6 research outputs found
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Entropy of Bernoulli convolutions and uniform exponential growth for linear groups
The exponential growth rate of non polynomially growing subgroups of
is conjectured to admit a uniform lower bound. This is known for non-amenable
subgroups, while for amenable subgroups it is known to imply the Lehmer
conjecture from number theory. In this note, we show that it is equivalent to
the Lehmer conjecture. This is done by establishing a lower bound for the
entropy of the random walk on the semigroup generated by the maps , where is an algebraic number. We give a bound
in terms of the Mahler measure of . We also derive a bound on the
dimension of Bernoulli convolutions.Simons Foundation
Royal Society
ER
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The structure theory of nilspaces I
This paper forms the first part of a series by the authors [GMV2,GMV3]
concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy.
A nilspace is a compact space together with closed collections of cubes
, satisfying some natural axioms.
Antol\'in Camarena and Szegedy proved that from these axioms it follows that
(certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of
nilmanifolds. The aim of our project is to provide a new self-contained
treatment of this theory and give new applications to topological dynamics.
This paper provides an introduction to the project from the point of view of
applications to higher order Fourier analysis. We define and explain the basic
definitions and constructions related to cubespaces and nilspaces and develop
the weak structure theory, which is the first stage of the proof of the main
structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace
can be built as a finite tower of extensions where each of the successive
fibers is a compact abelian group.
We also make some modest innovations and extensions to this theory. In
particular, we consider a class of maps that we term fibrations, which are
essentially equivalent to what are termed fiber-surjective morphisms by
Anatol\'in Camarena and Szegedy, and we formulate and prove a relative analogue
of the weak structure theory alluded to above for these maps. These results
find applications elsewhere in the project.Royal Societ
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The structure theory of nilspaces ii: Representation as nilmanifolds
This paper forms the second part of a series by the authors [GMV1,GMV3]
concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy.
A nilspace is a compact space together with closed collections of cubes
, satisfying some natural axioms. From
these axioms it follows that a nilspace can be built as a finite tower of
extensions where each of the successive fibers is a compact abelian group.
Our main result is a new proof of a result due to Antol\'in Camarena and
Szegedy [CS12], stating that if each of these groups is a torus then is
isomorphic (in a strong sense) to a nilmanifold . We also extend the
theorem to a setting where the nilspace arises from a dynamical system .
These theorems are a key stepping stone towards the general structure theorem
in [GMV3] (which again closely resembles the main theorem of [CS12]).
The main technical tool, enabling us to deduce algebraic information from
topological data, consists of existence and uniqueness results for solutions of
certain natural functional equations, again modelled on the theory in [CS12]
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On the dimension of Bernoulli convolutions
The Bernoulli convolution with parameter λ ∈ (0, 1) is the probability measure μλ that is the law of the random variable σn ≥ 0 ±λn, where the signs are independent unbiased coin tosses. We prove that each parameter λ ∈ (1/2, 1) with dimμλ < 1 can be approximated by algebraic parameters η ∈ (1/2, 1) within an error of order exp(-deg(η)A) such that dimμη < 1, for any number A. As a corollary, we conclude that dimμλ = 1 for each of λ = ln 2, e-1/2,π/4. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant a < 1 such that dimμλ = 1 for all λ ∈ (a, 1)