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PRODID:-//Google Inc//Google Calendar 70.9054//EN
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DESCRIPTION:One world numeration seminar seminar of IRIF
NAME:One world numeration seminar
REFRESH-INTERVAL:PT4H
TIMEZONE-ID:Europe/Paris
X-WR-CALDESC:One world numeration seminar seminar of IRIF
X-WR-CALNAME:One world numeration seminar
X-WR-TIMEZONE:Europe/Paris
BEGIN:VTIMEZONE
TZID:Europe/Paris
X-LIC-LOCATION:Europe/Paris
BEGIN:DAYLIGHT
DTSTART:19700329T020000
RRULE:FREQ=YEARLY;BYDAY=-1SU;BYMONTH=3
TZNAME:CEST
TZOFFSETFROM:+0100
TZOFFSETTO:+0200
END:DAYLIGHT
BEGIN:STANDARD
DTSTART:19701025T030000
RRULE:FREQ=YEARLY;BYDAY=-1SU;BYMONTH=10
TZNAME:CET
TZOFFSETFROM:+0200
TZOFFSETTO:+0100
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
SUMMARY:On the smallest base in which a number has a unique expansion - Pi
eter Allaart\, University of North Texas
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201110T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201110T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1208
DESCRIPTION:For x>0\, let U(x) denote the set of bases q in (1\,2] such th
at x has a unique expansion in base q over the alphabet {0\,1}\, and let f
(x)=inf U(x). I will explain that the function f(x) has a very complicated
structure: it is highly discontinuous and has infinitely many infinite le
vel sets. I will describe an algorithm for numerically computing f(x) that
often gives the exact value in just a small finite number of steps. The K
omornik-Loreti constant\, which is f(1)\, will play a central role in this
talk. This is joint work with Derong Kong\, and builds on previous work b
y Kong (Acta Math. Hungar. 150(1):194-208\, 2016).
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:The carry propagation of the successor function - Jacques Sakarovi
tch\, IRIF\, CNRS et Télécom Paris
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201117T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201117T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1209
DESCRIPTION:Given any numeration system\, the carry propagation at an inte
ger N is the number of digits that change between the representation of N
and N+1. The carry propagation of the numeration system as a whole is the
average carry propagations at the first N integers\, as N tends to infinit
y\, if this limit exists. \n\nIn the case of the usual base p numeration s
ystem\, it can be shown that the limit indeed exists and is equal to p/(p-
1). We recover a similar value for those numeration systems we consider an
d for which the limit exists. \nThe problem is less the computation of the
carry propagation than the proof of its existence. We address it for vari
ous kinds of numeration systems: abstract numeration systems\, rational ba
se numeration systems\, greedy numeration systems and beta-numeration. Thi
s problem is tackled with three different types of techniques: combinatori
al\, algebraic\, and ergodic\, each of them being relevant for different k
inds of numeration systems. \n\nThis work has been published in Advances i
n Applied Mathematics 120 (2020). In this talk\, we shall focus on the alg
ebraic and ergodic methods. \n\nJoint work with V. Berthé (Irif)\, Ch. Fr
ougny (Irif)\, and M. Rigo (Univ. Liège).
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Rauzy fractal unbounded in all directions of the plane - Mélodi
e Andrieu\, Aix-Marseille University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201027T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201027T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1210
DESCRIPTION:Until 2001 it was believed that\, as for Sturmian words\, the
imbalance of Arnoux-Rauzy words was bounded - or at least finite. Cassaign
e\, Ferenczi and Zamboni disproved this conjecture by constructing an Arno
ux-Rauzy word with infinite imbalance\, i.e. a word whose broken line devi
ates regularly and further and further from its average direction. Today\,
we hardly know anything about the geometrical and topological properties
of these unbalanced Rauzy fractals. The Oseledets theorem suggests that th
ese fractals are contained in a strip of the plane: indeed\, if the Lyapun
ov exponents of the matricial product associated with the word exist\, one
of these exponents at least is nonpositive since their sum equals zero. T
his talk aims at disproving this belief.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Distinct unit generated number fields and finiteness in number sys
tems - Tomáš Vávra\, University of Waterloo
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201103T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201103T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1212
DESCRIPTION:A distinct unit generated field is a number field K such that
every algebraic integer of the field is a sum of distinct units. In 2015\,
Dombek\, Masáková\, and Ziegler studied totally complex quartic fields\
, leaving 8 cases unresolved. Because in this case there is only one funda
mental unit u\, their method involved the study of finiteness in positiona
l number systems with base u and digits arising from the roots of unity in
K. \nFirst\, we consider a more general problem of positional representat
ions with base beta with an arbitrary digit alphabet D. We will show that
it is decidable whether a given pair (beta\, D) allows eventually periodic
or finite representations of elements of O_K. \nWe are then able to prove
the conjecture that the 8 remaining cases indeed are distinct unit genera
ted.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Representations for complex numbers with integer digits - Paul Sur
er\, University of Natural Resources and Life Sciences\, Vienna
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201020T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201020T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1213
DESCRIPTION:In this talk we present the zeta-expansion as a complex versio
n of the well-known beta-expansion. It allows us to expand complex numbers
with respect to a complex base by using integer digits. Our concepts fits
into the framework of the recently published rotational beta-expansions.
