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Home exam: course by Andrґ Navas es

Problem 1. a) Find a (non-minimal) map T : X X and a function : X R with bounded Birkhoff sums, x X, n N |Bn ()| C, for which the corresponding skew product admits no invariant continuous section. b) The same question, but now T has to be topologically transitive. Problem 2. Prove the Hedlund Theorem for group actions: if G acts on X minimally, and c : G в X R is an additive cocycle, then the following conditions are equivalent: i) x0 X, C > 0 s.t. |c(g , x0 )| C g G ii) x0 X, C > 0 s.t. |c(g , x0 )| C g G iii) The cohomological equation (g x) - (x) = c(g , x) has a continuous solution. Problem 3. Let G = Zd be a group, for an action of which we are given a bounded cocycle c(·, ·). Prove that then a sequence of almost invariant sections could be written explicitly as 1 i i n (x) = d c(T11 . . . Tdd , x) n 0i k

Problem 4. Prove that the following are equivalent: i)
S
1

log Dg (x)dµ(x) = 0 for any g -invariant measure µ;

ii) g has no hyperbolic periodic fixed points Problem 5. A center of a bounded set is the center of a ball of the smallest radius containing this set. a) Find the center of a triangle; b) Prove that the center belongs to convex closure of a set. Problem 6. a) Prove that for an irrational rotation of the circle for any continuous function C (S 1 ) one has a (uniform in x S 1 ) convergence Sn (x) n (x) dx.
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b) Prove that the cardinality of the set of {i : 0 i < n | + i I }, divided by n, tends to the length of an interval I . 1


Problem 7 (Denjoy-Koksma inequality). For a function on a circle with bounded variation show that , n |Sqn () - q
n S
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(x)dx| Var ,

n where pn is a sequence of good approximations of , that is, incomplete fractions correq sponding to the decomposition of into a continuous fraction.

Problem 8. Let G be a semigroup acting by isometries on a C AT (0)-space. Assume that there is a point with a bounded orbit. Then, the action has a fixed point. Problem 9. Prove the existence of a barycenter in the Cartan sense for the functional given by w H d2 (w, z )dµ(z ) Problem 10. Prove that the procedure of constructing a barycenter by induction converges to a point and prove the inequality d(barn (w1 , . . . wn ), barn (w1 , . . . wn )) 1 n
n

d(wi , wi )
1

Problem 11. Prove the existence and uniqueness of a center for uniformly convex Banach spaces. Problem 12 (Furstenberg example). Let f : T 2 T 2 be a map defined as (x, y ) (x + , y + (x)), where is irrational, and S 1 (x)dx = 0. Prove that a) The map f is minimal if and only if the cohomological equation (x) = h(x + ) - h(x) has no continuous solution; b) Show that there exists and such that the corresponding cohomological equation has a measurable solution, but no continuous one; c) Construct such a map f that is minimal, but not ergodic. Problem 13. Let G be a group of area-preserving diffeomorphisms of a compact orientable surface S . a) Associate to it a skew product action on the total space S в H, where H is a hyperbolic plane, such that the fiberwise actions are isometries. b) Translate to this framework the isometries version of Hedlund theorem: what is the resulting statement?

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