(Thurston-Sullivan?) Are all smooth actions of $\Phi _g$ on $\mathbb S^1$ which are topologically conjugate to a standard one differentiably conjugate to the standard one?
Hyperbolic
$\varphi _t: T^1M \to T^1M$ Anosov geodesic flow and $V: M \to \mathbb{R}$ such that $\int V(\pi \varphi_t x ) = 0 $ on every closed geodesic. Is $V$ identically $0$?
$T:M \to M \; C^\infty $ Anosov. $f \in C^r $ and $f(x) = u(x) - u(Tx) .$ Does $u \in C^r?$ ($r\geq 2$).
- Subshifts of finite type have good quotients with fixed points.
- Given a periodic point $p$ in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
- If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
- Do two subshifts of finite type with the same entropy have a common good quotient?
- $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B). $ Does $\sigma_A |\Sigma_A$ embed in $\sigma_B |\Sigma_B$?
Question 136 with expansive instead of hyperbolic.
For most $C^2$ maps $f : [0,1] \to [0,1]$, for all $\epsilon >0 $ there is a hyperbolic set $\Lambda$ such that $h(f|\Lambda) \geq h(f) - \epsilon.$ (See question 8c.)
Embedding algebraic variety over $Z_p$ into a basic set.
Cancellation theorem for basic sets. Analogue of cobordism theorem.
(Thom) $Grad F$ for real analytic $F:\mathbb{R}^n \to \mathbb{R}$. Stratification of orbits near a singular point.
(Sullivan) Show that $|\lambda| = 1$ for $f: M \to M$, where $f$ is $C^1$ and distal and $\lambda $ is an eigenvalue of $f_\ast$. Is $f \sim g$ for some Morse Smale $g$?
$det (I - A) $ as a group invariant for the weak foliation $W^{wu}$, seen as a subgroup lying in some $H_m (M, \mathbb S^1)$?
For a hyperbolic attractor $\Lambda$ of dimension $r$, does $W^s(x) \cap \Lambda $contain a disk of dimension $k := r- Dim W^u (x)?$
Central Limit Theorem for $\beta$-transform $x \mapsto (\beta x)$.
Embed automorphisms of compact groups as basic sets.
Note: \[ \frac {1}{1-t} \; = \; \Pi _{n= 0}^\infty (1 +t^{2^n}) \] in $Z[[ t]]$. Is there a $C^2$ map $\mathbb D^2 \to \mathbb D^2$ so that.....
How can you write \[ 1 +t +t^2 \; = \; \Pi _{i=0 }^\infty (1\pm t^{n_i}) \] in $Z[[ t]] ?$
Is the false zeta function of a basic set ....? Define false zeta function for a flow basic set
If a $C^1$ Anosov preserves a smooth measure, is it an equilibrium state for $ - \log \lambda ^u ?$
(Doug (Lind?)) Find open partitions in $\Sigma _{1/2,1/2}$ that are not weakly Bernoulli. Find invariants of finitary codes.
Example of a non-ergodic $C^1$ Anosov diffeo on $\mathbb{T}^2$ preserving Lebesgue measure.
Does every manifold $M^n, n\geq 3$ admit a smooth Bernoulli flow (Ruelle)?
Among degree $n$ polynomial maps of $[0,1]$ to itself, are Axiom A open and dense. Do bad ones form a stratified set? ...
If $f$ is Anosov on $M$ and $\tilde M$ contractible, what does $H^k(M)(\sim H^k (\pi _1(M)) )$ tell you via $f_\ast$ eigenvalue information? (See [1], pp. 200-202)
References
For Anosov flow $\varphi_t $ on $M$, try to approximate curves in $M$ by pseudo-orbits and compute $\pi_1(M)$ . . . as in Morse theory.
Unstable foliations of Anosov diffeos are given by some nilpotent group action.
Reddy examples of expansive maps. Related to Anosov diffeos. Are expansive diffeos likely to be Anosov?