But we also establish relations with piecewise affine maps of the torus an
d with shift radix systems.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Representations of real numbers on fractal sets - Kan Jiang\, Ning
bo University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201013T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201013T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1214
DESCRIPTION:There are many approaches which can represent real numbers. Fo
r instance\, the β-expansions\, the continued fraction and so forth. Repr
esentations of real numbers on fractal sets were pioneered by H. Steinhaus
who proved in 1917 that C+C=[0\,2] and C−C=[−1\,1]\, where C is the m
iddle-third Cantor set. Equivalently\, for any x ∈ [0\,2]\, there exist
some y\,z ∈ C such that x=y+z. In this talk\, I will introduce similar r
esults in terms of some fractal sets.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Finiteness and periodicity of continued fractions over quadratic n
umber fields - Francesco Veneziano\, University of Genova
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201006T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201006T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1215
DESCRIPTION:We consider continued fractions with partial quotients in the
ring of integers of a quadratic number field K\; a particular example of t
hese continued fractions is the β-continued fraction introduced by Bernat
. We show that for any quadratic Perron number β\, the β-continued fract
ion expansion of elements in Q(β) is either finite of eventually periodic
. We also show that for certain four quadratic Perron numbers β\, the β-
continued fraction represents finitely all elements of the quadratic field
Q(β)\, thus answering questions of Rosen and Bernat. \nBased on a joint
work with Zuzana Masáková and Tomáš Vávra.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Random matching for random interval maps - Marta Maggioni\, Leiden
University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200929T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200929T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1216
DESCRIPTION:In this talk we extend the notion of matching for deterministi
c transformations to random matching for random interval maps. For a large
class of piecewise affine random systems of the interval\, we prove that
this property of random matching implies that any invariant density of a s
tationary measure is piecewise constant. We provide examples of random mat
ching for a variety of families of random dynamical systems\, that include
s generalised beta-transformations\, continued fraction maps and a family
of random maps producing signed binary expansions. We finally apply the pr
operty of random matching and its consequences to this family to study min
imal weight expansions. \nBased on a joint work with Karma Dajani and Char
lene Kalle.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Multiscale Substitution Tilings - Yotam Smilansky\, Rutgers Univer
sity
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200922T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200922T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1217
DESCRIPTION:Multiscale substitution tilings are a new family of tilings of
Euclidean space that are generated by multiscale substitution rules. Unli
ke the standard setup of substitution tilings\, which is a basic object of
study within the aperiodic order community and includes examples such as
the Penrose and the pinwheel tilings\, multiple distinct scaling constants
are allowed\, and the defining process of inflation and subdivision is a
continuous one. Under a certain irrationality assumption on the scaling co
nstants\, this construction gives rise to a new class of tilings\, tiling
spaces and tiling dynamical system\, which are intrinsically different fro
m those that arise in the standard setup. In the talk I will describe thes
e new objects and discuss various structural\, geometrical\, statistical a
nd dynamical results. \nBased on joint work with Yaar Solomon.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lazy Ostrowski Numeration and Sturmian Words - Jeffrey Shallit\, U
niversity of Waterloo
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200915T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200915T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1218
DESCRIPTION:In this talk I will discuss a new connection between the so-ca
lled "lazy Ostrowski" numeration system\, and periods of the prefixes of S
turmian characteristic words. I will also give a relationship between peri
ods and the so-called "initial critical exponent". This builds on work of
Frid\, Berthé-Holton-Zamboni\, Epifanio-Frougny-Gabriele-Mignosi\, and ot
hers\, and is joint work with Narad Rampersad and Daniel Gabric.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Some fractal problems in beta-expansions - Bing Li\, South China U
niversity of Technology
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200908T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200908T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1219
DESCRIPTION:For greedy beta-expansions\, we study some fractal sets of rea
l numbers whose orbits under beta-transformation share some common propert
ies. For example\, the partial sum of the greedy beta-expansion converges
with the same order\, the orbit is not dense\, the orbit is always far fro
m that of another point etc. The usual tool is to approximate the beta-tra
nsformation dynamical system by Markov subsystems. We also discuss the sim
ilar problems for intermediate beta-expansions.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hotspot Lemmas for Noncompact Spaces - Bill Mance\, Adam Mickiewic
z University in Poznań
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200901T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200901T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1220
DESCRIPTION:We will explore a correction of several previously claimed gen
eralizations of the classical hotspot lemma. Specifically\, there is a com
mon mistake that has been repeated in proofs going back more than 50 years
. Corrected versions of these theorems are increasingly important as there
has been more work in recent years focused on studying various generaliza
tions of the concept of a normal number to numeration systems with infinit
e digit sets (for example\, various continued fraction expansions\, the L
üroth series expansion and its generalizations\, and so on). Also\, highl
ighting this (elementary) mistake may be helpful for those looking to stud
y these numeration systems further and wishing to avoid some common pitfal
ls.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On diophantine properties of generalized number systems - finite a
nd periodic representations - Attila Pethő\, University of Debrecen
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200714T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200714T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1221
DESCRIPTION:In this talk we investigate elements with special patterns in
their representations in number systems in algebraic number fields. We con
centrate on periodicity and on the representation of rational integers. We
prove under natural assumptions that there are only finitely many S-units
whose representation is periodic with a fixed period. We prove that the s
ame holds for the set of values of polynomials at rational integers.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Analogy of Lagrange spectrum related to geometric progressions - H
ajime Kaneko\, University of Tsukuba
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200707T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200707T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1222
DESCRIPTION:Classical Lagrange spectrum is defined by Diophantine approxim
ation properties of arithmetic progressions. The theory of Lagrange spectr
um is related to number theory and symbolic dynamics. In our talk we intro
duce significantly analogous results of Lagrange spectrum in uniform distr
ibution theory of geometric progressions. In particular\, we discuss the g
eometric sequences whose common ratios are Pisot numbers. For studying the
fractional parts of geometric sequences\, we introduce certain numeration
system. \nThis talk is based on a joint work with Shigeki Akiyama.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Continued fractions with two non integer digits - Niels Langeveld\
, Leiden University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200630T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200630T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1223
DESCRIPTION:In this talk\, we will look at a family of continued fraction
expansions for which the digits in the expansions can attain two different
(typically non-integer) values\, named α1 and α2 with α1α2 ≤ 1/2 .
If α1α2 < 1/2 we can associate a dynamical system to these expansions wi
th a switch region and therefore with lazy and greedy expansions. We will
explore the parameter space and highlight certain values for which we can
construct the natural extension (such as a family for which the lowest dig
it cannot be followed by itself). We end the talk with a list of open prob
lems.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Univoque bases of real numbers: local dimension\, Devil's staircas
e and isolated points - Derong Kong\, Chongqing University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200623T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200623T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1224
DESCRIPTION:Given a positive integer M and a real number x\, let U(x) be t
he set of all bases q in (1\,M+1] such that x has a unique q-expansion wit
h respect to the alphabet {0\,1\,...\,M}. We will investigate the local di
mension of U(x) and prove a 'variation principle' for unique non-integer b
ase expansions. We will also determine the critical values and the topolog
ical structure of U(x).
