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SUMMARY:Osama Khalil (Utah)
DTSTART;VALUE=DATE-TIME:20200409T180000Z
DTEND;VALUE=DATE-TIME:20200409T190000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/1
DESCRIPTION:Title: Random walks\, spectral gaps\, and Khinchine's theorem on fractals
\nby Osama Khalil (Utah) as part of Virtual lecture series in dynamics\n\n
\nAbstract\nIn 1984\, Mahler asked how well typical points on Cantor’s s
et can be approximated by\nrational numbers. His question fits within a pr
ogram\, set out by himself in the 1930s\, attempting to\ndetermine conditi
ons under which subsets of $\\mathbb R^d$\ninherit the Diophantine propert
ies of the ambient\nspace. Since the approximability of typical points in
Euclidean space by rational points is governed\nby Khinchine’s classical
theorem\, the ultimate form of Mahler’s question asks whether an analog
ous\nzero-one law holds for fractal measures. Significant progress has bee
n achieved in recent years\,\nalbeit\, almost all known results have been
of “convergence type”.\nIn this talk\, we will discuss the first insta
nces where a complete analogue of Khinchine’s theorem\nfor fractal measu
res is obtained. Our results hold for fractals generated by rational simil
arities of $\\mathbb R^d$\n\nand having sufficiently small Hausdorff co-di
mension. The main new ingredient is an effective\nequidistribution theorem
for certain fractal measures on the space of unimodular lattices. The lat
ter\nis established via a new technique involving the construction of $S$-
arithmetic Markov operators\npossessing a spectral gap and encoding the ar
ithmetic structure of the maps generating the fractal.\nThis is joint work
in progress with Manuel Luethi.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Quas (Victoria)
DTSTART;VALUE=DATE-TIME:20200416T180000Z
DTEND;VALUE=DATE-TIME:20200416T190000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/2
DESCRIPTION:Title: Stability and collapse of Oseledets spectrum for Perron-Frobenius cocy
cles\nby Anthony Quas (Victoria) as part of Virtual lecture series in
dynamics\n\n\nAbstract\nIt is known\, by work of Bochi\, Mañé\, Viana an
d others\nthat Lyapunov exponents are highly sensitive to perturbations of
a\ndynamical system. On the other hand\, work of Ledrappier\, Young\nand
my work with Froyland and Gonz´alez-Tokman has shown that\nin some situat
ions\, under “noise-like” perturbations\, Lyapunov exponents vary cont
inuously.\nWe are particularly interested in cocycles of Perron-Frobenius\
noperators\, as the Lyapunov exponents (and the corresponding Oseledets sp
aces) are related to rates of mixing (and the spaces can\nidentify obstruc
tions to mixing). We discuss a test case of a random composition of Blasch
ke products\, and their Perron-Frobenius\noperators acting on a Hardy spac
e of analytic functions. These operators are known to be compact. We ident
ify the full Lyapunov\nspectrum of these systems\, and give necessary and
sufficient conditions for the stability of the spectrum. [Joint work with
Cecilia\nGonz´alez-Tokman.]\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Davide Ravotti (Monash)
DTSTART;VALUE=DATE-TIME:20200423T210000Z
DTEND;VALUE=DATE-TIME:20200423T220000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/3
DESCRIPTION:Title: Quantitative equidistribution of horocycle push-forwards of transverse
arcs\nby Davide Ravotti (Monash) as part of Virtual lecture series in
dynamics\n\n\nAbstract\nFor several parabolic systems\, a technique often
used to prove mixing and\nother strong chaotic properties consists of a g
eometric shearing argument. In the case\nof the horocycle flow (both in co
nstant and in variable negative curvature\, as well as for\nits smooth tim
e-changes)\, this has been done successfully by analysing the action of th
