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HW 1.  (Classical Geometries. ) Due Monday, Oct 1, 2012, i.e.: Next class.  
\vskip .3cm

1. A gradient flow.  Let $\T$ be the flat torus
with standard coordinates $\theta_1, \theta_2$ mod $2 \pi$.
Let $V = - \cos(\theta_1) cos (\theta_2)$.     Locate the
equilibria.  Describe each type (source, saddle, sink). Sketch the flow lines .

\vskip .2cm
 2. Hamiltonian flows.     For $M = T^* \R = \R \times \R$ let
 $H = (1/2) p^2 + P(x)$ where $P$ is a polynomial. 
 Sketch the phase portraits in case
 
 a) $P$ is linear
 
 b) $P$ is  homogeneous  quadratic.
 
 c) $P$ is cubic. Do a few cases.
 
 d) $P = (x-1)^2 (x+1)^2$.
 \vskip .2cm
3.  Again as in 2. Again $P$ is polynomial.  Is the flow complete?
Find a proof or a counterexample.
\vskip .2cm
4.  N-dimensional oscillator.  This has for its Hamiltonian
$H(q,p) = (1/2) \langle p , p \rangle ^2 + \langle q, A q \rangle$
where $p, q \in \R^n$, where  we use the standard inner product $\langle \cdot , \cdot  \rangle$
to identify $\R^n$ with its dual, and where $A$ is a positive definite symmetric matrix.
    Prove that the closure of the typical orbit
is a $k$-torus, for some $k  \le N$.  Describe the maximal $k$ in terms
of the eigenvalues of $A$. 
\vskip .2cm
5. Guckenheimer-Holmes.  Exer. 5.1.2 and 5.1.3  of p. 234.

These exercises are on the   Smale Horseshoe and are best solved
using symbolic dynamics.   Let $\Gamma \subset I^2$ be the subset of
the square $S = I^2$  which never leaves the square in forward or backward time.

5.1.2. Show that all the periodic orbits are of saddle type.  Locate the periodic
orbits with period 4 or less and write out their symbol sequence.
Show that $\Lambda$
contains a countable infinity of heteroclinic and homoclinic orbits. Show that $\Lambda$
contains an uncountable number of orbits  which are not periodic.  

5.1.3 Show that $\Lambda$ contains a dense orbit.
\vskip .2cm
6.  A gradient system in $\R^n$ is given by $\dot x = -\nabla V (x), x \in \R^n$
where $V$ is a smooth function.  What is special about the linearization of a gradient
system at an equilibrium, in comparison to  a general linear system $\dot x = A x$
with $A$ a general n by n matrix.
\vskip .2cm
7.    Newton's equations on   $\R^n$ are equations of the form  $\ddot x = -\nabla V (x), x \in \R^n$
where $V$ is a smooth function, called the potential. 

a) First orderize the system by introducing $v = \dot x$ so as to make
it an ODE on $\R^n \times \R^n = T \R^n$.   

b.  What is special about the linearization of Newton's equations
at an equilibrium in in comparison to  a general linear system $\dot x = A x$
with $A$ a general $ 2n by 2n$ matrix.
 
 \vskip .2cm
 8.  Take the Cantor set to be the product space $\Z_2 ^{\Np}$,
( $\Np$ is the set of all positive integers)
 endowed with the product topology.  
 An element of the Cantor set is then an infinite sequence $(\sigma_i)_{i \in \N}$ of 1's and 0's;
 $\sigma_i \in \{0, 1 \}$. 
   the sequence labelled by the positive  integers. Consider the map
 $F$ whcih sends  to
 $$; _2: C_2  \to [0,1];  F_2 ( \sigma) = \Sigma_{i \in \Np} \sigma_i 2 ^{-i}$$
 
 a) Show that $F_2$ is continous and onto.
 
 b) If we give   $\Z_2$ the ``coin-flip'' measure (each element has probability $1/2$)
  then the Cantor set inherits
 a probability measure.  (The product of probability spaces is a probability space, so that the Cantor set 
 has a probability measure on it. )
 Show that $F_2$ is an isomorphism in the sense
 of measure theory: it is onto, and the map is measure preserving:
 $\mu (F_2 ^{-1} (I) = | I |$
 for any interval $I$.
 
 hint: consider dyadic intervals.
 
 d)Show that $F_2$ is a measure preserving semi-conjugacy between the Bernoulli shift
 on the Cantor set and the doubling map (mod 1) on the interval.
 
 e) Repeat  (a)-(c) for $F_N: \Z_N ^{\Np} \to I$. 
\vskip .3cm

f) Use the fact that there is a bijective map from two disjoint copies of $\Np$ to $\Np$
to define an onto map $C \to I \times I$.  Repeat, to establish the existence of an onto
map from the Cantor set ONTO the $n$-cube.  Onto any compact $n$-manifold.


\vskip .2cm
9. Show that the doubling map $S^1 \to S^1$ is measure preserving.
\vskip .2cm
10. Prove that rotation of the circle is NOT mixing.
\vskip .2cm
11.  Prove that the suspension of a map is NOT a mixing flow.
\vskip .2cm
12. Construct a homeomorphism of the plane $\R^2 = \C$
which maps the spiral $exp(1+ i )t), t \in \R$
to the ray $y = 0 , x> 0$.



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