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Approximations of the Lagrange and Markov spectra - Carlos Matheus
\, CNRS\, École Polytechnique
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200616T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200616T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1225
DESCRIPTION:The Lagrange and Markov spectra are closed subsets of the posi
tive real numbers defined in terms of diophantine approximations. Their to
pological structures are quite involved: they begin with an explicit discr
ete subset accumulating at 3\, they end with a half-infinite ray of the fo
rm [4.52...\,∞)\, and the portions between 3 and 4.52... contain complic
ated Cantor sets. In this talk\, we describe polynomial time algorithms to
approximate (in Hausdorff topology) these spectra.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Equidistribution results for self-similar measures - Simon Baker\,
University of Birmingham
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200609T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200609T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1226
DESCRIPTION:A well known theorem due to Koksma states that for Lebesgue al
most every x>1 the sequence (x^n) is uniformly distributed modulo one. In
this talk I will discuss an analogue of this statement that holds for frac
tal measures. As a corollary of this result we show that if C is equal to
the middle third Cantor set and t≥1\, then almost every x in C+t is such
that (x^n) is uniformly distributed modulo one. Here almost every is with
respect to the natural measure on C+t.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linear repetition in polytopal cut and project sets - Henna Koivus
alo\, University of Vienna
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200602T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200602T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1227
DESCRIPTION:Cut and project sets are aperiodic point patterns obtained by
projecting an irrational slice of the integer lattice to a subspace. One w
ay of classifying aperiodic sets is to study repetition of finite patterns
\, where sets with linear pattern repetition can be considered as the most
ordered aperiodic sets. \nRepetitivity of a cut and project set depends o
n the slope and shape of the irrational slice. The cross-section of the sl
ice is known as the window. In an earlier work it was shown that for cut a
nd project sets with a cube window\, linear repetitivity holds if and only
if the following two conditions are satisfied: (i) the set has minimal co
mplexity and (ii) the irrational slope satisfies a certain Diophantine con
dition. In a new joint work with Jamie Walton\, we give a generalisation o
f this result for other polytopal windows\, under mild geometric condition
s. A key step in the proof is a decomposition of the cut and project schem
e\, which allows us to make sense of condition (ii) for general polytopal
windows.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ergodic behavior of transformations associated with alternate base
expansions - Célia Cisternino\, University of Liège
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200526T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200526T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1228
DESCRIPTION:We consider a p-tuple of real numbers greater than 1\, beta=(b
eta_1\,…\,beta_p)\, called an alternate base\, to represent real numbers
. Since these representations generalize the beta-representation introduce
d by Rényi in 1958\, a lot of questions arise. In this talk\, we will stu
dy the transformation generating the alternate base expansions (greedy rep
resentations). First\, we will compare the beta-expansion and the (beta_1*
…*beta_p)-expansion over a particular digit set and study the cases when
the equality holds. Next\, we will talk about the existence of a measure
equivalent to Lebesgue\, invariant for the transformation corresponding to
the alternate base and also about the ergodicity of this transformation.
\nThis is a joint work with Émilie Charlier and Karma Dajani.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On singular substitution Z-actions - Boris Solomyak\, University o
f Bar-Ilan
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200519T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200519T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1229
DESCRIPTION:We consider primitive aperiodic substitutions on d letters and
the spectral properties of associated dynamical systems. In an earlier wo
rk we introduced a spectral cocycle\, related to a kind of matrix Riesz pr
oduct\, which extends the (transpose) substitution matrix to the d-dimensi
onal torus. The asymptotic properties of this cocycle provide local inform
ation on the (fractal) dimension of spectral measures. In the talk I will
discuss a sufficient condition for the singularity of the spectrum in term
s of the top Lyapunov exponent of this cocycle. \nThis is a joint work wit
h A. Bufetov.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preservation of normality by selection - Olivier Carton\, Universi
té de Paris
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200512T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200512T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1230
DESCRIPTION:We first recall Agafonov's theorem which states that finite st
ate selection preserves normality. We also give two slight extensions of t
his result to non-oblivious selection and suffix selection. We also propos
e a similar statement in the more general setting of shifts of finite type
by defining selections which are compatible with the shift.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ostrowski numeration and repetitions in words - Narad Rampersad\,
University of Winnipeg
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20200505T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20200505T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1231
DESCRIPTION:One of the classical results in combinatorics on words is Deje
an's Theorem\, which specifies the smallest exponent of repetitions that a
re avoidable on a given alphabet. One can ask if it is possible to determi
ne this quantity (called the repetition threshold) for certain families of
infinite words. For example\, it is known that the repetition threshold f
or Sturmian words is 2+phi\, and this value is reached by the Fibonacci wo
rd. Recently\, this problem has been studied for balanced words (which gen
eralize Sturmian words) and rich words. The infinite words constructed to
resolve this problem can be defined in terms of the Ostrowski-numeration s
ystem for certain continued-fraction expansions. They can be viewed as Ost
rowski-automatic sequences\, where we generalize the notion of k-automatic
sequence from the base-k numeration system to the Ostrowski numeration sy
stem.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rigid fractal tilings - Michael Barnsley\, Australian National Uni
versity
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201201T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201201T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1242
DESCRIPTION:I will describe recent work\, joint with Louisa Barnsley and A
ndrew Vince\, concerning a symbolic approach to self-similar tilings. This
approach uses graph-directed iterated function systems to analyze both cl
assical tilings and also generalized tilings of what may be unbounded frac
tal subsets of R^n. A notion of rigid tiling systems is defined. Our key t
heorem states that when the system is rigid\, all the conjugacies of the t
ilings can be described explicitly. In the seminar I hope to prove this fo
r the case of standard IFSs.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Limit theorems on counting large continued fraction digits - Tanja
Isabelle Schindler\, Scuola Normale Superiore di Pisa
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201208T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201208T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1243
DESCRIPTION:We establish a central limit theorem for counting large contin
ued fraction digits (a_n)\, that is\, we count occurrences {a_n>b_n}\, whe
re (b_n) is a sequence of positive integers. Our result improves a similar
result by Philipp\, which additionally assumes that bn tends to infinity.
Moreover\, we also show this kind of central limit theorem for counting t
he number of occurrences entries such that the continued fraction entry li
es between d_n and d_n (1+1/c_n) for given sequences (c_n) and (d_n). For
such intervals we also give a refinement of the famous Borel–Bernstein t
heorem regarding the event that the nth continued fraction digit lying inf
initely often in this interval. As a side result\, we explicitly determine
the first φ-mixing coefficient for the Gauss system - a result we actual
ly need to improve Philipp's theorem. This is joint work with Marc Kesseb
öhmer.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:The digits of n+t - Lukas Spiegelhofer\, Montanuniversität Leoben
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20201215T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20201215T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1262
DESCRIPTION:We study the binary sum-of-digits function s_2 under addition
of a constant t. For each integer k\, we are interested in the asymptotic
density δ(k\,t) of integers t such that s_2(n+t) - s_2(n) = k. In this ta
lk\, we consider the following two questions.\n\n(1) Do we have c_t = δ(0
\,t) + δ(1\,t) + ... > 1/2? This is a conjecture due to T. W. Cusick (201
1).\n\n(2) What does the probability distribution defined by k → δ(k\,t
) look like?\n\nWe prove that indeed c_t > 1/2 if the binary expansion of
t contains at least M blocks of contiguous ones\, where M is effective. Ou
r second theorem states that δ(j\,t) usually behaves like a normal distri
bution\, which extends a result by Emme and Hubert (2018). \n\nThis is joi
nt work with Michael Wallner (TU Wien).
