We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurf... more We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurface of flxed points, and we apply it to obtain a complete list of formal normal forms for 2-dimensional holomorphic maps tangential to a curve of flxed points.
... della meccanica statistica, tanto che è possibile definire anche una pressione di un sistema ... more ... della meccanica statistica, tanto che è possibile definire anche una pressione di un sistema dinamico, e questi concetti di entropia e pressione ... 2007), in dimensione 2 i sistemi con tangenza omoclina, benché generici da un punto di vista topologico, capitano con probabilità ...
We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. T... more We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. The results obtained are applied to the case of the Kobayashi metric for which, under suitable hypotheses, we describe the exponential map and the relationship between the indicatrix and small geodesic balls. Finally, exploiting the connection between intrinsic metrics and the complex Monge-AmpSre equation, we give characterizations for circular domains in (Y~.
Journal of the Australian Mathematical Society, 2002
We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class o... more We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behavior of Kobayashi distance and metric, and a new Lindelöf principle.
We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoco... more We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C n , generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing results obtained in the unit ball by Jevtić, Massaneda and Thomas, by Duren and Weir, and by MacCluer.
From this result it is also possible to obtain a geometric characterization of the Kobayashi metr... more From this result it is also possible to obtain a geometric characterization of the Kobayashi metric. The following corollary (Corollary 3.10) improves results of Pang [P] and it is closely related to those of Faran [F]: Corollary 0.3: Let F be a strongly pseudoconvex smooth complete Finsler metric with constant holomorphic curvature K F ≡ −4 and such that H α ≡ 0 for all α. Then F is the Kobayashi metric of M .
We study mapping properties of Toeplitz operators associated to a finite positive Borel measure o... more We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D ⊂⊂ C n . In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A p (D) into A r (D) with r > p, generalizing and making more precise results by Cučković and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoglu for the unit ball.
We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface... more We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré-Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 (C). We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n−1 (C) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighbourhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighbourhood of the origin for a substantial class of 2dimensional holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .
The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the no... more The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls in C n , and by the author to holomorphic maps between strongly (pseudo)convex domains.
... As already remarked by Koranyi [Ko] (and fully exploited by Stein |S] and many others) in sev... more ... As already remarked by Koranyi [Ko] (and fully exploited by Stein |S] and many others) in several complex variables the non-tangential ... However, the generalization of the Lindel6f principle requires another inter-mediate notion of limit, worked out by ~irka [C] (see also [D], [DZ ...
We describe a general construction providing index theorems localizing the Chern classes of the n... more We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.
Let f be a (germ of) holomorphic self-map of C 2 such that the origin is an isolated fixed point,... more Let f be a (germ of) holomorphic self-map of C 2 such that the origin is an isolated fixed point, and such that df O = id. Let ν(f ) be the degree of the first non-vanishing term in the homogeneous expansion of f − id. We generalize to C 2 the classical Leau-Fatou Flower Theorem proving that there exist ν(f ) − 1 holomorphic curves f -invariant, with the origin in their boundary, and attracted by O under the action of f .
We describe a general renormalization procedure for germs of holomorphic (or even formal) self-ma... more We describe a general renormalization procedure for germs of holomorphic (or even formal) self-maps, producing a formal normal form simpler than the classical Poincaré-Dulac normal form. As an example of application of our method we provide a complete list of normal forms for quadratic bi-dimensional superattracting germs, that could not be simplified using the classical Poincaré-Dulac normalization only. Finally, we also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.
Proof. The coherence of the sheaf of ideals of S implies that if { 1 , . . . , k } are generators... more Proof. The coherence of the sheaf of ideals of S implies that if { 1 , . . . , k } are generators of I S,p then the corresponding germs are generators of I S,q for ) = M f ∂ 1 .
We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurf... more We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurface of flxed points, and we apply it to obtain a complete list of formal normal forms for 2-dimensional holomorphic maps tangential to a curve of flxed points.
