Seminário Marco López-UFC-20.05.2022.pdf
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                    Shrinking target sets for nonautonomous
systems
May 19, 2022

For an autonomous dynamical system T : X → X the shrinking target set
D is the set of points in X whose orbits hit a sequence of balls B (xn, rn ), with
rn → 0, innitely many times. That is,
D=

∞ \
∞
[

T −n (B (xn , rn )) .

m=1 n=m

This type of sets are dynamical counterparts to sets of well-approximable numbers in Diophantine approximation.
A nonautonomous dynamical system consists of a sequence of maps Tn : X →
X where iteration is dened as
T n = Tn ◦ Tn−1 ◦ · · · ◦ T1 .

In this talk we will consider two dierent kinds of problems related to shrinking targets: One regarding dimension and one regarding measure. For nonautonomous systems the story begins with work related to Cantor series expansions
by Fishman, Mance, Simmons, and Urba«ski (regarding dimensions); and Sun
and Cao (regarding measures). We will then talk about the more general setting
of conformal (nonautonomous) iterated function systems in higher dimensions.

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