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:alpha-odd continued fractions - Claire Merriman\, Ohio State Unive
rsity
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210105T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210105T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1269
DESCRIPTION:The standard continued fraction algorithm come from the Euclid
ean algorithm. We can also describe this algorithm using a dynamical syste
m of [0\,1)\, where the transformation that takes x to the fractional part
of 1/x is said to generate the continued fraction expansion of x. From th
ere\, we ask two questions: What happens to the continued fraction expansi
on when we change the domain to something other than [0\,1)? What happens
to the dynamical system when we impose restrictions on the continued fract
ion expansion\, such as finding the nearest odd integer instead of the flo
or? This talk will focus on the case where we first restrict to odd intege
rs\, then start shifting the domain [α-2\, α). \n\nThis talk is based on
joint work with Florin Boca and animations done by Xavier Ding\, Gustav J
ennetten\, and Joel Rozhon as part of an Illinois Geometry Lab project.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernoulli Convolutions and Measures on the Spectra of Algebraic In
tegers - Tom Kempton\, University of Manchester
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210119T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210119T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1270
DESCRIPTION:Given an algebraic integer beta and alphabet A = {-1\,0\,1}\,
the spectrum of beta is the set \n\n \\Sigma(\\beta) := \\{ \\sum_{i=
1}^n a_i \\beta^i : n \\in \\mathbb{N}\, a_i \\in A \\}. \n\nIn the case t
hat beta is Pisot one can study the spectrum of beta dynamically using sub
stitutions or cut and project schemes\, and this allows one to see lots of
local structure in the spectrum. There are higher dimensional analogues f
or other algebraic integers. \nIn this talk we will define a random walk o
n the spectrum of beta and show how\, with appropriate renormalisation\, t
his leads to an infinite stationary measure on the spectrum. This measure
has local structure analagous to that of the spectrum itself. Furthermore\
, this measure has deep links with the Bernoulli convolution\, and in part
icular new criteria for the absolute continuity of Bernoulli convolutions
can be stated in terms of the ergodic properties of these measures.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prevalence of matching for families of continued fraction algorith
ms: old and new results - Carlo Carminati\, Università di Pisa
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210126T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210126T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1291
DESCRIPTION:We will give an overview of the phenomenon of matching\, which
was first observed in the family of Nakada's α-continued fractions\, but
is also encountered in other families of continued fraction algorithms. \
n\nOur main focus will be the matching property for the family of Ito-Tana
ka continued fractions: we will discuss the analogies with Nakada's case (
such as prevalence of matching)\, but also some unexpected features which
are peculiar of this case. \n\nThe core of the talk is about some recent r
esults obtained in collaboration with Niels Langeveld and Wolfgang Steiner
.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Interplay between finite topological rank minimal Cantor systems\,
S-adic subshifts and their complexity - Samuel Petite\, Université de Pi
cardie Jules Verne
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210202T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210202T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1292
DESCRIPTION:The family of minimal Cantor systems of finite topological ran
k includes Sturmian subshifts\, coding of interval exchange transformation
s\, odometers and substitutive subshifts. They are known to have dynamical
rigidity properties. In a joint work with F. Durand\, S. Donoso and A. Ma
ass\, we provide a combinatorial characterization of such subshifts in ter
ms of S-adic systems. This enables to obtain some links with the factor co
mplexity function and some new rigidity properties depending on the rank o
f the system.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Multiplicative automatic sequences - Clemens Müllner\, TU Wien
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210209T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210209T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1293
DESCRIPTION:It was shown by Mariusz Lemańczyk and the author that automat
ic sequences are orthogonal to bounded and aperiodic multiplicative functi
ons. This is a manifestation of the disjointedness of additive and multipl
icative structures. We continue this path by presenting in this talk a com
plete classification of complex-valued sequences which are both multiplica
tive and automatic. This shows that the intersection of these two worlds h
as a very special (and simple) form. This is joint work with Mariusz Lema
ńczyk and Jakub Konieczny.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Good's Theorem for Hurwitz Continued Fractions - Gerardo González
Robert\, Universidad Nacional Autónoma de México
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210216T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210216T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1294
DESCRIPTION:In 1887\, Adolf Hurwitz introduced a simple procedure to write
any complex number as a continued fraction with Gaussian integers as part
ial denominators and with partial numerators equal to 1. While similaritie
s between regular and Hurwitz continued fractions abound\, there are impor
tant differences too (for example\, as shown in 1974 by R. Lakein\, Serret
's theorem on equivalent numbers does not hold in the complex case). In th
is talk\, after giving a short overview of the theory of Hurwitz continued
fractions\, we will state and sketch the proof of a complex version of I.
J. Good's theorem on the Hausdorff dimension of the set of real numbers w
hose regular continued fraction tends to infinity. Finally\, we will discu
ss some open problems.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Odd-odd continued fraction algorithm - Seulbee Lee\, Scuola Normal
e Superiore di Pisa
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210223T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210223T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1295
DESCRIPTION:The classical continued fraction gives the best approximating
rational numbers of an irrational number. We define a new continued fracti
on\, say odd-odd continued fraction\, which gives the best approximating r
ational numbers whose numerators and denominators are odd. We see that a j
ump transformation associated to the Romik map induces the odd-odd continu
ed fraction. We discuss properties of the odd-odd continued fraction expan
sions. This is joint work with Dong Han Kim and Lingmin Liao.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Normal sets in (ℕ\,+) and (ℕ\,×) - Vitaly Bergelson\, Ohio St
ate University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210302T160000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210302T170000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1310
DESCRIPTION:We will start with discussing the general idea of a normal set
in a countable cancellative amenable semigroup\, which was introduced and
developed in the recent paper "A fresh look at the notion of normality" (
joint work with Tomas Downarowicz and Michał Misiurewicz). We will move t
hen to discussing and juxtaposing combinatorial and Diophantine properties
of normal sets in semigroups (ℕ\,+) and (ℕ\,×). We will conclude the
lecture with a brief review of some interesting open problems.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:The flow view and infinite interval exchange transformation of a r
ecognizable substitution - Natalie Priebe Frank\, Vassar College
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210309T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210309T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1311
DESCRIPTION:A flow view is the graph of a measurable conjugacy between a s
ubstitution or S-adic subshift or tiling space and an exchange of infinite
ly many intervals in [0\,1]. The natural refining sequence of partitions o
f the sequence space is transferred to [0\,1] with Lebesgue measure using
a canonical addressing scheme\, a fixed dual substitution\, and a shift-in
variant probability measure. On the flow view\, sequences are shown horizo
ntally at a height given by their image under conjugacy. \n\nIn this talk
I'll explain how it all works and state some results and questions. There
will be pictures.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Double rotations and their ergodic properties - Alexandra Skripche
nko\, Higher School of Economics
