How Long to Read Quasianalytic Monogenic Solutions of a Cohomological Equation

How Long Does it Take to Read Quasianalytic Monogenic Solutions of a Cohomological Equation?

It takes the average reader 1 hour and 23 minutes to read Quasianalytic Monogenic Solutions of a Cohomological Equation by S. Marmi

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Description

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 quasi analyticity. 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 $q$. Borel's theory of non-analytic monogenic functions has been first investigated by Arnold and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation.Herman raised the question whether the solutions of the cohomological equation had a quasi analytic dependence on the parameter $q$. Indeed they are analytic for $q\in\mathbb{C}\setminus\mathbb{S}^1$, the unit circle $\S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of $\mathbb{S}^1$ 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 of our equation belong to the associated space of monogenic functions; some general properties of these monogenic functions and particular properties of the solutions are then studied.For instance the solutions are defined and admit asymptotic expansions at the points of $\mathbb{S}^1$ which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasi analyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasi analyticity is obtained as a particular case of Ecalle's theory of resurgent functions.And at constant-type points, where no quasi analytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasi analyticity. Our results are obtained by reducing the problem, by means of Hadamard's product, to the study of a fundamental solution (which turns out to be the so-called $q$-logarithm or 'quantum logarithm'). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasi analytic properties of a certain number-theoretical Borel-Wolff-Denjoy series.

How long is Quasianalytic Monogenic Solutions of a Cohomological Equation?

Quasianalytic Monogenic Solutions of a Cohomological Equation by S. Marmi is 83 pages long, and a total of 20,999 words.

This makes it 28% the length of the average book. It also has 26% more words than the average book.

How Long Does it Take to Read Quasianalytic Monogenic Solutions of a Cohomological Equation Aloud?

The average oral reading speed is 183 words per minute. This means it takes 1 hour and 54 minutes to read Quasianalytic Monogenic Solutions of a Cohomological Equation aloud.

What Reading Level is Quasianalytic Monogenic Solutions of a Cohomological Equation?

Quasianalytic Monogenic Solutions of a Cohomological Equation is suitable for students ages 10 and up.

Note that there may be other factors that effect this rating besides length that are not factored in on this page. This may include things like complex language or sensitive topics not suitable for students of certain ages.

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