Let $$\{f_\nu \}\subset \mathop {\mathrm {Hol}}\nolimits (X,X)$$ { f ν } ⊂ Hol ( X , X ) be a sequence of holomorphic self-maps of a hyperbolic Riemann surface X. In this paper we shall study the asymptotic behaviour of the sequences obtained by iteratively left-composing or right-composing the maps $$\{f_\nu \}$$ { f ν } ; the sequences of self-maps of X so obtained are called left (respectively, right) iterated function systems. We shall obtain the analogue for left iterated function systems of the theorems proved by Beardon, Carne, Minda and Ng for right iterated function systems with value in a Bloch domain; and we shall extend to the setting of general hyperbolic Riemann surfaces results obtained by Short and the second author in the unit disk $$\mathbb {D}$$ D for iterated function systems generated by maps close enough to a given self-map.
The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map f of the unit disc D. Far less well understood are nonautonomous dynamical systems Fn = fn. . , where fi and gj are holomorphic self-maps of D. Here we obtain a thorough understanding of such systems (Fn) and (Gn) under the assumptions that fn → f and gn → f . We determine when the dynamics of (Fn) and (Gn) mirror that of (f n ), as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map f .
The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map f of the unit disc D. Far less well understood are nonautonomous dynamical systemswhere f i and g j are holomorphic self-maps of D. Here we obtain a thorough understanding of such systems (F n ) and (G n ) under the assumptions that f n → f and g n → f . We determine when the dynamics of (F n ) and (G n ) mirror that of (f n ), as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map f .
This article concerns the locus of locally constant $\textrm{SL}(2,\mathbb{R})$-valued cocycles that have a dominated splitting, called the hyperbolic locus. By developing the theory of Möbius semigroups we show that cocycles on the boundary of the hyperbolic locus, apart from a few exceptions, exhibit some form of hyperbolic behaviour. This behaviour is used to answer a question posed by Avila, Bochi and Yoccoz. Our approach introduces a new locus of cocycles, closely related to the hyperbolic locus, and motivates a line of investigation on the subject.
We prove an inequality which quantifies the idea that a holomorphic self-map of the disc that perturbs two points is close to the identity function.
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