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210316T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210316T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1312
DESCRIPTION:Double rotations are the simplest subclass of interval transla
tion mappings. A double rotation is of finite type if its attractor is an
interval and of infinite type if it is a Cantor set. It is easy to see tha
t the restriction of a double rotation of finite type to its attractor is
simply a rotation. It is known due to Suzuki - Ito - Aihara and Bruin - Cl
ark that double rotations of infinite type are defined by a subset of zero
measure in the parameter set. We introduce a new renormalization procedur
e on double rotations\, which is reminiscent of the classical Rauzy induct
ion. Using this renormalization we prove that the set of parameters which
induce infinite type double rotations has Hausdorff dimension strictly sma
ller than 3. Moreover\, we construct a natural invariant measure supported
on these parameters and show that\, with respect to this measure\, almost
all double rotations are uniquely ergodic. In my talk I plan to outline t
his proof that is based on the recent result by Ch. Fougeron for simplicia
l systems. I also hope to discuss briefly some challenging open questions
and further research plans related to double rotations. \n\nThe talk is ba
sed on a joint work with Mauro Artigiani\, Charles Fougeron and Pascal Hub
ert.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arithmetic averages and normality in continued fractions - Godofre
do Iommi\, Pontificia Universidad Católica de Chile
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210323T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210323T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1313
DESCRIPTION:Every real number can be written as a continued fraction. Ther
e exists a dynamical system\, the Gauss map\, that acts as the shift in th
e expansion. In this talk\, I will comment on the Hausdorff dimension of t
wo types of sets: one of them defined in terms of arithmetic averages of t
he digits in the expansion and the other related to (continued fraction) n
ormal numbers. In both cases\, the non compactness that steams from the fa
ct that we use countable many partial quotients in the continued fraction
plays a fundamental role. Some of the results are joint work with Thomas J
ordan and others together with Aníbal Velozo.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:(Logarithmic) Densities for Automatic Sequences along Primes and S
quares - Michael Drmota\, TU Wien
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210330T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210330T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1314
DESCRIPTION:It is well known that the every letter α of an automatic sequ
ence a(n) has a logarithmic density -- and it can be decided when this log
arithmic density is actually a density. For example\, the letters 0 and 1
of the Thue-Morse sequences t(n) have both frequences 1/2. [The Thue-Morse
sequence is the binary sum-of-digits functions modulo 2.] \n\nThe purpose
of this talk is to present a corresponding result for subsequences of gen
eral automatic sequences along primes and squares. This is a far reaching
generalization of two breakthrough results of Mauduit and Rivat from 2009
and 2010\, where they solved two conjectures by Gelfond on the densities o
f 0 and 1 of t(p_n) and t(n^2) (where p_n denotes the sequence of primes).
\n\nMore technically\, one has to develop a method to transfer density re
sults for primitive automatic sequences to logarithmic-density results for
general automatic sequences. Then as an application one can deduce that t
he logarithmic densities of any automatic sequence along squares (n^2)_{n
≥0} and primes (p_n)_{n≥1} exist and are computable. Furthermore\, if
densities exist then they are (usually) rational. \n\nThis is a joint work
with Boris Adamczewski and Clemens Müllner.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Measure theoretic entropy of random substitutions - Andrew Mitchel
l\, University of Birmingham
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210413T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210413T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1364
DESCRIPTION:
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Metrical theory for the set of points associated with the generali
zed Jarnik-Besicovitch set - Ayreena Bakhtawar\, La Trobe University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210420T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210420T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1365
DESCRIPTION:From Lagrange's (1770) and Legendre's (1808) results we conclu
de that to find good rational approximations to an irrational number we on
ly need to focus on its convergents. Let [a_1(x)\,a_2(x)\,…] be the cont
inued fraction expansion of a real number x ∈ [0\,1). The Jarnik-Besicov
itch set in terms of continued fraction consists of all those x ∈ [0\,1)
which satisfy a_{n+1}(x) ≥ e^{τ (log|T'x|+⋯+log|T'(T^{n-1}x)|)} for
infinitely many n ∈ N\, where a_{n+1}(x) is the (n+1)-th partial quotien
t of x and T is the Gauss map. In this talk\, I will focus on determining
the Hausdorff dimension of the set of real numbers x ∈ [0\,1) such that
for any m ∈ N the following holds for infinitely many n ∈ N: a_{n+1}(x
)a_{n+2}(x)⋯a_{n+m}(x) ≥ e^{τ(x)(f(x)+⋯+f(T^{n-1}x))}\, where f and
τ are positive continuous functions. Also we will see that for appropria
te choices of m\, τ(x) and f(x) our result implies various classical resu
lts including the famous Jarnik-Besicovitch theorem.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Expansions of numbers in multiplicatively independent bases: Furst
enberg's conjecture and finite automata - Boris Adamczewski\, CNRS\, Unive
rsité Claude Bernard Lyon 1
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210427T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210427T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1377
DESCRIPTION:It is commonly expected that expansions of numbers in multipli
catively independent bases\, such as 2 and 10\, should have no common stru
cture. However\, it seems extraordinarily difficult to confirm this naive
heuristic principle in some way or another. In the late 1960s\, Furstenber
g suggested a series of conjectures\, which became famous and aim to captu
re this heuristic. The work I will discuss in this talk is motivated by on
e of these conjectures. Despite recent remarkable progress by Shmerkin and
Wu\, it remains totally out of reach of the current methods. While Furste
nberg’s conjectures take place in a dynamical setting\, I will use inste
ad the language of automata theory to formulate some related problems that
formalize and express in a different way the same general heuristic. I wi
ll explain how the latter can be solved thanks to some recent advances in
Mahler’s method\; a method in transcendental number theory initiated by
Mahler at the end of the 1920s. This a joint work with Colin Faverjon.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hausdorff Hensley Good & Gauss - Tushar Das\, University of Wiscon
sin - La Crosse
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210504T160000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210504T170000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1381
DESCRIPTION:Several participants of the One World Numeration Seminar (OWNS
) will know Hensley's haunting bounds (c. 1990) for the dimension of irrat
ionals whose regular continued fraction expansion partial quotients are al
l at most N\; while some might remember Good's great bounds (c. 1940) for
the dimension of irrationals whose partial quotients are all at least N. W
e will report on relatively recent results in https://arxiv.org/abs/2007.1
0554 that allow one to extend such fabulous formulae to unexpected expansi
ons. Our technology may be utilized to study various systems arising from
numeration\, dynamics\, or geometry. The talk will be accessible to studen
ts and beyond\, and I hope to present a sampling of open questions and res
earch directions that await exploration.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:The bifurcation locus for numbers of bounded type - Giulio Tiozzo\
, University of Toronto
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210511T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210511T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1383
DESCRIPTION:We define a family B(t) of compact subsets of the unit interva
l which provides a filtration of the set of numbers whose continued fracti
on expansion has bounded digits. This generalizes to a continuous family t
he well-known sets of numbers whose continued fraction expansion is bounde
d above by a fixed integer. \n\nWe study how the set B(t) changes as the p
arameter t ranges in [0\,1]\, and describe precisely the bifurcations that
occur as the parameters change. Further\, we discuss continuity propertie
s of the Hausdorff dimension of B(t) and its regularity. \n\nFinally\, we
establish a precise correspondence between these bifurcations and the bifu
