Smooth dynamics

Smooth dynamics is a subfield of dynamical systems and often studied with tools from ergodic theory and Topological dynamics.

Smooth dynamics ^[1] assumes a differential structure on top of manifold M or generically on Banach spaces over M. This can be piece-wise affine (I.e with constant derivatives), differentiable (e.g. ${\displaystyle C^{1}(M)}$ or ${\displaystyle C^{n}(M)}$, or smooth I.e. ${\displaystyle C^{\infty }(M)}$. Typically one studies dynamical systems in order of complexity and with different techniques, for example the typical setup of newtonian mechanics requires time evolution maps ${\displaystyle C^{2}(M)}$, a fractal attractor or a brownian motion may be continuous but not differentiable in a countable number of points and a piece-wise affine map may be studied with techniques of algebraic geometry.

The hyperbolicity condition, I.e having stretching and contracting directions, has a relation with chaos theory and therefore was studied extensively. Important subbranches are Hyperbolic dynamics, non uniform hyperbolicity^[2] and partial hyperbolicity.^[3]

Assume a two dimensional manifold, and a real (or complex) smooth map on it, after diagonalization I.e after a coordinate transformation, the map can be approximated locally by a linear map of the plane given by the matrix

$${\displaystyle {\begin{bmatrix}\lambda &0\\0&\mu \end{bmatrix}}}$$

There are 4 important cases:

1. The dissipative case,where 0<λ<μ <1 and the orbits converge towards a stable point or an attractor
2. The case where both eigenvalues are divergent 1<λ<μ and dynamics is divergent like for an unstable point
3. The case where one or more the eigenvalues have modulus 1: if all eigenvalues have modulus one, the matrix is unitary and there is typically a conserved quantity.
4. The hyperbolic case where the eigenvalues are 0<λ<1<μ and in one direction there is convergence and in the other direction there is divergence. 

To give a visual local intuition of the motion the unitary case (3) is a stable orbit such as a circle, in the dissipative case (1) the orbit is spiralling inwards towards the centre, and in the divergent case spiralling outwards (2), the hyperbolic case (4) is a mixed scenario which depends on the direction of motion.

Given a manifold that splits into a stable (s) and unstable (u) part ${\displaystyle T_{\Lambda }M=E^{s}\oplus E^{u}}$ and given the diffeomorphism splits also naturally into two endomorphisms ${\displaystyle (Df)_{x}E_{x}^{s}=E_{f(x)}^{s}}$ and ${\displaystyle (Df)_{x}E_{x}^{u}=E_{f(x)}^{u}}$ for all ${\displaystyle x\in \Lambda }$

uniform hyperbolic vs non-uniform hyperbolic vs partial hyperbolic

*	Stable manifold & forward	Unstable manifold & backward	Condition on eigenvalues
uniform hyperbolic[1]	${\displaystyle \|Df^{n}(v)\|\leq c\lambda ^{n}\|v\|}$, ${\displaystyle \forall v\in E^{s}}$ and ${\displaystyle n>0}$	${\displaystyle \|Df^{-n}(v)\|\leq c\lambda ^{n}\|v\|}$, ${\displaystyle \forall v\in E^{u}}$ and ${\displaystyle n>0}$	${\displaystyle \lambda <1<\mu }$
non uniform hyperbolic[2]	${\displaystyle \|Df_{x}^{n}(v)\|\leq C(x)e^{(\lambda _{s}+\epsilon )n}\|v\|,\forall v\in E_{x}^{s}}$ and ${\displaystyle n\geq 0}$	${\displaystyle \|Df_{x}^{-n}(v)\|\leq C(x)e^{(-\lambda _{u}+\epsilon )n}\|v\|,\forall v\in E_{x}^{u}}$ and ${\displaystyle n\geq 0}$	${\displaystyle \lambda _{s}<0<\lambda _{u}}$
partial hyperbolic[3]	${\displaystyle \|Df^{n}(v)\|\leq c\lambda ^{n}\|v\|}$ ${\displaystyle \forall v\in E^{s}}$ and ${\displaystyle n>0}$	${\displaystyle \|Df^{-n}(v)\|\leq c\mu ^{n}\|v\|}$, ${\displaystyle \forall v\in E^{u}}$ and ${\displaystyle n>0}$	${\displaystyle \lambda \neq \mu }$

Uniform hyperbolic is the case of Anosov diffeomorphisms and Hyperbolic sets, the definition is based on the boundness of the diffeomorphism, this case shows all common properties of Hyperbolic dynamics. The non uniform case generalizes evidencing the exponential divergence of close orbits and the dependence of the boundness condition on the position. The partial hyperbolic case is still based on boundness but generalizes instead for general eigenvalues. These last two cases instead are relevant for applications.

An initial goal of studying chaotic systems was to clarify the relationships between entropy and chaos. The initial intuition behind the Kolmogorov-Sinai entropy was that there were two classes of systems, the probabilitistic ones with entropy non zero and the deterministic ones with entropy zero, this is actually not true and hyperbolicity is linked to entropy. This also leads to the theory of deterministic chaos, which is "rigid" in structure but still unpredictable.^[4]

A second goal from the Smale school was to identify chaotic systems that are robust to perturbation, and the relationships between chaos and dynamical billiards which are in general hyperbolic. This has lead to the concept of structurally stable systems. Anosov diffeomorphism are proven to be structurally stable, i.e. the dynamics is robust to perturbations, by a theorem of Anosov.

A third goal was to understand the relationships between mixing, eigenvalues and phase transitions, this has lead to the "Thermodynamic formalism" from David Ruelle,^[5] where Topological entropy is defined formally starting from measure theory. This has allowed to understand it's relationships with information entropy through the variational principle. The work of Ruelle and Sinai was expanded by Rufus Bowen on anosov systems,^[6] their entropy and the introduction of Sinai-Ruelle-Bowen measures, ultimately leading Donald Ornstein to prove that Bernoulli shifts with the same entropy are isomorphic.^[7]

• Katok Hasselblatt An introduction to dynamical systems