The Fitzrovia Forum

8 Oct 2024

Dynamics.Day@UCL

Time: 14:00 – 17:00

G01, Euston Road (222)

• 12PM-Lunch–Euston Flyer

  • No reservation (there is usually lots of room on a Tuesday)

• Drinks+dinner afterwards

  • Location TBD

Speakers

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2:00-2:50

Anthony Quas

Universal gaps between Lyapunov exponents in noisy cocycles

Consider a sequence of $d\times d$ matrices $(A_n)$, and write $A_n^\epsilon$ for $A_n + \epsilon N_n$ where $N_n$ are independent Gaussian matrices. We study the products ${A^\epsilon}^{(n)}=A^\epsilon_n\cdots A^\epsilon_1$, and consider gaps between the Lyapunov exponents. We show that there exist strictly positive functions $f_d(\epsilon)$ with the property that for any choice of $(A_n)$, and for any $\epsilon$, the gaps are at least $f_d(\epsilon)$.

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3:00-3:50

Richard Sharp

An abelian Livsic theorem for amenable covers

Consider a transitive Anosov flow generated by a vector field $X$. A famous theorem of Livsic says that if a Hölder continuous function $f$ integrates to zero around every periodic orbit of a hyperbolic flow then the function is a dynamical coboundary, i.e. $f = L_Xu$, where $L_X$ is the Lie derivative. More recently, A. Gogolev and F. Rodriguez Hertz established an “abelian Livsic theorem”: if the flow is homologically full and $f$ integrates to zero around all null-homologous periodic orbits then $f = L_Xu + \omega(X)$, where $\omega$ is some closed $1$-form. We will discuss a new proof of this result, based on thermodynamic formalism, and give an extension to amenable covers.

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3:50-4:10

Break

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4:10-5:00

Reem Yassawi

An application of Veech’s structure theorem to automorphism groups

Veech’s structure theorem is a generalisation of Furstenberg’s structure theorem for distal systems. It tells us that a minimal system with a residual set of distal points has an almost automorphic extension which is an inverse limit of almost isometric extensions. We use Veech’s theorem to give an improved description of the automorphism group for some systems with a nontrivial equicontinuous factor. This is joint work with Clemens Müllner.

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Schedule and other details to appear

Contact

Please let me know if you are interested in attending.

Terry Soo
t.soo@ucl.ac.uk

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