rcations for the classical family of real quadratic polynomials.\n\nJoint
with C. Carminati.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Solved and unsolved problems in normal numbers - Joseph Vandehey\,
University of Texas at Tyler
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210518T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210518T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1384
DESCRIPTION:We will survey a variety of problems on normal numbers\, some
old\, some new\, some solved\, and some unsolved\, in the hope of spurring
some new directions of study. Topics will include constructions of normal
numbers\, normality in two different systems simultaneously\, normality s
een through the lens of informational or logical complexity\, and more.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dynamics of simplicial systems and multidimensional continued frac
tion algorithms - Charles Fougeron\, IRIF
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210525T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210525T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1392
DESCRIPTION:Motivated by the richness of the Gauss algorithm which allows
to efficiently compute the best approximations of a real number by rationa
ls\, many mathematicians have suggested generalisations to study Diophanti
ne approximations of vectors in higher dimensions. Examples include Poinca
ré's algorithm introduced at the end of the 19th century or those of Brun
and Selmer in the middle of the 20th century. Since the beginning of the
90's to the present day\, there has been many works studying the convergen
ce and dynamics of these multidimensional continued fraction algorithms. I
n particular\, Schweiger and Broise have shown that the approximation sequ
ence built using Selmer and Brun algorithms converge to the right vector w
ith an extra ergodic property. On the other hand\, Nogueira demonstrated t
hat the algorithm proposed by Poincaré almost never converges. \n\nStarti
ng from the classical case of Farey's algorithm\, which is an "additive" v
ersion of Gauss's algorithm\, I will present a combinatorial point of view
on these algorithms which allows to us to use a random walk approach. In
this model\, taking a random vector for the Lebesgue measure will correspo
nd to following a random walk with memory in a labelled graph called sympl
icial system. The laws of probability for this random walk are elementary
and we can thus develop probabilistic techniques to study their generic dy
namical behaviour. This will lead us to describe a purely graph theoretic
criterion to check the convergence of a continued fraction algorithm.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Automorphisms and factors of finite topological rank systems - Bas
tián Espinoza\, Université de Picardie Jules Verne and Universidad de Ch
ile
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210601T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210601T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1393
DESCRIPTION:Finite topological rank systems are a type of minimal S-adic s
ubshift that includes many of the classical minimal systems of zero entrop
y (e.g. linearly recurrent subshifts\, interval exchanges and some Toeplit
z sequences). In this talk I am going to present results concerning the nu
mber of automorphisms and factors of systems of finite topological rank\,
as well as closure properties of this class with respect to factors and re
lated combinatorial operations.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Counting balanced words and related problems - Shigeki Akiyama\, U
niversity of Tsukuba
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210608T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210608T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1407
DESCRIPTION:Balanced words and Sturmian words are ubiquitous and appear in
the intersection of many areas of mathematics. In this talk\, I try to ex
plain an idea of S. Yasutomi to study finite balanced words. His method gi
ves a nice way to enumerate number of balanced words of given length\, slo
pe and intercept. Applying this idea\, we can obtain precise asymptotic fo
rmula for balanced words. The result is connected to some classical topics
in number theory\, such as Farey fraction\, Riemann Hypothesis and Large
sieve inequality.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dyadic approximation in the Cantor set - Sam Chow\, University of
Warwick
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210615T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210615T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1408
DESCRIPTION:We investigate the approximation rate of a typical element of
the Cantor set by dyadic rationals. This is a manifestation of the times-t
wo-times-three phenomenon\, and is joint work with Demi Allen and Han Yu.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simultaneous Diophantine approximation of the orbits of the dynami
cal systems x2 and x3 - Lingmin Liao\, Université Paris-Est Créteil Val
de Marne
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210622T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210622T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1409
DESCRIPTION:We study the sets of points whose orbits of the dynamical syst
ems x2 and x3 simultaneously approach to a given point\, with a given spee
d. A zero-one law for the Lebesgue measure of such sets is established. Th
e Hausdorff dimensions are also determined for some special speeds. One di
mensional formula among them is established under the abc conjecture. At t
he same time\, we also study the Diophantine approximation of the orbits o
f a diagonal matrix transformation of a torus\, for which the properties o
f the (negative) beta transformations are involved. This is a joint work w
ith Bing Li\, Sanju Velani and Evgeniy Zorin.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hausdorff dimension of Gauss-Cantor sets and their applications to
the study of classical Markov spectrum - Polina Vytnova\, University of W
arwick
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210629T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210629T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1416
DESCRIPTION:The classical Lagrange and Markov spectra are subsets of the r
eal line which arise in connection with some problems in theory Diophantin
e approximation theory. In 1921 O. Perron gave a definition in terms of co
ntinued fractions\, which allowed to study the Markov and Lagrange spectra
using limit sets of iterated function schemes. \n\nIn this talk we will s
ee how the first transition point\, where the Markov spectra acquires the
full measure can be computed by the means of estimating Hausdorff dimensio
n of the certain Gauss-Cantor sets. \n\nThe talk is based on a joint work
with C. Matheus\, C. G. Moreira and M. Pollicott.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Littlewood and Duffin-Schaeffer-type problems in diophantine appro
ximation - Niclas Technau\, University of Wisconsin - Madison
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210706T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210706T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1417
DESCRIPTION:Gallagher's theorem describes the multiplicative diophantine a
pproximation rate of a typical vector. Recently Sam Chow and I establish a
fully-inhomogeneous version of Gallagher's theorem\, and a diophantine fi
bre refinement. In this talk I outline the proof\, and the tools involved
in it.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On Hermite's problem\, Jacobi-Perron type algorithms\, and Dirichl
et groups - Oleg Karpenkov\, University of Liverpool
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210907T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210907T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1424
DESCRIPTION:In this talk we introduce a new modification of the Jacobi-Per
ron algorithm in the three dimensional case. This algorithm is periodic fo
r the case of totally-real conjugate cubic vectors. To the best of our kno
wledge this is the first Jacobi-Perron type algorithm for which the cubic
periodicity is proven. This provides an answer in the totally-real case to
the question of algebraic periodicity for cubic irrationalities posed in
1848 by Ch.Hermite.\n\nWe will briefly discuss a new approach which is bas
ed on geometry of numbers. In addition we point out one important applicat
ion of Jacobi-Perron type algorithms to the computation of independent ele
ments in the maximal groups of commuting matrices of algebraic irrationali
ties.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Descriptive complexity in numeration systems - Steve Jackson\, Uni
versity of North Texas
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210914T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210914T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1429
DESCRIPTION:Descriptive set theory gives a means of calibrating the comple
xity of sets\, and we focus on some sets occurring in numerations systems.
