Strange non-chaotic attractors in quasi-periodically forced circle maps: Diophantine forcing (Q2873993)

The author considers the space \({\mathcal F}\) of \(C^1\)-diffeomorphisms \(F: \mathbb{T}^2 \to {\mathbb{T}^2}\) such that \(\pi_1 \circ F = \pi_1\), where \(\pi_1:{\mathbb{T}^2} \to {\mathbb{T}^1}\) denotes the projection onto the first coordinate. Namely, \(F(\theta, x) = (\theta, f_{\theta}(x))\), where \(\theta \in \mathbb{T}^1\) and so, \(F\) can be viewed as a \(C^1\) collection \(\{f_{\theta}\}_{\theta \in \mathbb{T}^1}\) of \(C^1\) fibre maps \(f_{\theta}\).NEWLINENEWLINETaking into account the vertical Lyapunov exponent, which is computed by \(\lambda = \int _{\mathbb{T}^1} \log |\partial_x f_{\theta}| d \theta\), the author calls the measurable attractor of the map a ``strange non-chaotic attractor'' if \(\lambda <0\).NEWLINENEWLINEThe subject of research lies on the abundance in the space of parameters of strange non-chaotic attractors, when a map of the space \({\mathcal F}\) is quasi-periodically forced by certain frequency \(\omega \in \mathbb{T}^1\). NEWLINENEWLINENEWLINETo explain his results, the author first considers the space \({\mathcal P}\) of \(C^1\) one-parameter families \(\{F_{\tau}\}_{\tau}\) of maps \(F_{\tau} \in {\mathcal F}\). So \(F_{\tau}(\theta, \cdot) = (\theta, f_{\tau, \theta})\). Then, for each value of the parameter \(\tau\), he constructs the quasi-periodically forced (qpf) map \((\theta, x) \mapsto (\theta + \omega, f_{\tau, \theta}(x) )\), where \(\omega\) is the forcing frequency. By hypothesis the forcing frequency satisfies a Diophantine condition.NEWLINENEWLINEThe main purpose of the paper is to prove its Theorem 1.1, which states that for any one-parameter family in a non-empty open set of \({\mathcal P}\), and for any frequency \(\omega\) satisfying the Diophantine condition, the set of parameter values for which the qpf map has a unique strange non-chaotic attractor has positive Lebesgue measure.NEWLINENEWLINEThe proof is performed by a recurrent parameter exclusion method, which takes into account certain dynamically defined critical sets and regions.NEWLINENEWLINEThe author also applies his result to study a qpf modification of the Arnold circle maps.