BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Osama Khalil (Utah) DTSTART:20200409T180000Z DTEND:20200409T190000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/1 DESCRIPTION:Title: Random walks\, spectral gaps\, and Khinchine's theorem on fract als\nby Osama Khalil (Utah) as part of Virtual lecture series in dynam ics\n\n\nAbstract\nIn 1984\, Mahler asked how well typical points on Canto r’s set can be approximated by\nrational numbers. His question fits with in a program\, set out by himself in the 1930s\, attempting to\ndetermine conditions under which subsets of $\\mathbb R^d$\ninherit the Diophantine properties of the ambient\nspace. Since the approximability of typical poi nts in Euclidean space by rational points is governed\nby Khinchine’s cl assical theorem\, the ultimate form of Mahler’s question asks whether an analogous\nzero-one law holds for fractal measures. Significant progress has been achieved in recent years\,\nalbeit\, almost all known results hav e been of “convergence type”.\nIn this talk\, we will discuss the firs t instances where a complete analogue of Khinchine’s theorem\nfor fracta l measures is obtained. Our results hold for fractals generated by rationa l similarities of $\\mathbb R^d$\n\nand having sufficiently small Hausdorf f co-dimension. The main new ingredient is an effective\nequidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter\nis established via a new technique involving the construction of $S$-arithmetic Markov operators\npossessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal.\nThis is joi nt work in progress with Manuel Luethi.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Anthony Quas (Victoria) DTSTART:20200416T180000Z DTEND:20200416T190000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/2 DESCRIPTION:Title: Stability and collapse of Oseledets spectrum for Perron-Frobeni us cocycles\nby Anthony Quas (Victoria) as part of Virtual lecture ser ies in dynamics\n\n\nAbstract\nIt is known\, by work of Bochi\, Mañé\, V iana and others\nthat Lyapunov exponents are highly sensitive to perturbat ions of a\ndynamical system. On the other hand\, work of Ledrappier\, Youn g\nand my work with Froyland and Gonz´alez-Tokman has shown that\nin some situations\, under “noise-like” perturbations\, Lyapunov exponents va ry continuously.\nWe are particularly interested in cocycles of Perron-Fro benius\noperators\, as the Lyapunov exponents (and the corresponding Osele dets spaces) are related to rates of mixing (and the spaces can\nidentify obstructions to mixing). We discuss a test case of a random composition of Blaschke products\, and their Perron-Frobenius\noperators acting on a Har dy space of analytic functions. These operators are known to be compact. W e identify the full Lyapunov\nspectrum of these systems\, and give necessa ry and sufficient conditions for the stability of the spectrum. [Joint wor k with Cecilia\nGonz´alez-Tokman.]\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Davide Ravotti (Monash) DTSTART:20200423T210000Z DTEND:20200423T220000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/3 DESCRIPTION:Title: Quantitative equidistribution of horocycle push-forwards of tra nsverse arcs\nby Davide Ravotti (Monash) as part of Virtual lecture se ries in dynamics\n\n\nAbstract\nFor several parabolic systems\, a techniqu e often used to prove mixing and\nother strong chaotic properties consists of a geometric shearing argument. In the case\nof the horocycle flow (bot h in constant and in variable negative curvature\, as well as for\nits smo oth time-changes)\, this has been done successfully by analysing the actio n of the\nhorocycle flow on geodesic arcs. The quantitative estimates one can obtain following this\napproach\, however\, are not optimal\, since\, in the constant curvature case\, do not match the\nones obtained by Ratner .\nIn this talk\, we will discuss an effective equidistribution result for the horocycle pushforwards of homogeneous arcs which are transverse to th e weak-stable leaves of the\ngeodesic flow. As a corollary\, we derive a g eometric proof of Ratner’s quantitative mixing\nresult for the horocycle flow\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Florian Richter (Northwestern) DTSTART:20200430T180000Z DTEND:20200430T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/4 DESCRIPTION:Title: Dynamical generalizations of the Prime Number Theorem and disjo intness of additive and multiplicative actions\nby Florian Richter (No rthwestern) as part of Virtual lecture series in dynamics\n\n\nAbstract\nO ne of the fundamental challenges in number theory is to understand the int ricate\nway in which the additive and multiplicative structures in the int egers intertwine. We will\nexplore a dynamical approach to this topic. Aft er introducing a new dynamical framework for\ntreating questions in multip licative number theory\, we will present an ergodic theorem which\ncontain s various classical number-theoretic results\, such as the Prime Number Th eorem\, as\nspecial cases. This naturally leads to a formulation of an ext ended form of Sarnak's conjecture\,\nwhich deals with the disjointness of actions of (N\, +) and (N\, ·). This talk is based on joint\nwork with Vi taly Bergelson.