Also\, the descriptive complexity of the difference of two sets gives a n
otion of the logical independence of the sets. A classic result of Ki and
Linton says that the set of normal numbers for a given base is a Π_3^0 co
mplete set. In work with Airey\, Kwietniak\, and Mance we extend to other
numerations systems such as continued fractions\, ????-expansions\, and GL
S expansions. In work with Mance and Vandehey we show that the numbers whi
ch are continued fraction normal but not base b normal is complete at the
expected level of D_2(Π_3^0). An immediate corollary is that this set is
uncountable\, a result (due to Vandehey) only known previously assuming th
e generalized Riemann hypothesis.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:The distribution of reduced quadratic irrationals arising from con
tinued fraction expansions - Maria Siskaki\, University of Illinois at Urb
ana-Champaign
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210921T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210921T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1430
DESCRIPTION:It is known that the reduced quadratic irrationals arising fro
m regular continued fraction expansions are uniformly distributed when ord
ered by their length with respect to the Gauss measure. In this talk\, I w
ill describe a number theoretical approach developed by Kallies\, Ozluk\,
Peter and Snyder\, and then by Boca\, that gives the error in the asymptot
ic behavior of this distribution. Moreover\, I will present the respective
result for the distribution of reduced quadratic irrationals that arise f
rom even (joint work with F. Boca) and odd continued fractions.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:A strong version of Cobham's theorem - Philipp Hieronymi\, Univers
ität Bonn
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20210928T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20210928T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1438
DESCRIPTION:Let k\,l>1 be two multiplicatively independent integers. A sub
set X of N^n is k-recognizable if the set of k-ary representations of X is
recognized by some finite automaton. Cobham’s famous theorem states tha
t a subset of the natural numbers is both k-recognizable and l-recognizabl
e if and only if it is Presburger-definable (or equivalently: semilinear).
We show the following strengthening. Let X be k-recognizable\, let Y be l
-recognizable such that both X and Y are not Presburger-definable. Then th
e first-order logical theory of (N\,+\,X\,Y) is undecidable. This is in co
ntrast to a well-known theorem of Büchi that the first-order logical theo
ry of (N\,+\,X) is decidable. Our work strengthens and depends on earlier
work of Villemaire and Bès.\n\nThe essence of Cobham's theorem is that re
cognizability depends strongly on the choice of the base k. Our results st
rengthens this: two non-Presburger definable sets that are recognizable in
multiplicatively independent bases\, are not only distinct\, but together
computationally intractable over Presburger arithmetic.\n\nThis is joint
work with Christian Schulz.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On upper and lower fast Khintchine spectra in continued fractions
- Lulu Fang\, Nanjing University of Science and Technology
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211005T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211005T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1439
DESCRIPTION:Let $\\psi: \\mathbb{N} \\to \\mathbb{R}^+$ be a function sati
sfying $\\psi(n)/n \\to \\infty$ as $n \\to \\infty$. We investigate from
a multifractal analysis point of view the growth speed of the sums $\\sum_
{k=1}^n \\log a_k(x)$ with respect to $\\psi(n)$\, where $x = [a_1(x)\,a_2
(x)\,\\dots]$ denotes the continued fraction expansion of $x \\in (0\,1)$.
The (upper\, lower) fast Khintchine spectrum is defined as the Hausdorff
dimension of the set of points $x \\in (0\,1)$ for which the (upper\, lowe
r) limit of $\\frac{1}{\\psi(n)} \\sum_{k=1}^n \\log a_k(x)$ is equal to 1
. These three spectra have been studied by Fan\, Liao\, Wang & Wu (2013\,
2016)\, Liao & Rams (2016). In this talk\, we will give a new look at the
fast Khintchine spectrum\, and provide a full description of upper and low
er fast Khintchine spectra. The latter improves a result of Liao and Rams
(2016).
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On the Existence of Numbers with Matching Continued Fraction and D
ecimal Expansion - Taylor Jones\, University of North Texas
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211005T150000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211005T160000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1440
DESCRIPTION:A Trott number in base 10 is one whose continued fraction expa
nsion agrees with its base 10 expansion in the sense that [0\;a_1\,a_2\,..
