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On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Christian Camilo Silva Villamil

TL;DR

The paper studies how the Hausdorff dimension of the dynamical Lagrange spectrum $\mathcal{L}_{\varphi,f}$ changes with the threshold $t$ for generic perturbations of a conservative system with a mixing horseshoe. It defines the dimension function $L_{\varphi,f}(t)=HD(\mathcal{L}_{\varphi,f}\cap(-\infty,t))$ and shows, on a canonical interval $I_{\varphi,f}$ bounded by endpoints $c_{\varphi,f}$ and $\tilde{c}_{\varphi,f}$, that for generic $\varphi$ and $f$ the discontinuities of $L_{\varphi,f}$ occur at most at finitely many points away from the endpoints; in particular, at most two accumulation points of discontinuities can exist. The proof differentiates the regimes $HD(\Lambda)<1$ and $HD(\Lambda)\ge 1$, uses subhorseshoe geometry, stable/unstable Cantor sets, and connections via invariant manifolds, and reduces the high-complexity dynamics to finite-type decompositions when necessary. The results extend previous conservative-case continuity phenomena to generic non-conservative perturbations, with clear geometric implications for the structure of the dynamical Lagrange spectrum in hyperbolic dynamics on surfaces.

Abstract

Let $\varphi_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a mixing horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $\varphi$, close to $\varphi_0$, let $\mathcal{L}_{\varphi, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $Λ(\varphi)$ of the horseshoe $Λ_0$ and $f$. We show that, for generic choices of $\varphi$ and $f$, if $L_{\varphi, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{\varphi, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{\varphi, f}$.

On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

TL;DR

The paper studies how the Hausdorff dimension of the dynamical Lagrange spectrum changes with the threshold for generic perturbations of a conservative system with a mixing horseshoe. It defines the dimension function and shows, on a canonical interval bounded by endpoints and , that for generic and the discontinuities of occur at most at finitely many points away from the endpoints; in particular, at most two accumulation points of discontinuities can exist. The proof differentiates the regimes and , uses subhorseshoe geometry, stable/unstable Cantor sets, and connections via invariant manifolds, and reduces the high-complexity dynamics to finite-type decompositions when necessary. The results extend previous conservative-case continuity phenomena to generic non-conservative perturbations, with clear geometric implications for the structure of the dynamical Lagrange spectrum in hyperbolic dynamics on surfaces.

Abstract

Let be a -conservative diffeomorphism of a compact surface and let be a mixing horseshoe of . Given a smooth real function defined in and some diffeomorphism , close to , let be the Lagrange spectrum associated to the hyperbolic continuation of the horseshoe and . We show that, for generic choices of and , if is the map that gives the Hausdorff dimension of the set for , then there are at most two points that can be limit of a infinite sequence of discontinuities of .
Paper Structure (16 sections, 24 theorems, 186 equations, 2 figures)

This paper contains 16 sections, 24 theorems, 186 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical spectra
  3. Dynamical spectra
  4. Statement of the main theorem
  5. Preliminary results
  6. Stable and unstable dimensions
  7. Sets of finite type and connection of subhorseshoes
  8. Proof of Theorem \ref{['principal0']}
  9. The residual subsets
  10. A technical proposition
  11. First accumulation point of the Lagrange spectrum
  12. Geometric consequences of having a discontinuity
  13. Sequences of subhorseshoes
  14. Subhorseshoes and connection by periodic orbits
  15. End of the proof of Theorem \ref{['principal0']} when the dimension is less than $1$
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $\mathcal{U}\subset\textrm{Diff}^{2}(S)$ is sufficiently small, then there exists a residual subset $\mathcal{U}^{*}\subset \mathcal{U}$ with the property that for every $\varphi\in\mathcal{U}^{*}$ and any $r\geq2$, there exists a $C^r$-residual set $\mathcal{P}_{\varphi,\Lambda}\subset C^r(S,\ma and Even more,

Figures (2)

  • Figure 1: The subhorseshoes $\Lambda^s(t_0)$ and $\Lambda^u(t_0)$ in Proposition \ref{['conection']}.
  • Figure 2: Construction of $O(n)$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 32 more