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Felipe García Ramos (Universidad Autónoma de San Luis Potosí) DTSTART:20200507T180000Z DTEND:20200507T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/5 DESCRIPTION:Title: On topological models of zero entropy loosely Bernoulli systems \nby Felipe García Ramos (Universidad Autónoma de San Luis Potosí) as part of Virtual lecture series in dynamics\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alena Erchenko (Stonybrook) DTSTART:20200514T180000Z DTEND:20200514T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/6 DESCRIPTION:Title: Flexibility of Lyapunov exponents with respect to two classes o f measures\nby Alena Erchenko (Stonybrook) as part of Virtual lecture series in dynamics\n\n\nAbstract\nWe give an overview of the flexibility p hilosophy proposed by Anatole Katok and concentrate\non questions (answere d and opened) connected to Lyapunov exponents with respect to various\nmea sures. There are several interesting classes of measures. We will look at the invariant\nmeasure that is absolutely continuous with respect to the L ebesgue measure and the measure\nof maximal entropy. We show that positive Lyapunov exponents with respect to these two\nprobability measures for An osov area-preserving diffeomorphisms on a two-torus that are homotopic to a fixed area-preserving Anosov automorphism take on all values that satisf y some well-known inequalities.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Wenyu Pan (Chicago) DTSTART:20200521T180000Z DTEND:20200521T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/7 DESCRIPTION:Title: Exponential mixing of gedoesic flow for geometrically finite ma nifolds 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 $\\mathrm{Iso m}_+(\\mathbb H^n)$ with cusps.\nIn the forthcoming joint work with Jialun Li\, we establish exponential mixing of the geodesic flow over the unit t angent bundle $T^1(\\Gamma\\backslash \\mathbb H^n)$. Previously\, such re sults were proved by Stoyanov for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for $\\Gamma$ with large critical exponent. We obtain our\nresult by constructing a nice coding for the geodesic flow\, which in particular satisfies the exponential tail condition.\nIn the firs t 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 discuss part of the process on how to prove a spectral bound for the tran sfer operator building\non Dolgopyat’s framework.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Amir Algom (Penn State) DTSTART:20200528T180000Z DTEND:20200528T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/8 DESCRIPTION:Title: Furstenberg-Marstrand slicing Theorems for $(\\times m\, \\time s n)$ invariant sets\nby Amir Algom (Penn State) as part of Virtual le cture series in dynamics\n\n\nAbstract\nIn 1970 Furstenberg proposed the f ollowing $\\times 2\, \\times 3$ type conjecture: Let $m\, n \\gt 1$ be in dependent integers\, and let $X\, Y \\subseteq [0\, 1]$ be two closed sets that are ×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 $Z = X \\times Y$\, this is equivalent to\n$$\n\\dim_H Z \\cap \\ell \\le \\max {\\dim_H Z − 1\, 0}\, \\qquad\\text{for any line $\\ell$ not parallel to the majo r axes.}\\qquad\\qquad(1)\n$$\nIn 2016\, Pablo Shmerkin and Meng Wu (indep endently) proved the conjecture to be correct.\nWe shall present a general ization 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 for product sets). This talk is based on joint work with Meng Wu.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Ayşe Şahin (Wright State) DTSTART:20200604T180000Z DTEND:20200604T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/9 DESCRIPTION:Title: The complexity threshold for the loosely Bernoulli property \nby Ayşe Şahin (Wright State) as part of Virtual lecture series in dyna mics\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Nyima Kao (Chicago) DTSTART:20200611T180000Z DTEND:20200611T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/10 DESCRIPTION:Title: Pressure metrics for deformation spaces of quasifuchsian group s with parabolics\nby Nyima Kao (Chicago) as part of Virtual lecture s eries in dynamics\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Scott Schmieding (Denver) DTSTART:20200618T180000Z DTEND:20200618T200000Z DTSTAMP:20260404T110653Z UID:VLSDynamics/11 DESCRIPTION:Title: Local P entropy and stabilized automorphism groups\nby Sco tt Schmieding (Denver) as part of Virtual lecture series in dynamics\n\n\n Abstract\nFor a homeomorphism of a compact metric space T : X → X\, the\ nstabilized automorphism group Aut(∞)\n(T) consists of all self-homeomor phisms 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 dis cuss a certain entropy for\ngroups called P local entropy. We will show ho w P local entropy can be used to\ngive a complete classification of the st abilized automorphisms groups of full shifts\;\nin particular\, we show th e stabilized groups for the 2-shift and the 3-shift are not\nisomorphic.\n LOCATION:https://stable.researchseminars.org/talk/VLSDynamics/11/ END:VEVENT END:VCALENDAR