In this post, we discuss the Freire-Mane conjecture.

In this post, we discuss a question about flows with dominated splittings.

In this post, we talk about a proof of Klingenberg’s fact: an Anosov geodesic flow is without conjugate points.

In this post, we talk about the Gauss-Bonnet theorem for disks.

In this post, we talk about the Livsic theorem and how we can use it to upgrade orbit equivalences to conjugacies.

In this post I discuss a calculation involving topological entropy and expansive maps.

In this post I prove a result on absolutely continuous ergodic measures.

In this post I prove the existence of generic vectors.

In this post I discuss geodesic currents and intersection numbers

In this post I discuss Cartan’s moving frame on a surface.

In this post I discuss variational fields.

In this post I discuss a result on the spectrum of diffeomorphisms. Credit to Thomas O’Hare for discussing the result with me and showing me a proof.

In this post I discuss fibered partially hyperbolic diffeomorphisms.

In this post I discuss a way of pushing forward forms assuming a vertical condition.

In this post I discuss pushing forward differential forms, inspired by this stackexchange post.

In this post I discuss the Ledrappier-Walters formula.

In this post I discuss a variation on the closing lemma.

In this post I discuss a variation on Poincare Recurrence.

In this post I discuss the Perron-Frobenius operator, also known as the transfer operator.

In this post I continue the discussion from the last post.

In this post, I begin discussing the magnetic boundary rigidity problem, following this paper by Dairbekov, Paternain, Stefanov, and Uhlmann.

In this post, I discuss how to prove the existence of a Poincare map.

In this post, I outline some results on linearly independent vector fields on the \(n\)-sphere.

In this post we continue discussing/translating a rigidty result proven in Grognet for magnetic flows.

In this post I begin discussing/translating a rigidty result proven in Grognet for magnetic flows.

In this post, I discuss a the proof of isospectral rigidty on negatively curved surfaces following the argument of Guillemin and Kazhdan.

In this post, I review the matrix exponential.

In this post, I discuss a variety of definitions for the magnetic flow.

In this post, I talk about the Sasaki metric.

In this post, I talk about the fat Cantor set.

In this post, I talk a bit about the semester.

In this post, I talk a bit about the semester.

In this post, we discuss a basic set theory problem.

In this post, we discuss a new file uploaded.

In this post, I hope to outline one of the big theorems in ergodic theory – Poincare recurrence.

In this post, I describe a nice example of a homeomorphism of a compact space which is not minimal but is uniquely ergodic, as well as a condition for a complex matrix to have a square root.

In this post, I described what I’ve been doing this week.

In this post, we continue the discussion from here and Katok & Hasselblatt Section 2.1.

In this post, we explore the notion of degree for circles. This will mostly come from Hatcher and Katok & Hasselblatt (A First Course in Dynamical System).

In this post, I give a quick update on what I am up to this week (which is vacation).

In this post, I discuss a little about moduli and equivalences (see Katok & Hasselblatt 2.1) and a little on duality in Banach spaces (see chapter 4 of Rudin).

In this post, we’ll explore shadow orbits and Ryll-Nardzewski.

In this post, we explore more on topological entropy (Katok & Hasselblatt section 3.2) and discuss the Krein-Milman theorem (Rudin chapter 3).

In this post, we discuss topological entropy (Katok & Hasselblatt section 3.1) and weak topologies (Rudin chapter 3).

In this post, we discuss Katok & Hasselblatt section 1.7 and 1.8, as well as Rudin chapter 2.

In this post, we discuss a problem from dynamical systems that I’ve been thinking about, as well as the end of Rudin chapter 1.

In my last post (found here) I mentioned going through the metrization theorem. Today, I’ll actually go through and prove the result.

In this post, we discuss a little on Katok and Hasselblatt’s chapter 0, as well as Rudin chapter 1.

I’ve successfully added a blog component to the website (and also made it more modern with Jekyll implementation). The goal is to hopefully use the blog to discuss interesting things I’ve been thinking about in a more relaxed fashion, rather than having to write out a TeXed up note.