e\nhorocycle flow on geodesic arcs. The quantitative estimates one can obt
ain following this\napproach\, however\, are not optimal\, since\, in the
constant curvature case\, do not match the\nones obtained by Ratner.\nIn t
his talk\, we will discuss an effective equidistribution result for the ho
rocycle pushforwards of homogeneous arcs which are transverse to the weak-
stable leaves of the\ngeodesic flow. As a corollary\, we derive a geometri
c proof of Ratner’s quantitative mixing\nresult for the horocycle flow\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Richter (Northwestern)
DTSTART;VALUE=DATE-TIME:20200430T180000Z
DTEND;VALUE=DATE-TIME:20200430T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/4
DESCRIPTION:Title: Dynamical generalizations of the Prime Number Theorem and disjointness
of additive and multiplicative actions\nby Florian Richter (Northwest
ern) as part of Virtual lecture series in dynamics\n\n\nAbstract\nOne of t
he fundamental challenges in number theory is to understand the intricate\
nway in which the additive and multiplicative structures in the integers i
ntertwine. We will\nexplore a dynamical approach to this topic. After intr
oducing a new dynamical framework for\ntreating questions in multiplicativ
e number theory\, we will present an ergodic theorem which\ncontains vario
us classical number-theoretic results\, such as the Prime Number Theorem\,
as\nspecial cases. This naturally leads to a formulation of an extended f
orm of Sarnak's conjecture\,\nwhich deals with the disjointness of actions
of (N\, +) and (N\, ·). This talk is based on joint\nwork with Vitaly Be
rgelson.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/4/
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BEGIN:VEVENT
SUMMARY:Felipe García Ramos (Universidad Autónoma de San Luis Potosí)
DTSTART;VALUE=DATE-TIME:20200507T180000Z
DTEND;VALUE=DATE-TIME:20200507T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/5
DESCRIPTION:Title: On topological models of zero entropy loosely Bernoulli systems\nb
y Felipe García Ramos (Universidad Autónoma de San Luis Potosí) as part
of Virtual lecture series in dynamics\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alena Erchenko (Stonybrook)
DTSTART;VALUE=DATE-TIME:20200514T180000Z
DTEND;VALUE=DATE-TIME:20200514T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/6
DESCRIPTION:Title: Flexibility of Lyapunov exponents with respect to two classes of measu
res\nby Alena Erchenko (Stonybrook) as part of Virtual lecture series
in dynamics\n\n\nAbstract\nWe give an overview of the flexibility philosop
hy proposed by Anatole Katok and concentrate\non questions (answered and o
pened) connected to Lyapunov exponents with respect to various\nmeasures.
There are several interesting classes of measures. We will look at the inv
ariant\nmeasure that is absolutely continuous with respect to the Lebesgue
measure and the measure\nof maximal entropy. We show that positive Lyapun
ov exponents with respect to these two\nprobability measures for Anosov ar
ea-preserving diffeomorphisms on a two-torus that are homotopic to a fixed
area-preserving Anosov automorphism take on all values that satisfy some
well-known inequalities.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenyu Pan (Chicago)
DTSTART;VALUE=DATE-TIME:20200521T180000Z
DTEND;VALUE=DATE-TIME:20200521T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/7
DESCRIPTION:Title: Exponential mixing of gedoesic flow for geometrically finite manifolds
with cusps\nby Wenyu Pan (Chicago) as part of Virtual lecture series
in dynamics\n\n\nAbstract\nLet $\\mathbb H^n$ be the hyperbolic $n$-space
and Γ be a geometrically finite discrete subgroup in $\\mathrm{Isom}_+(\\
mathbb H^n)$ with cusps.\nIn the forthcoming joint work with Jialun Li\, w
e establish exponential mixing of the geodesic flow over the unit tangent
