arXiv subjects

Todd Fisher

Publications and source records attributed to Todd Fisher.

At least 19 recordsLinked to original sources

arXiv2021

Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature

We study the one parameter family of potential functions $qφ^u$ associated with the geometric potential $φ^u$ for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that there is a unique equilibrium state associated with $qφ^u$, and it has full support. For $q > 1$ it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value $q=1$ and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure, or measures supported on the singular set. In particular, when~$q = 1$, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

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arXiv2020

A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows

We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.

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arXiv2020

Accessibility and centralizers for partially hyperbolic flows

Stable accessibility for partially hyperbolic diffeomorphisms is central to their ergodic theory, and we establish its $C^1$-density among 1. all, 2. volume-preserving, 3. symplectic, and 4. contact partially hyperbolic flows. As applications, we obtain in each of these 4 categories $C^1$-density of $C^1$-stable topological transitivity, ergodicity, and triviality of the centralizer.

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arXiv2020

Equilibrium states for natural extensions of non-uniformly expanding local homeomorphisms

We examine uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a Hölder continuous potential function. We do this by applying general techniques developed by Climenhaga and Thompson, and show there is a natural condition on decompositions that guarantees that a unique equilibrium state exists. We then show how to apply these results to partially hyperbolic attractors.

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arXiv2019

Equilibrium states for certain partially hyperbolic attractors

We prove that a class of partially hyperbolic attractors introduced by Castro and Nascimento have unique equilibrium states for natural classes of potentials. We also show if the attractors are $C^2$ and have invariant stable and centerunstable foliations, then there is a unique equilibrium state for the geometric potential and its 1-parameter family. We do this by applying general techniques developed by Climenhaga and Thompson.

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arXiv2019

Centralizers of hyperbolic and kinematic-expansive flows

We show generic $C^\infty$ hyperbolic flows (Axiom A and no cycles, but not transitive Anosov) commute with no $C^\infty$-diffeomorphism other than a time-t map of the flow itself. Kinematic expansivity, a substantial weakening of expansivity, implies that $C^0$ flows have quasi-discrete $C^0$-centralizer, and additional conditions broader than transitivity then give discrete $C^0$-centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set.

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arXiv2018

Unique equilibrium states for geodesic flows in nonpositive curvature

We study geodesic flows over compact rank 1 manifolds and prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not carry full pressure. In dimension 2, this proves uniqueness for scalar multiples of the geometric potential on the interval $(-\infty,1)$, which is optimal. In higher dimensions, we obtain the same result on a neighborhood of 0, and give examples where uniqueness holds on all of $\mathbb{R}$. For general potential functions $φ$, we prove that the pressure gap holds whenever $φ$ is locally constant on a neighborhood of the singular set, which allows us to give examples for which uniqueness holds on a $C^0$-open and dense set of Hölder potentials.

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arXiv2017

Entropy of $C^1$ diffeomorphisms without a dominated splitting

A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound deriving from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). Adapting classical techniques, we use perturbations that are local and can be chosen to preserve volume or symplectic form or a homoclinic connection. This optimal entropy creation yields a number of consequences for $C^1$-generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism and we prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero.

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arXiv2016

Local perturbations of conservative $C^1$-diffeomorphisms

A number of techniques have been developed to perturb the dynamics of $C^1$-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create local homoclinic orbits. These techniques have been crucial for the understanding of $C^1$ dynamics, but their most precise forms have mostly been shown in the dissipative setting. This work extends these results to volume-preserving and especially symplectic systems. These tools underlie our study of the entropy of $C^1$-diffeomorphisms in (arxiv:1606.01765). We also give an application to the approximation of transitive invariant sets without genericity assumptions.

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arXiv2014

Open sets of diffeomorphisms with trivial centralizer in the $C^1$ topology

On the torus of dimension $2$, $3$, or $4$, we show that the subset of diffeomorphisms with trivial centralizer in the $C^1$ topology has nonempty interior. We do this by developing two approaches, the fixed point and the odd prime periodic point, to obtain trivial centralizer for an open neighbourhood of Anosov diffeomorphisms arbitrarily near certain irreducible hyperbolic toral automorphism.

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arXiv2013

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.

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arXiv2011

Entropy-expansiveness for partially hyperbolic diffeomorphisms

We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.

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arXiv2011

Entropic stability beyond partial hyperbolicity

We analyze a class of deformations of Anosov diffeomorphisms: these $C^0$-small, but $C^1$-macroscopic deformations break the topological conjugacy class but leave the high entropy dynamics unchanged. More precisely, there is a partial conjugacy between the deformation and the original Anosov system that identifies all invariant probability measures with entropy close to the maximum. We also establish expansiveness around those measures. This class of deformations contains many of the known nonhyperbolic robustly transitive diffeomorphisms. In particular, we show that it includes a class of nonpartially hyperbolic, robustly transitive diffeomorphisms described by Bonatti and Viana.

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arXiv2009

Resolving extensions of finitely presented systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map $π_+: X\to Y$. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.

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arXiv2009

Symbolic extensions for partially hyperbolic diffeomorphisms

We show that every partially hyperbolic diffeomorphism with a 1-dimensional center bundle has a principal symbolic extension. On the other hand, we show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing non-hyperbolic robustly transitive sets with center indecomposable bundle of dimension at least 2.

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arXiv2009

Intrinsic ergodicity for certain nonhyperbolic robustly transitive systems

We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic, but nevertheless have constant entropy and isomorphic unique measures of maximal entropy.

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arXiv2008

Trivial centralizers for codimension-one attractors

We show that if $Λ$ is a codimension-one hyperbolic attractor for a $C^r$ diffeomorphism $f$, where $2\leq r\leq \infty$, and $f$ is not Anosov, then there is a neighborhood $\mathcal{U}$ of $f$ in $\mathrm{Diff}^r(M)$ and an open and dense set $\mathcal{V}$ of $\mathcal{U}$ such that any $g\in\mathcal{V}$ has a trivial centralizer on the basin of attraction for $Λ$.

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arXiv2008

Trivial centralizers for Axiom A diffeomorphisms

We show there is a residual set of non-Anosov $C^{\infty}$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If $M$ is a surface and $2\leq r\leq \infty$, then we will show there exists an open and dense set of of $C^r$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a compact invariant set $Λ$ where $Λ$ is a hyperbolic chain recurrent class for one of the diffeomorphisms.

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