arXiv subjects

Carlo Carminati

Publications and source records attributed to Carlo Carminati.

16 recordsLinked to original sources

arXiv2026

On the minimum of $σ$-Brjuno functions

$σ$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/σ},$ $(σ>0).$ As in the classical case, $B_σ$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_σ$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $σ=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $ [0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $σ$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $σ$ varies.

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arXiv2025

Global and local minima of $α$-Brjuno functions

The main goal of this article is to analyze some peculiar features of the global (and local) minima of $α$-Brjuno functions $B_α$ where $α\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_α$ for other values of $α$: we show that for $α\in (g,1)$ the global minimum is again attained at $g$, while for $α$ in a neighbourhood of $1/2$ the function $B_α$ attains its minimum at $γ:=\sqrt{2}-1$. The fact that the minimum of $B_α$ is attained when $α$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_α$ is lower semicontinuous for all rational $α,$ but we also exhibit an irrational $α$ for which $B_α$ is not lower semicontinuous. %We also prove that if $α$ is rational then $B_α$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_α$ is not lower semicontinuous for a suitable irrational $α.$

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arXiv2024

Regularity properties of the $α$-Wilton functions

The aim of this article is to study the regularity properties of the Wilton functions $W_α$ associated with $α$-continued fractions. We prove that the Wilton function is BMO for $α\in[1-g,g]$ (where $g:=\frac{\sqrt{5}-1}{2}$ denotes the golden number), and we show that this result is optimal, since we find that on any left neighbourhood of $1-g$ and on any right neighbourhood of $g$ there are values $α$ for which $W_α$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $α$-continued fractions called ``matching''. Our results complete those of Marmi--Moussa--Yoccoz (1997) and of Lee--Marmi--Petrykiewicz--Schindler (2024), where it is proven that Wilton function is BMO for, respectively, $α=1/2$ (\cite{MaMoYo_97}) and $α\in[\frac{1}{2},g]$ (\cite{LeMar_24}).

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arXiv2021

The bifurcation locus for numbers of bounded type

We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of a corresponding section of the bifurcation set E.

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arXiv2020

Tanaka-Ito $α$-continued fractions and matching

Two closely related families of $α$-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter $α$ has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito $α$-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada's $α$-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka-Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients.

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arXiv2020

On the regularity of Mather's $β$-function for standard-like twist maps

We consider the minimal average action (Mather's $β$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $β$-function associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.

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arXiv2017

Matching in a family of piecewise affine maps

We consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching

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arXiv2016

Matching for generalised $β$-transformations

We investigate matching for the family $T_α(x) = βx + α\pmod 1$, $α\in [0,1]$, for fixed $β> 1$. Matching refers to the property that there is an $n \in \mathbb N$ such that $T_α^n(0) = T_α^n(1)$. We show that for various Pisot numbers $β$, matching occurs on an open dense set of $α\in [0,1]$ and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent.

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arXiv2015

The local Hoelder exponent for the dimension of invariant subsets of the circle

We consider for each t the set K(t) of points of the circle whose forward orbit for the doubling map does not intersect (0,t), and look at the dimension function eta(t) := H.dim K(t). We prove that at every bifurcation parameter t, the local Hoelder exponent of the dimension function equals the value of the function eta(t) itself. The same statement holds by replacing the doubling map with the map g(x) := dx mod 1 for d >2.

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arXiv2013

Continued fractions with SL(2, Z)-branches: combinatorics and entropy

We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

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arXiv2012

Tuning and plateaux for the entropy of $α$-continued fractions

The entropy $h(T_α)$ of $α$-continued fraction transformations is known to be locally monotone outside a closed, totally disconnected set $\EE$. We will exploit the explicit description of the fractal structure of $\EE$ to investigate the self-similarities displayed by the graph of the function $α\mapsto h(T_α)$. Finally, we completely characterize the plateaux occurring in this graph, and classify the local monotonic behaviour.

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arXiv2012

Dynamics of continued fractions and kneading sequences of unimodal maps

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

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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.

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arXiv2010

A canonical thickening of Q and the dynamics of continued fractions

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of alpha-continued fractions, thus proving a conjecture of Nakada and Natsui.

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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.

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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.

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