\relax \@writefile{toc}{\contentsline {section}{\numberline {0}Aims and Objectives}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {0.1}Acknowledgment}{2}} \@writefile{toc}{\contentsline {section}{\numberline {1}Circle Maps}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Rotation Numbers and Poincar\'e's Theorem}{3}} \newlabel{dense}{{1.1}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Denjoy's Theorem}{5}} \newlabel{denjoy1}{{1.3}{5}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Family of Circle Maps and Arnol'd Tongues}{7}} \newlabel{increasing}{{1.5}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Structural Stability}{8}} \newlabel{MSstable}{{1.8}{8}} \newlabel{MSapprox}{{1.9}{8}} \@writefile{toc}{\contentsline {section}{\numberline {2}Interval Maps}{9}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Sharkovskii's Theorem}{9}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Doubling Map and the Full Quadratic Map}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Some Ergodic Theory}{10}} \@writefile{toc}{\contentsline {section}{\numberline {3}Examples in Two Dimensions}{11}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}The Horseshoe Map}{11}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces There is a vertical strip in $S$ mapped across itself by $f^n$}}{12}} \newlabel{stilde}{{1}{12}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}The Linear Anosov Map}{12}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The Smale-Horseshoe in its 'original habitat', the sphere}}{13}} \newlabel{Horseshoe}{{2}{13}} \@writefile{toc}{\contentsline {section}{\numberline {4}Hyperbolic Dynamics}{13}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}The Theorem of Hartman-Grobman}{14}} \newlabel{HG}{{4.2}{14}} \newlabel{Phiconjug}{{4.4}{15}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Stability of Hyperbolic Periodic Points}{18}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Stable and Unstable Manifolds}{18}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.4}$\lambda $-Lemma}{19}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.5}Stable Manifold Theorem for General Invariant Sets}{19}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.6}The Shadowing Lemma}{20}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.7}An Application of the Shadowing Lemma}{21}} \newlabel{shadappl}{{4.11}{21}} \@writefile{toc}{\contentsline {section}{\numberline {5}Examples}{22}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}The Smale Attractor (Solenoid)}{22}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Homoclinic Intersections: The Occurrence of Smale's Horseshoe}{23}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}$\Omega $-Explosion}{23}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}The Markov Partition Associated to a Linear Anosov Map}{23}} \@writefile{toc}{\contentsline {section}{\numberline {6}Exercises}{24}} \@writefile{toc}{\contentsline {section}{\numberline {7}Beyond the Scope of This Course}{30}}