, An introduction to hyperbolic dynamical systems, I.E.P.I. Pisa 2001.
F. Bracci, Local holomorphic dynamics of diffeomorphisms in dimension one, in ''Five lectures in complex analysis'', Contemp. Math., 525, Amer. Math. Soc., Providence, RI, 2010, 1-42. MR2683218
X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds. III. Overview and applications, J. Differential Equations 218 (2005), 444-515. MR2177465
G. Della Sala, Nowhere analytic smooth curves with non-trivial analytic isotropy, preprint, 2013.
K.-T. Kim and J.-C. Yoccoz, CR manifolds admitting a CR contraction, J. Geom. Anal. 21 (2011), 476-493. MR2772081
D. Repovš, A. B. Skopenkov and E. V. Ščepin, \(C^1\)-homogeneous compacta in \({\mathbb R}^n\) are \(C^1\)-submanifolds of \({\mathbb R}^n\), Proc. Amer. Math. Soc. 124 (1996), 1219-1226. MR1301046
A. B. Skopenkov, A characterization of submanifolds by a homogeneity condition, Topology Appl. 154 (2007), 1894-1897. MR2319261
A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque 358 (2013), 75-165. MR3203217

Home Riv.Mat.Univ.Parma