One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant cu... more One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of \begin{document}$ d\alpha $\end{document}, where \begin{document}$ d\in \mathbb{N}^* $\end{document} and \begin{document}$ \alpha $\end{document} is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.
The interconnections between financial markets and macroeconomic stability of sovereigns is curre... more The interconnections between financial markets and macroeconomic stability of sovereigns is currently under the spot light. In fact, all members of the European Monetary Union (EMU) experienced in the past years some degree of economic distress caused by chain-reaction mechanisms; however, reactions to financial shocks differ greatly among them. The credit risk score attributed to a country by rating agencies express indeed also the resiliency to financial disturbances. The first goal of this work is to build a quantitative credit risk score that combines macroeconomic and financial variables, which can also help in explaining the country-specific reactions. We select as macroeconomic variables the primary fiscal deficit and the ratio public debt/GDP, that are also subject to limitations after the Maastricht Treaty (1992). We add market yields of government bonds, that provide a timely insight into financial markets’ expectations. Their trend is moreover at the origin of chain-reaction mechanisms: e.g. Greece’s default (debt restructuring) became unavoidable since the country debt was excluded from secondary market. We chose to follow the ideas of Minsky (1992) to connect these quantities by an economic model. In the government budget equation, we forecast future interest expenses thanks to the current market yield, so as to infer whether the state is a Ponzi debtor. Based on this, we define a non-linear credit-score, from which a default probability (PD) is derived. We compare it to the PD implied from the CDS market for a panel of EU countries and we assess whether a cointegration relationship exists between the two time series. In fact, the econometrics of integrated VARs provides long-run equilibrium relationship to be the natural economic interpretation of cointegrating relations (Johansen, 1995). We rely therefore on the existence of such a relation as an indicator of the country weakness in resist against financial shocks, since it shows how a temporary lack of public balance equilibrium affects market confidence in the long run.
The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs an... more The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs and other one-dimensional small divisor problems. The Brjuno functions associated with various continued fractions including the by-excess continued fraction were subsequently investigated, and it was conjectured that the difference between the classical Brjuno function and the even part of the Brjuno function associated with the by-excess continued fraction extends to a Hölder continuous function of the whole real line. In this paper, we prove this conjecture and we extend its validity to the more general case of Brjuno functions with positive exponents. Moreover, we study the Brjuno functions associated to the odd and even continued fractions introduced by Schweiger. We show that they belong to all L spaces, p ≥ 1, and we prove that they differ from the classical Brjuno function by a Hölder continuous function. The odd-odd continued fraction, introduced in the study of the best approxima...
Abstract. We give an explicit arithmetical condition which guarantees the existence of the unstab... more Abstract. We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.
Irrational numbers of bounded type have several equivalent characterizations. They have bounded p... more Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to r-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.
We discuss the quasianalytic properties of various spaces of functions suitable for one-dimension... more We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K_j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K_j's.
We prove that the generator of the renormalization group of Potts models on hierarchical lattices... more We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.
We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.
Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and... more Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for
Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ... more Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ≤ xn < 1, x −1 n = an+1 + xn+1, an+1 ∈ IN, for n ≥ 0. The Brjuno function is then B(x) = ∑∞ x0x1... xn−1 ln(x−1 n=0 n), and the number x satisfies the Brjuno diophantine condition whenever B(x) is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value + ∞ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also...
We prove that the solutions of a cohomological equation of complex dimension one and in the analy... more We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasianalyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by q.
We prove that skew systems with a sufficiently expanding base have approximate exponential decay ... more We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.
We present a simple dynamical model of stock index returns grounded on the ability of the Cyclica... more We present a simple dynamical model of stock index returns grounded on the ability of the Cyclically Adjusted Price Earning valuation ratio devised by Robert Shiller to predict longhorizon performances of the market. Specifically, within the model returns are driven by a fundamental term and an autoregressive component perturbed by external random disturbances. The autoregressive component arises from the agentsi¯ belief that expected returns are higher in bullish markets than in bearish ones. The fundamental value, towards which fundamentalists expect that the current price should revert, varies in time and depends on the initial averaged Price-to-Earnings ratio. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of the stylized dynamics agrees with empirical evidences reported in literature.
One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant cu... more One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of \begin{document}$ d\alpha $\end{document}, where \begin{document}$ d\in \mathbb{N}^* $\end{document} and \begin{document}$ \alpha $\end{document} is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.
