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MATH 4290 Homework Assignment 1
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Due on Thursday, September 20th, at the BEGINNING of class.\\

\noindent $\bullet$
If $(X, T)$ is a non-invertible dynamical system and $x \in X$ is recurrent, prove that $Tx$ is recurrent.\\

\noindent $\bullet$
If $(X, T)$ is a non-invertible dynamical system, $x \in X$ is recurrent, and $U$ is an open set containing $x$, prove that $R_U(x) := \{n \in \mathbb{N} \ : \ T^n x \in U\}$ is infinite.\\

\noindent $\bullet$
If $(X, T)$ is a minimal dynamical system and $(Y, S)$ is conjugate to $(X, T)$, prove that $(Y, S)$ is also minimal.\\

\noindent $\bullet$
Prove that there exists $n \in \mathbb{N}$ so that the decimal expansion of $2^n$ begins with $777$, and give a provable upper bound on $n$.\\

\noindent $\bullet$
Define $(X, T)$ by $X = \mathbb{T}^2$, which you can think of as $[0,1)^2$ with the top/bottom edges identified and left/right edges identified, and $T: (x, y) \mapsto (x + \alpha, y + \alpha)$ for $\alpha \notin \mathbb{Q}$. Describe, with proof, all minimal subsystems of $(X, T)$.\\

\noindent $\bullet$
If $(X, T)$ is a dynamical system and $(M, T)$ is a nonempty subsystem, prove that $(M, T)$ is the unique minimal subsystem of $(X, T)$ if and only if every subsystem of $(X, T)$ contains $(M, T)$. 

%\noindent $\bullet$
%Choose any $\alpha \notin \mathbb{Q}$, and take $(X, T)$ to be the irrational circle rotation given by $X = \mathbb{T}$ and $T = T_{\alpha}$. Prove that for any %$x \in X$ and open $U$ containing $x$, the set $R_U(x) = \{n \in \mathbb{N} \ : \ T^n x \in U\}$ has \textbf{bounded gaps}, i.e. there exists $N$ so that for
% every $i \in \mathbb{N}$, at least one of $i, i+1, \ldots, i+N$ is in $R_U(x)$.\\



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