... della meccanica statistica, tanto che è possibile definire anche una pressione di un sistema ... more ... della meccanica statistica, tanto che è possibile definire anche una pressione di un sistema dinamico, e questi concetti di entropia e pressione ... 2007), in dimensione 2 i sistemi con tangenza omoclina, benché generici da un punto di vista topologico, capitano con probabilità ...
We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. T... more We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. The results obtained are applied to the case of the Kobayashi metric for which, under suitable hypotheses, we describe the exponential map and the relationship between the indicatrix and small geodesic balls. Finally, exploiting the connection between intrinsic metrics and the complex Monge-AmpSre equation, we give characterizations for circular domains in (Y~.
Journal of the Australian Mathematical Society, 2002
We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class o... more We describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behavior of Kobayashi distance and metric, and a new Lindelöf principle.
We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoco... more We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in C n , generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing results obtained in the unit ball by Jevtić, Massaneda and Thomas, by Duren and Weir, and by MacCluer.
From this result it is also possible to obtain a geometric characterization of the Kobayashi metr... more From this result it is also possible to obtain a geometric characterization of the Kobayashi metric. The following corollary (Corollary 3.10) improves results of Pang [P] and it is closely related to those of Faran [F]: Corollary 0.3: Let F be a strongly pseudoconvex smooth complete Finsler metric with constant holomorphic curvature K F ≡ −4 and such that H α ≡ 0 for all α. Then F is the Kobayashi metric of M .
We study mapping properties of Toeplitz operators associated to a finite positive Borel measure o... more We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D ⊂⊂ C n . In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A p (D) into A r (D) with r > p, generalizing and making more precise results by Cučković and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoglu for the unit ball.
We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface... more We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré-Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P 1 (C). We then show how to associate to a homogeneous vector field Q in C n a rank 1 singular holomorphic foliation F of P n−1 (C) and a (partial) meromorphic connection ∇ o along F so that integral curves of Q are described by the geodesic flow of ∇ o along the leaves of F , which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of Q, and of the behavior of the geodesic flow in a neighbourhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighbourhood of the origin for a substantial class of 2dimensional holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in C 2 .
The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the no... more The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls in C n , and by the author to holomorphic maps between strongly (pseudo)convex domains.
... As already remarked by Koranyi [Ko] (and fully exploited by Stein |S] and many others) in sev... more ... As already remarked by Koranyi [Ko] (and fully exploited by Stein |S] and many others) in several complex variables the non-tangential ... However, the generalization of the Lindel6f principle requires another inter-mediate notion of limit, worked out by ~irka [C] (see also [D], [DZ ...
We describe a general construction providing index theorems localizing the Chern classes of the n... more We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.
Let f be a (germ of) holomorphic self-map of C 2 such that the origin is an isolated fixed point,... more Let f be a (germ of) holomorphic self-map of C 2 such that the origin is an isolated fixed point, and such that df O = id. Let ν(f ) be the degree of the first non-vanishing term in the homogeneous expansion of f − id. We generalize to C 2 the classical Leau-Fatou Flower Theorem proving that there exist ν(f ) − 1 holomorphic curves f -invariant, with the origin in their boundary, and attracted by O under the action of f .
We describe a general renormalization procedure for germs of holomorphic (or even formal) self-ma... more We describe a general renormalization procedure for germs of holomorphic (or even formal) self-maps, producing a formal normal form simpler than the classical Poincaré-Dulac normal form. As an example of application of our method we provide a complete list of normal forms for quadratic bi-dimensional superattracting germs, that could not be simplified using the classical Poincaré-Dulac normalization only. Finally, we also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.
Proof. The coherence of the sheaf of ideals of S implies that if { 1 , . . . , k } are generators... more Proof. The coherence of the sheaf of ideals of S implies that if { 1 , . . . , k } are generators of I S,p then the corresponding germs are generators of I S,q for ) = M f ∂ 1 .
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