The interconnections between financial markets and macroeconomic stability of sovereigns is curre... more The interconnections between financial markets and macroeconomic stability of sovereigns is currently under the spot light. In fact, all members of the European Monetary Union (EMU) experienced in the past years some degree of economic distress caused by chain-reaction mechanisms; however, reactions to financial shocks differ greatly among them. The credit risk score attributed to a country by rating agencies express indeed also the resiliency to financial disturbances. The first goal of this work is to build a quantitative credit risk score that combines macroeconomic and financial variables, which can also help in explaining the country-specific reactions. We select as macroeconomic variables the primary fiscal deficit and the ratio public debt/GDP, that are also subject to limitations after the Maastricht Treaty (1992). We add market yields of government bonds, that provide a timely insight into financial markets’ expectations. Their trend is moreover at the origin of chain-reaction mechanisms: e.g. Greece’s default (debt restructuring) became unavoidable since the country debt was excluded from secondary market. We chose to follow the ideas of Minsky (1992) to connect these quantities by an economic model. In the government budget equation, we forecast future interest expenses thanks to the current market yield, so as to infer whether the state is a Ponzi debtor. Based on this, we define a non-linear credit-score, from which a default probability (PD) is derived. We compare it to the PD implied from the CDS market for a panel of EU countries and we assess whether a cointegration relationship exists between the two time series. In fact, the econometrics of integrated VARs provides long-run equilibrium relationship to be the natural economic interpretation of cointegrating relations (Johansen, 1995). We rely therefore on the existence of such a relation as an indicator of the country weakness in resist against financial shocks, since it shows how a temporary lack of public balance equilibrium affects market confidence in the long run.
The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs an... more The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs and other one-dimensional small divisor problems. The Brjuno functions associated with various continued fractions including the by-excess continued fraction were subsequently investigated, and it was conjectured that the difference between the classical Brjuno function and the even part of the Brjuno function associated with the by-excess continued fraction extends to a Hölder continuous function of the whole real line. In this paper, we prove this conjecture and we extend its validity to the more general case of Brjuno functions with positive exponents. Moreover, we study the Brjuno functions associated to the odd and even continued fractions introduced by Schweiger. We show that they belong to all L spaces, p ≥ 1, and we prove that they differ from the classical Brjuno function by a Hölder continuous function. The odd-odd continued fraction, introduced in the study of the best approxima...
Abstract. We give an explicit arithmetical condition which guarantees the existence of the unstab... more Abstract. We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.
Irrational numbers of bounded type have several equivalent characterizations. They have bounded p... more Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to r-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.
We discuss the quasianalytic properties of various spaces of functions suitable for one-dimension... more We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K_j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K_j's.
We prove that the generator of the renormalization group of Potts models on hierarchical lattices... more We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.
We consider the one-parameter family of interval maps arising from generalized continued fraction... more We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.
Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and... more Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for
Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ... more Abstract. The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z, is given by 0 ≤ xn < 1, x −1 n = an+1 + xn+1, an+1 ∈ IN, for n ≥ 0. The Brjuno function is then B(x) = ∑∞ x0x1... xn−1 ln(x−1 n=0 n), and the number x satisfies the Brjuno diophantine condition whenever B(x) is bounded. Invariant circles under a complex rotation persist when the map is analytically perturbed, if and only if the rotation number satisfies the Brjuno condition, and the same holds for invariant circles in the semi-standard and standard maps cases. In this lecture, we will review some properties of the Brjuno function, and give some generalisations related to familiar diophantine conditions. The Brjuno function is highly singular and takes value + ∞ on a dense set including rationals. We present a regularisation leading to a complex function holomorphic in the upper half plane. Its imaginary part tends to the Brjuno function on the real axis, the real part remaining bounded, and we also...
We prove that the solutions of a cohomological equation of complex dimension one and in the analy... more We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasianalyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by q.
We prove that skew systems with a sufficiently expanding base have approximate exponential decay ... more We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.
We present a simple dynamical model of stock index returns grounded on the ability of the Cyclica... more We present a simple dynamical model of stock index returns grounded on the ability of the Cyclically Adjusted Price Earning valuation ratio devised by Robert Shiller to predict longhorizon performances of the market. Specifically, within the model returns are driven by a fundamental term and an autoregressive component perturbed by external random disturbances. The autoregressive component arises from the agentsi¯ belief that expected returns are higher in bullish markets than in bearish ones. The fundamental value, towards which fundamentalists expect that the current price should revert, varies in time and depends on the initial averaged Price-to-Earnings ratio. We demonstrate both analytically and by means of numerical experiments that the long-run behavior of the stylized dynamics agrees with empirical evidences reported in literature.
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