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$C^1$ Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point

Rubio Gunawan

TL;DR

The paper constructs a $C^1$ circle covering map that is topologically conjugate to the doubling map and has a physical measure supported at a hyperbolic repelling fixed point. By carefully controlling regularity at a single point $q$, the authors produce two regimes: a full-basin example with $|B_{\delta_p}|=1$ when the map is $C^\infty$ off $q$, and a $C^1$-regular example at $q$ with $|B_{\delta_p}|>0$ (potentially full under relaxation at $q$). The core method uses a Realization Lemma to realize prescribed first return maps $F$ on an interval $I_2$ as the induced dynamics of a full-branch circle map in $\mathcal{D}$, enabling the application of wild-attractor techniques to obtain a physical measure at the repeller. It also develops a regularity-by-gluing framework and provides explicit constructions of $f_1$ and $F$ to realize the desired properties, yielding a canonical bridge from induced dynamics to full-circle maps. Overall, the work demonstrates that repelling fixed points can support physical measures in one-dimensional dynamics and offers a systematic procedure to realize the induced maps as full-branch circle maps.

Abstract

We construct an example of a $C^1$ circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point. By relaxing the $C^1$ condition at a single point, we also construct an example where the basin of the physical measure has full measure. A key technical step is a realization lemma of independent interest, which gives a canonical way to construct a full branch map given its induced map.

$C^1$ Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point

TL;DR

The paper constructs a circle covering map that is topologically conjugate to the doubling map and has a physical measure supported at a hyperbolic repelling fixed point. By carefully controlling regularity at a single point , the authors produce two regimes: a full-basin example with when the map is off , and a -regular example at with (potentially full under relaxation at ). The core method uses a Realization Lemma to realize prescribed first return maps on an interval as the induced dynamics of a full-branch circle map in , enabling the application of wild-attractor techniques to obtain a physical measure at the repeller. It also develops a regularity-by-gluing framework and provides explicit constructions of and to realize the desired properties, yielding a canonical bridge from induced dynamics to full-circle maps. Overall, the work demonstrates that repelling fixed points can support physical measures in one-dimensional dynamics and offers a systematic procedure to realize the induced maps as full-branch circle maps.

Abstract

We construct an example of a circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point. By relaxing the condition at a single point, we also construct an example where the basin of the physical measure has full measure. A key technical step is a realization lemma of independent interest, which gives a canonical way to construct a full branch map given its induced map.
Paper Structure (24 sections, 12 theorems, 53 equations, 3 figures)

This paper contains 24 sections, 12 theorems, 53 equations, 3 figures.

Table of Contents

  1. Main Result and Discussion
  2. Statement of Main Results
  3. Literature Study
  4. Known Results in $\mathcal{D}$
  5. Similar Results in Other Settings
  6. Outline of Proof
  7. Induced Dynamics
  8. Dynamics of F
  9. Realization Lemma
  10. Realization Lemma
  11. Graph Pieces
  12. Regularity by Gluing
  13. Construction for Proposition \ref{['prop_result_full']}
  14. Construction of $f_1$
  15. Construction of $F$
  16. ...and 9 more sections

Key Result

Theorem 1.2

$\exists f \in \mathcal{D}: f'(p) > 1$, $f$ is $C^\infty$ on $\mathbb{S}^1 \setminus \{q\}$, and $|B_{\delta_p}| = 1$.

Figures (3)

  • Figure 1: Example of $f$ and $\phi_{a,b}$ with $a \leq 4$.
  • Figure 2: Sketch of $(f_1,F)$ with $q = 1/2, b = 3/8$.
  • Figure 3: Sketch of the three pieces of $F|_{K_n}$

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 2.6: Realization Lemma
  • ...and 15 more