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Stefano Marmi

Publications and source records attributed to Stefano Marmi.

50 records · Page 3Linked to original sources
arXiv2012

Linearization of generalized interval exchange maps

A standard interval exchange map is a one-to-one map of the interval which is locally a translation except at finitely many singularities. We define for such maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine arithmetical condition called restricted Roth type which is almost surely satisfied in parameter space. Let $T_0$ be a standard interval exchange map of restricted Roth type, and let $r$ be an integer $\geq 2$. We prove that, amongst $C^{r+3}$ deformations of $T_0$ which are $C^{r+3}$ tangent to $T_0$ at the singularities, those which are conjugated to $T_0$ by a $C^r$ diffeomorphism close to the identity form a $C^1$ submanifold of codimension $(g-1)(2r+1) +s$. Here, $g$ is the genus and $s$ is the number of marked points of the translation surface obtained by suspension of $T_0$. Both $g$ and $s$ can be computed from the combinatorics of $T_0$.

math.DS↗
arXiv2011

Reducibility of cocycles under a Brjuno-Rüssmann arithmetical condition

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Russmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Poschel-Russmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.

math.DS↗
arXiv2011

There is only one KAM curve

We consider the standard family of area-preserving twist maps of the annulus and the corresponding KAM curves. Addressing a question raised by Kolmogorov, we show that, instead of viewing these invariant curves as separate objects, each of which having its own Diophantine frequency, one can encode them in a single function of the frequency which is naturally defined in a complex domain containing the real Diophantine frequencies and which is monogenic in the sense of Borel; this implies a remarkable property of quasianalyticity, a form of uniqueness of the monogenic continuation, although real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances.

math.DS↗
arXiv2011

A quasianalyticity property for monogenic solutions of small divisor problems

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.

math.DS↗
arXiv2009

The entropy of alpha-continued fractions: numerical results

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.

math.DS↗
arXiv2008

Potts models on hierarchical lattices and Renormalization Group dynamics

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.

cond-mat.stat-mech↗
arXiv2008

Linearization of germs: regular dependence on the multiplier

We prove that the linearization of a germ of holomorphic map of the type $F_λ(z)=λ(z+O(z^2))$ has a $ C^1$--holomorphic dependence on the multiplier $λ$. $C^1$--holomorphic functions are $ C^1$--Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar{\partial}$ operator. The linearization is analytic for $|λ|\not= 1$ and the unit circle $S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of $S^1$, namely those points which lie ``far enough from resonances'', i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated spaces of ${\cal C}^1$--holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions, and the corresponding function space is arcwise-quasianalytic. Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points.

math.DS↗
arXiv2007

Generalized Brjuno functions associated to $α$-continued fractions

For αin the interval [0,1], we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α=1) and the nearest integer (α=1/2) and by-excess (α=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_α=(0,max(α,1-α)) we associate a generalized Brjuno function B_(α,u)(x). For α=1/2 or α=1, and u(x)=-\log(x), these functions were introduced by Yoccoz in his work on the linearization of holomorphic maps. Their regularity properties, including BMO regularity and their extension to the complex plane, have been thoroughly investigated. We compare the functions obtained with different values of αand we prove that the set of (α,u)-Brjuno numbers does not depend on the choice of αprovided that α>0. We then consider the case α=0, u(x)=-\log(x) and we prove that x is a Brjuno number (for α> 0) if and only if both x and -x are Brjuno numbers for α=0.

math.DS↗
arXiv2006

On the entropy of Japanese continued fractions

We consider a one-parameter family of expanding interval maps $\{T_α\}_{α\in [0,1]}$ (japanese continued fractions) which include the Gauss map ($α=1$) and the nearest integer and by-excess continued fraction maps ($α={1/2},α=0$). We prove that the Kolmogorov-Sinai entropy $h(α)$ of these maps depends continuously on the parameter and that $h(α) \to 0$ as $α\to 0$. Numerical results suggest that this convergence is not monotone and that the entropy function has infinitely many phase transitions and a self-similar structure. Finally, we find the natural extension and the invariant densities of the maps $T_α$ for $α=\frac{1}{n}$.

math.DS↗
arXiv2004

The cohomological equation for Roth type interval exchange maps

We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation $Ψ-Ψ\circ T=Φ$ has a bounded solution $Ψ$ provided that the datum $Φ$ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The class of interval exchange maps is characterized in terms of a diophantine condition of ``Roth type'' imposed to an acceleration of the Rauzy--Veech--Zorich continued fraction expansion associated to T. CONTENTS 0. Introduction 1. The continued fraction algorithm for interval exchange maps 1.1 Interval exchnge maps 1.2 The continued fraction algorithm 1.3 Roth type interval exchange maps 2. The cohomological equation 2.1 The theorem of Gottschalk and Hedlund 2.2 Special Birkhoff sums 2.3 Estimates for functions of bounded variation 2.4 Primitives of functions of bounded variation 3. Suspensions of interval exchange maps 3.1 Suspension data 3.2 Construction of a Riemann surface 3.3 Compactification of $M_ζ^*$ 3.4 The cohomological equation for higher smoothness 4. Proof of full measure for Roth type 4.1 The basic operation of the algorithm for suspensions 4.2 The Teichmüller flow 4.3 The absolutely continuous invariant measure 4.4 Integrability of $\log\Vert Z_{(1)}\Vert$ 4.5 Conditions (b) and (c) have full measure 4.6 The main step 4.7 Condition (a) has full measure 4.8 Proof of the Proposition Appendix A Roth--type conditions in a concrete family of i.e.m. Appendix B A non--uniquely ergodic i.e.m. satsfying condition (a) References

math.DS↗
arXiv2003

On the cohomological equation for interval exchange maps

We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation $Ψ-Ψ\circ T=Φ$ has a bounded solution $Ψ$ provided that the datum $Φ$ belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The class of interval exchange maps is characterized in terms of a diophantine condition of ``Roth type'' imposed to an acceleration of the Rauzy--Veech--Zorich continued fraction expansion associated to T. Contents 0. French abridged version 1. Interval exchange maps and the cohomological equation. Main Theorem 2. Rauzy--Veech--Zorich continued fraction algorithm and its acceleration 3. Special Birkhoff sums 4. The Diophantine condition 5. Sketch of the proof of the theorem

math.DS↗
arXiv2001

Quasianalytic monogenic solutions of a cohomological equation

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 equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue $q$ of the linear part. This problem has been first investigated by Arnol'd and Herman. Herman raised the question whether the solutions of the cohomological equation had a quasianalytic dependence on the parameter. Indeed they are analytic outside $§^1$ which is a natural boundary but the solutions are still defined at points of $§^1$ which lie ``far enough from resonances''. We adapt to our case Herman's construction of an increasing sequence of compacts which avoid resonances and prove that the solutions belong to the associated space of monogenic functions. The solutions admit asymptotic expansions at the points of $§^1$ which satisfy some arithmetical condition, and Carleman's Theorem allows us to answer negatively to the question of quasianalyticity at these points. But resonances (roots of unity) lead to asymptotic expansions, for which quasianalyticity is obtained as a particular case of Écalle's theory of resurgent functions. At constant-type points one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasianalyticity. Our results are obtained by reducing the problem to the study of a fundamental solution (which is the ``quantum logarithm''). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy series.

math.DS↗