Fractal Weyl law for skew extensions of expanding maps
A natural class of partially hyperbolic dynamical systems can be constructed as follows. Let \(E:\mathbb{S}^1\rightarrow\mathbb{S}^1\) be a smooth expanding map, let \(G\) be a compact Lie group with normalized Haar measure \(m\), and let \(\tau:\mathbb{S}^1\rightarrow G\) be a smooth map. One can define NEWLINE\[NEWLINE\forall(x,g)\in\mathbb{S}^1\times G,\;\hat{E}_{\tau}(x,g):=(E(x),\tau(x)g).NEWLINE\]NEWLINE This extension of the map \(E\) has a \(1\)-dimensional expanding direction, and a neutral direction corresponding to \(G\). It provides one of the simplest models of partially hyperbolic dynamical systems, and one can try to understand the ergodic properties of this map, such as decay of correlations. For instance, it is known that the smooth invariant SRB measures associated to \(E\) is mixing, and it is natural to ask if mixing still holds for the extension with respect to the invariant measure \(\mu\times m\) (at least under some dynamical conditions). Some quantitative results on these questions were first obtained by \textit{D. Dolgopyat} [Isr. J. Math. 130, 157--205 (2002; Zbl 1005.37005)], and many improvements have been obtained since this article.NEWLINENEWLINEThe article under review deals with this question in the case where \(G\) is the simplest non-abelian Lie group, i.e., \(G=\mathrm{SU}(2)\). The case \(G=\mathrm{U}(1)\) was treated in [\textit{F. Faure}, Nonlinearity 24, No. 5, 1473--1498 (2011; Zbl 1233.37018)]. The main objective of the article is to describe spectral properties of the transfer operator \(\mathcal{F}_{\tau}\) associated to the map \(\hat{E}_{\tau}\) using semiclassical techniques. For that purpose, the author uses harmonic analysis to decompose the operator \(\mathcal{F}_{\tau}\) into a sum of operators \(\hat{F}_{j}\) acting on simpler spaces \(L^2(\mathbb{S}^1)\times \mathcal{D}_j\), indexed by the irreducible finite-dimensional unitary representations of \(G\). Then, he studies the spectral properties of the operators \(\hat{F}_{j}\) on the Hilbert spaces \(H^m(\mathbb{S}^1)\times\mathcal{D}_{j}\), where \(H^m(\mathbb{S}^1)\) is the standard Sobolev space with \(m<0\).NEWLINENEWLINEUsing semiclassical techniques, he shows that, for every \(j\) and every \(m\), the operator \(\hat{F}_{\alpha}\) acting on \(H^m(\mathbb{S}^1)\times\mathcal{D}_{j}\) has discrete spectrum outside a disc of radius \(r_m\). Moreover, \(r_m\rightarrow 0\) as \(m\rightarrow -\infty\), and the discrete sprectrum is independent of \(m\) (Theorem \(1\)). Then, he proves that if the map is partially captive, then there is a spectral gap in the spectrum in the semiclassical limit \(j\rightarrow +\infty\), and the spectral gap can be expressed in terms of a minimal expanding rate (Theorem \(2\)). From this last result, one can deduce exponential mixing of the map \(\hat{E}_{\tau}\) (Corollary \(3\)). Finally, he obtains a fractal Weyl upper bound (involving the dimension of the trapped set) for the counting function of resonances in the semiclassical limit \(j\rightarrow +\infty\) (Theorem \(5\)).NEWLINENEWLINEThe proofs are based on techniques which were developped in the context of chaotic scattering, and which were recently used to study spectral properties of transfer operators [Faure, loc. cit.; \textit{F. Faure} et al., Open Math. J. 1, 35--81 (2008; Zbl 1177.37032); Commun. Math. Phys. 308, No. 2, 325--364 (2011; Zbl 1260.37016)].
scientific article; zbMATH DE number 6052374