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John Smillie

Publications and source records attributed to John Smillie.

At least 19 recordsLinked to original sources
arXiv2025

Tremors and horocycle dynamics on the moduli space of translation surfaces

We introduce a "tremor" deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horocycle flow orbits in strata of translation surfaces. In the genus two stratum with two singular points, we find orbits which are generic for a measure whose support is strictly contained in the orbit and find orbits which are not generic for any measure. We also describe a horocycle orbit-closure whose Hausdorff dimension is not an integer.

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arXiv2023

On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by $\mathrm{SL}_2(\mathbb{R})$. As a consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle flow orbits do not equidistribute towards any invariant measure.

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arXiv2023

Horospherical dynamics in invariant subvarieties

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety, and geometrically, it is induced by the canonical splitting of $\mathbb{C}$-valued cohomology into its real and imaginary parts. We define a natural volume form on the leaves of this foliation, and define horospherical measures as those measures whose conditional measures on leaves are given by the volume form. We show that the natural measures on invariant subvarieties, and in particular, the Masur-Veech measures on strata, are horospherical. We show that these measures are the unique horospherical measures giving zero mass to the set of surfaces with no horizontal saddle connections, extending work of Lindenstrauss-Mirzakhani and Hamenstaedt for principal strata. We describe all the leaf closures for the horospherical foliation.

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arXiv2020

Horocycle dynamics: new invariants and eigenform loci in the stratum H(1,1)

We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. We classify the orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution statements regarding limits of sequences of measures, some of which have applications to counting problems.

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arXiv2014

A Symbolic Characterization of the Horseshoe Locus in the Hénon Family

We consider the family of quadratic Hénon diffeomorphisms of the plane ${\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a 2-D analog of the familiar "${1\over 2}$-wake" for the quadratic family $p_c(z) = z^2$.

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arXiv2012

Parabolic Bifurcations in Complex Dimension 2

Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions. Specifically we consider a two variable family of diffeomorphisms $F_ε: M\to M$ given locally by $$F_ε(x,y) = (x + x^2 + ε^2+ ..., b_εy+...)$$ where $|b_ε|<1$, and the `$...$' terms involve $x$, $y$ and $ε$.

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arXiv2010

Homotopy shadowing

Michael Shub proved in 1969 that the topological conjugacy class of an expanding endomorphism on a compact manifold is determined by its homotopy type. In this article we generalize this result in two directions. In one direction we consider certain expanding maps on metric spaces. In a second direction we consider maps which are hyperbolic with respect to product cone fields on a product manifold. A key step in the proof is to establish a shadowing theorem for pseudo--orbits with some additional homotopy information.

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arXiv2010

Geodesic flow on the Teichmueller disk of the regular octagon, cutting sequences and octagon continued fractions maps

In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichmueller disk of the octagon and explain the relation with Teichmueller geodesic flow. This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic flow on the modular surface. We use this connection to construct the natural extension and the invariant measure for the continued fraction map. We also define an acceleration of the continued fraction map which has a finite invariant measure.

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arXiv2009

Symbolic coding for linear trajectories in the regular octagon

We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well studied Sturmian sequences which can be analyzed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.

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arXiv2008

Characterizations of lattice surfaces

We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property.

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arXiv2008

Finiteness results for flat surfaces: large cusps and short geodesics

For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any non-elementary Veech group can appear only finitely many times in a fixed stratum, that any non-elementary Veech group is of finite index in its normalizer, and that the quotient of the upper half plane by a non-lattice Veech group contains arbitrarily large embedded disks. These are proved using the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T.

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arXiv2007

Algebraically periodic translation surfaces

Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms provide examples. In this case the periodic direction field is the holonomy field. We show that every algebraic field arises as the periodic direction field of a translation surface arising from a right-angled billiard table. The J-invariant of a translation surface was introduced by Kenyon and Smillie. We analyze the $J$ invariants of algebraically periodic surfaces and show that in some cases they are determined by the periodic direction field. We give explicit formulas for $J$ invariants in these cases. The Homological Affine Group was introduced by McMullen in his study of translation surfaces in genus two. We calculate this group for many algebraically periodic surfaces and relate it to the automorphism group of the J-invariant. We show that surfaces which admit certain decompositions into squares have totally real periodic direction field. This is related to a result of Hubert and Lanneau.

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arXiv2003

Dynamics in two complex dimensions

We describe results on the dynamics of polynomial diffeomorphisms of ${\bf C^2}$ and draw connections with the dynamics of polynomial maps of ${\bf C}$ and the dynamics of polynomial diffeomorphisms of ${\bf R^2}$ such as the Hénon family.

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arXiv2001

Polynomial Diffeomorphisms of $\C^2$. VIII: Quasi-Expansion

This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C^2. We introduce a dynamical property of polynomial diffeomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes hyperbolicity for polynomial maps of the complex plane.

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