If $f$ is Axiom A, is there an Axiom A $g$, $C^0$ near $f$ with dim$\Omega (g) = d$ and $h(f) = h(g) ?$
If $f$ is Anosov and $g \sim f$, does $h(g) \geq h(f) ?$
Cancellation of $\Omega_i$. Simplest $f$ in an iosotopy class.
Classify all Anosov systems or attractors (which $\Omega_i$ can occur as attractors?)
Electric circuits
- Analogue computer for finding Axiom A examples
- Is noise sometimes due to hyperbolicity in the dynamics?
$\ell (f^n \gamma) $ grows slowly with $n$ for many curves $\gamma$ and Axiom A diffeos $f$.
$C$-dense (mixing) Axiom A flows
- speed of mixing
- asymptotic expression for the number of periodic orbits
- is $\varphi_1$ intrinsically ergodic?
- direct proof of mixing of measures
- analogue of $h(f) \geq \log |\lambda| $
- understand det$(Id - A) $ as an invariant; relation to $\zeta (0)$
- stability of $C$-density for attractors
- condition on $g$ so that $\Sigma_A (g) $ is analytically or $C^\infty $ embeddable as a basic set.
- can a closed orbit of an Anosov flow be null homotopic?
Is $\varphi_1$ a continuity point for the entropy as a function of diffeos when $\varphi_t $ is Axiom A flow? an Anosov flow?
If a geodesic flow is expansive, is it an Anosov flow?
Anosov diffeos
- Hypothesis on $H_1(M)$
- Fixed points
- $\Omega = M$
Find Axiom A infinite attractor in some O.D.E. on $\mathbb{R}^3$ (quadratic).
Kupka-Smale plus $h(f) >0 $ forces homoclinic points.
Bifurcation of Axiom A in terms of symbols.
Canonical $C^0$ perturbation of Anosov diffeo to 0-dimensional $\Omega _i$'s with the same entropy
Canonical embedding of Axiom A $\Omega _i$
Assume $\varphi _t $ $C$-dense. If $\nu $ is $\varphi _1 $ invariant is $\nu $ $\varphi $-invariant?
Interpret $-\log \lambda^u$ as a potential function (Kolmogorov's idea on surfaces of negative curvature)?
Renewal theorems for dependent random variables.
- Derive as a motivation for Axiom A flow mixingness
- How fast is the mixing for Axiom A flows?
Suspensions of diffeos. - Are they generically not (conjugated to) constant time suspensions? what is the strongest statement for Axiom A attractors?
Shub's entropy conjecture: $h(f) \geq \log |\lambda| $
- for diffeos
- $\Omega $ finite plus hyperbolic
- Axiom A with cycles.
Nonalgebraic Anosov diffeos. Classify 3-dimensional Anosov flows; Is the variable curvature surface geodesic flow conjugate to constant curvature?
Unique ergodicity of $W^u$ for partially Anosov diffeos.
Non Axiom A examples. Newhouse, Abraham-Smale, Simon, Lorenz, billiards.
- Axiomatic description
- Statistical properties
- For all $\epsilon$, there exists a horseshoe $X_\epsilon$ inside with $ h(f|X_\epsilon) \geq h(f) - \epsilon$
- Statistical properties of Lorenz in particular
- Any specification type property
Structure of basic sets
- Classification via $(R,A)$
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Local Axiom A implies embeddable
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existence of canonical coordinates implies embeddable (compact abelian group actions ? are $\Omega $'s).
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Phantom homology groups -shift equivalence of induced maps!
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dim $\Omega$?; when is the quotient a manifold?
Zeta function for Axiom A flows and systems
- topological identification (try 1-dimensional $\Omega $ first); conjugacy invariance of $\zeta (0)$.
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For $C^\infty $ flows, $\zeta (s) $ has a meromorphic extension to the complex plane.
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Connection with Laplacian vs. geodesic results; automorphic forms.
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Anosov actions.
Classification of singularities by the local properties of the gradient flow.