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SUMMARY:Michał Rams (IM PAN)
DTSTART:20251216T130000Z
DTEND:20251216T140000Z
DTSTAMP:20260423T040046Z
UID:OWNS/157
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/157/">T
 he entropy of Lyapunov-optimizing measures for some matrix cocycles</a>\nb
 y Michał Rams (IM PAN) as part of One World Numeration seminar\n\n\nAbstr
 act\nConsider a simple (to formulate...) mathematical object: you are give
 n a finite collection of matrices $A_i\\in GL(2\,\\mathbb R)\; i=1\,\\ldot
 s\,k$ and you are allowed to multiply them\, in any order. The notion you 
 are interested in is the exponential rate of speed of growth of the norm: 
 given $\\omega\\in \\{1\,\\ldots\,k\\}^{\\mathbb N}$\, let\n\\[\n\\lambda(
 \\omega) = \\lim_{n\\to\\infty} \\frac 1n \\log ||A_{\\omega_n} \\cdot \\l
 dots \\cdot A_{\\omega_1}||.\n\\] \nThis object has many names\, in dynami
 cal systems we call it the Lyapunov exponent. \n\nWe are in particular int
 erested in the set of those $\\omega$'s that give the extremal (maximal\, 
 minimal) value of the Lyapunov exponent. A long-standing conjecture states
  that for a generic matrix collection those sets ought to be {\\it small}\
 , in some sense. In the result I will present we (Jairo Bochi and me) are 
 proving that for certain open set of collections of matrices those $\\omeg
 a$'s that maximize/minimize Lyapunov exponent have zero topological entrop
 y.\n
LOCATION:https://researchseminars.org/talk/OWNS/157/
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