Searcharxiv⌕ Search

arXiv subjects

Mario Shannon

Publications and source records attributed to Mario Shannon.

7 recordsLinked to original sources
arXiv2026

Hyperbolicity of complements of orbits in Anosov flows

We show that if an Anosov flow on a 3-dimensional manifold has orientable stable and unstable foliations, then the complement of any filling periodic orbit is a hyperbolic manifold. This generalizes the known case of the complement of a closed, filling geodesic orbit in the unit tangent bundle of a hyperbolic surface. Furthermore, we show that the orientability condition on invariant foliations is necessary, by constructing a counterexample in the absence of this property. In the case of the suspension flows we obtain that the complement of every collection of periodic orbits is hyperbolic, while for the geodesic flow (regardless of orientability of the invariant foliations) the complement of every filling and anannular collection of periodic orbits is hyperbolic.

math.DS↗
arXiv2026

Fourier decay and non-decay for pseudo-affine self-conformal measures

We study the sharpness of recent sufficient conditions for polynomial Fourier decay of self conformal measures on the line. First, we construct a $C^\infty$ iterated function system which is not $C^1$-conjugate to self-similar, but which nevertheless admits a stationary measure that is not Rajchman. Second, for every strongly separated Bernoulli convolution $\mu$ and every $1\leq r<\infty$, we construct a $C^r$-diffeomorphism $h$ such that $h'$ is constant on $\operatorname{supp}\mu$, yet the image measure $h\mu$ has polynomial Fourier decay. All constructions are within the framework of pseudo-affine iterated function systems, previously introduced by the authors.

math.DS↗
arXiv2024

Infinitely many closed paths in the graph of Anosov flows

Given an Anosov flow on a closed 3-manifold, we are interested in the problem of whether or not making non-trivial Fried surgeries along a finite set of periodic orbits can produce a flow equivalent to itself. We show that for some suspension Anosov flows, there exist infinitely many pairs of periodic orbits satisfying this property.

math.DS↗
arXiv2024

How linear can a non-linear hyperbolic IFS be?

Motivated by a question of M. Hochman, we construct examples of hyperbolic IFSs $\Phi$ on $[0,1]$ where linear and non-linear behaviour coexist. Namely, for every $2\leq r \leq \infty$ we exhibit the existence of a $C^r$-smooth IFS such that $f'\equiv c(\Phi)$ on the attractor and $f''\equiv 0$ for every $f \in \Phi$, yet $\Phi$ is not $C^t$-smooth for any $t>r$, nor $C^r$-conjugate to self-similar. We provide a complete classification of these systems. Furthermore, when $r>1$, we give a necessary and sufficient Livsic-like matching condition for a self-conformal $C^r$-smooth IFS to be conjugated to one of these systems having $f''=0$ on the attractor, for every $f\in \Phi$. We also show that this condition fails to ensure the existence of a $C^1$-conjugacy in mere $C^1$-regularity.

math.DS↗
arXiv2019

Almost equivalence of suspension Anosov flows

We provide a written proof of a result due to H. Minakawa, which states that all suspension Anosov flows generated by hyperbolic matrices with positive trace are pairwise almost equivalent. The proof relies on constructing, for any given suspension flow, a genus-one Birkhoff section whose first-return map has fewer fixed points than the original map. We improve Minakawa's result by explicitly calculating the first return map onto this section, which leads to explicit bounds on the distances between suspension Anosov flows within the graph of Anosov flows.

math.DS↗
arXiv2018

Seifert manifolds admitting partially hyperbolic diffeomorphisms

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

math.DS↗