bundle $T^1(\\Gamma\\backslash \\mathbb H^n)$. Previously\, such results w
ere proved by Stoyanov for convex cocompact discrete subgroups and Mohamma
di-Oh and Edwards-Oh for $\\Gamma$ with large critical exponent. We obtain
our\nresult by constructing a nice coding for the geodesic flow\, which i
n particular satisfies the exponential tail condition.\nIn the first part
of the talk\, I am going to explain the construction of the coding\, which
is partly inspired by the works of Lai-Sang Young and Burns-Masur-Matheus
-Wilkinson. In the second part of the talk\, Jialun\nLi is going to discus
s part of the process on how to prove a spectral bound for the transfer op
erator building\non Dolgopyat’s framework.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Algom (Penn State)
DTSTART;VALUE=DATE-TIME:20200528T180000Z
DTEND;VALUE=DATE-TIME:20200528T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/8
DESCRIPTION:Title: Furstenberg-Marstrand slicing Theorems for $(\\times m\, \\times n)$ i
nvariant sets\nby Amir Algom (Penn State) as part of Virtual lecture s
eries in dynamics\n\n\nAbstract\nIn 1970 Furstenberg proposed the followin
g $\\times 2\, \\times 3$ type conjecture: Let $m\, n \\gt 1$ be independe
nt integers\, and let $X\, Y \\subseteq [0\, 1]$ be two closed sets that a
re ×m and ×n invariant\, respectively.\nThen for every invertible affine
map $g$\, the Hausdorff dimension of $g(X)\\cap Y$ is at most the maximum
\nof $\\dim_H X + \\dim_H Y − 1$ and $0$. Writing $Z = X \\times Y$\, th
is is equivalent to\n$$\n\\dim_H Z \\cap \\ell \\le \\max {\\dim_H Z − 1
\, 0}\, \\qquad\\text{for any line $\\ell$ not parallel to the major axes.
}\\qquad\\qquad(1)\n$$\nIn 2016\, Pablo Shmerkin and Meng Wu (independentl
y) proved the conjecture to be correct.\nWe shall present a generalization
of this result: The slicing bound (1) holds whenever $Z \\subseteq [0\, 1
]^2$ is a closed $(\\times m\, \\times n)$ invariant set (i.e. not only fo
r product sets). This talk is based on joint work with Meng Wu.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayşe Şahin (Wright State)
DTSTART;VALUE=DATE-TIME:20200604T180000Z
DTEND;VALUE=DATE-TIME:20200604T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/9
DESCRIPTION:Title: The complexity threshold for the loosely Bernoulli property\nby Ay
şe Şahin (Wright State) as part of Virtual lecture series in dynamics\n\
nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nyima Kao (Chicago)
DTSTART;VALUE=DATE-TIME:20200611T180000Z
DTEND;VALUE=DATE-TIME:20200611T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/10
DESCRIPTION:Title: Pressure metrics for deformation spaces of quasifuchsian groups with
parabolics\nby Nyima Kao (Chicago) as part of Virtual lecture series i
n dynamics\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Scott Schmieding (Denver)
DTSTART;VALUE=DATE-TIME:20200618T180000Z
DTEND;VALUE=DATE-TIME:20200618T200000Z
DTSTAMP;VALUE=DATE-TIME:20241112T121728Z
UID:VLSDynamics/11
DESCRIPTION:Title: Local P entropy and stabilized automorphism groups\nby Scott Schm
ieding (Denver) as part of Virtual lecture series in dynamics\n\n\nAbstrac
t\nFor a homeomorphism of a compact metric space T : X → X\, the\nstabil
ized automorphism group Aut(∞)\n(T) consists of all self-homeomorphisms
of\nX which commute with some power of T. Motivated by studying Aut(∞)\n
(T) in\nthe setting of symbolic systems\, we will introduce and discuss a
certain entropy for\ngroups called P local entropy. We will show how P loc
al entropy can be used to\ngive a complete classification of the stabilize
d automorphisms groups of full shifts\;\nin particular\, we show the stabi
lized groups for the 2-shift and the 3-shift are not\nisomorphic.\n
LOCATION:https://researchseminars.org/talk/VLSDynamics/11/
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