.] = 0.(a_1)(a_2)... where (a_i) represents the string of digits of a_i. A
s an example [0\;3\,29\,54\,7\,...] = 0.329547... An analogous definition
may be given for a Trott number in any integer base b>1\, the set of which
we denote by T_b. The first natural question is whether T_b is empty\, an
d if not\, for which b? We discuss the history of the problem\, and give a
heuristic process for constructing such numbers. We show that T_{10} is i
ndeed non-empty\, and uncountable. With more delicate techniques\, a compl
ete classification may be given to all b for which T_b is non-empty. We al
so discuss some further results\, such as a (non-trivial) upper bound on t
he Hausdorff dimension of T_b\, as well as the question of whether the int
ersection of T_b and T_c can be non-empty.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Inflection points in the Lyapunov spectrum for IFS on intervals -
Liangang Ma\, Binzhou University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211012T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211012T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1441
DESCRIPTION:We plan to present the audience a general picture about regula
rity of the Lyapunov spectrum for some iterated function systems\, with em
phasis on its inflection points in case the spectrum is smooth. Some sharp
or moderate relationship between the number of Lyapunov inflections and (
essential) branch number of a linear system is clarified. As most numerati
on systems are non-linear ones\, the corresponding relationship for these
systems are still mysterious enough comparing with the linear systems.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:q-analog of the Markoff injectivity conjecture - Mélodie Lapointe
\, IRIF
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211019T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211019T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1472
DESCRIPTION:The Markoff injectivity conjecture states that $w\\mapsto\\mu(
w)_{12}$ is injective on the set of Christoffel words where $\\mu:\\{\\mat
htt{0}\,\\mathtt{1}\\}^*\\to\\mathrm{SL}_2(\\mathbb{Z})$ is a certain homo
morphism and $M_{12}$ is the entry above the diagonal of a $2\\times2$ mat
rix $M$. Recently\, Leclere and Morier-Genoud (2021) proposed a $q$-analog
$\\mu_q$ of $\\mu$ such that $\\mu_{q\\to1}(w)_{12}=\\mu(w)_{12}$ is the
Markoff number associated to the Christoffel word $w$. We show that there
exists an order $<_{radix}$ on $\\{\\mathtt{0}\,\\mathtt{1}\\}^*$ such tha
t for every balanced sequence $s \\in \\{\\mathtt{0}\,\\mathtt{1}\\}^\\mat
hbb{Z}$ and for all factors $u\, v$ in the language of $s$ with $u <_{radi
x} v$\, the difference $\\mu_q(v)_{12} - \\mu_q(u)_{12}$ is a nonzero poly
nomial of indeterminate $q$ with nonnegative integer coefficients. Therefo
re\, for every $q>0$\, the map $\\{\\mathtt{0}\,\\mathtt{1}\\}^*\\to\\math
bb{R}$ defined by $w\\mapsto\\mu_q(w)_{12}$ is increasing thus injective o
ver the language of a balanced sequence. The proof uses an equivalence b
etween balanced sequences satisfying some Markoff property and indistingui
shable asymptotic pairs.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spectral aspects of aperiodic dynamical systems - Michael Baake\,
Universität Bielefeld
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211026T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211026T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1473
DESCRIPTION:One way to analyse aperiodic systems employs spectral notions\
, either via dynamical systems theory or via harmonic analysis. In this ta
lk\, we will look at two particular aspects of this\, after a quick overvi
ew of how the diffraction measure can be used for this purpose. First\, we
consider some concequences of inflation rules on the spectra via renormal
isation\, and how to use it to exclude absolutely continuous componenta. S
econd\, we take a look at a class of dynamical systems of number-theoretic
origin\, how they fit into the spectral picture\, and what (other) method
s there are to distinguish them.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:On the existence of Trott numbers relative to multiple bases - Pie
ter Allaart\, University of North Texas
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211102T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211102T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1488
DESCRIPTION:Trott numbers are real numbers in the interval (0\,1) whose co
ntinued fraction expansion equals their base-b expansion\, in a certain li
beral but natural sense. They exist in some bases\, but not in all. In a p
revious OWNS talk\, T. Jones sketched a proof of the existence of Trott nu
mbers in base 10. In this talk I will discuss some further properties of t
hese Trott numbers\, and focus on the question: Can a number ever be Trott
in more than one base at once? While the answer is almost certainly "no"\
, a full proof of this seems currently out of reach. But we obtain some in
teresting partial answers by using a deep theorem from Diophantine approxi
mation.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:How inhomogeneous Cantor sets can pass a point - Zhiqiang Wang\, E
ast China Normal University
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211109T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211109T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1489
DESCRIPTION:Abstract: For $x > 0$\, we define $\\Upsilon(x) = \\{ (a\,b):
x\\in E_{a\,b}\, a>0\, b>0\, a+b \\le 1 \\}$\, where the set $E_{a\,b}$ is
the unique nonempty compact invariant set generated by the inhomogeneous
IFS $\\{ f_0(x) = a x\, f_1(x) = b(x+1) \\}$. We show the set $\\Upsilon(
x)$ is a Lebesgue null set with full Hausdorff dimension in $\\mathbb{R}^2
$\, and the intersection of sets $\\Upsilon(x_1)\, \\Upsilon(x_2)\, \\cdot
s\, \\Upsilon(x_\\ell)$ still has full Hausdorff dimension $\\mathbb{R}^2$
for any finitely many positive real numbers $x_1\, x_2\, \\cdots\, x_\\el
l$.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Extensions of the random beta-transformation - Younès Tierce\, Un
iversité de Rouen Normandie
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211109T150000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211109T160000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1490
DESCRIPTION:Let $\\beta \\in (1\,2)$ and $I_\\beta := [0\,\\frac{1}{\\beta
-1}]$. Almost every real number of $I_\\beta$ has infinitely many expansio
ns in base $\\beta$\, and the random $\\beta$-transformation generates all
these expansions. We present the construction of a "geometrico-symbolic"
extension of the random $\\beta$-transformation\, providing a new proof of
the existence and unicity of an absolutely continuous invariant probabili
ty measure\, and an expression of the density of this measure. This extens
ion shows off some nice renewal times\, and we use these to prove that the
natural extension of the system is a Bernoulli automorphism.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rational self-affine tiles associated to (nonstandard) digit syste
ms - Lucía Rossi\, Montanuniversität Leoben
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211116T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211116T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1491
DESCRIPTION:In this talk we will introduce the notion of rational self-aff
ine tiles\, which are fractal-like sets that arise as the solution of a se
t equation associated to a digit system that consists of a base\, given by
an expanding rational matrix\, and a digit set\, given by vectors. They c
an be interpreted as the set of “fractional parts” of this digit syste
m\, and the challenge of this theory is that these sets do not live in a E
uclidean space\, but on more general spaces defined in terms of Laurent se
ries. Steiner and Thuswaldner defined rational self-affine tiles for the c
ase where the base is a rational matrix with irreducible characteristic po
lynomial. We present some tiling results that generalize the ones obtained
by Lagarias and Wang: we consider arbitrary expanding rational matrices a
s bases\, and simultaneously allow the digit sets to be nonstandard (meani
ng they are not a complete set of residues modulo the base). We also state
some topological properties of rational self-affine tiles and give a crit
erion to guarantee positive measure in terms of the digit set.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:Analogues of Khintchine's theorem for random attractors - Sascha T
roscheit\, Universität Wien
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211123T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211123T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1514
DESCRIPTION:Khintchine’s theorem is an important result in number theory
which links the Lebesgue measure of certain limsup sets with the converge
nce/divergence of naturally occurring volume sums. This behaviour has been
observed for deterministic fractal sets and inspired by this we investiga
te the random settings. Introducing randomisation into the problem makes s
ome parts more tractable\, while posing separate new challenges. In this t
alk\, I will present joint work with Simon Baker where we provide sufficie
nt conditions for a large class of stochastically self-similar and self-af
fine attractors to have positive Lebesgue measure.
LOCATION:Online
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA - Jamie Walton\, University of Nottingham
DTSTART;TZID=Europe/Paris;VALUE=DATE-TIME:20211207T143000
DTEND;TZID=Europe/Paris;VALUE=DATE-TIME:20211207T153000
DTSTAMP;VALUE=DATE-TIME:20211128T180302Z
UID:1515
DESCRIPTION:
LOCATION:Online
END:VEVENT
END:VCALENDAR