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Dirac Physical Measures on Saddle-Type Fixed Points

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Abstract

In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a \(C^1\) generic diffeomorphism, a Dirac invariant measure whose statistical basin of attraction is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a \(C^1\)-diffeomorphism having a Dirac invariant measure, supported on a saddle-type hyperbolic fixed point, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure. Our technique can be applied also to construct a \(C^1\) diffeomorphism whose set of points with historic behaviour has positive measure and is nowhere dense.

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Notes

  1. This is a non-trivial consequence of the specification property, see [36, Theorem 4]. The fact that specification holds for transitive Anosov diffeomorphisms was proved by Bowen [6].

  2. The same statement holds for periodic points, with a similar proof.

  3. The adjective exterior refers to the fact the boxes \(S_n^e\) will be contained in the exterior connected components of the complement of the figure-eight attractor we shall produce in this section.

  4. The dependence of \({\mathcal {S}}\) and \({\tilde{{\mathcal {S}}}}\) on \(n_0\) is not explicit in our notation because we shall fix once and for all the integer \(n_0\).

  5. See [30] for instance.

  6. Recall that this means that, inside V, the map f is topologically conjugate to its linear part Df at O.

  7. We have used here the well known fact that every periodic orbit of a planar vector field bounds a disk containing a zero of the vector field inside. This can be proved combining Poincaré-Bendixon’s theorem with Zorn’s lemma.

  8. Just as an example, fix some \(R>1\) and let \(p=(0,0) \in U=B(0,1) \subset W_0=B(0,R)\). Consider first the real function \(g:W_0{\setminus }{\overline{U}}\rightarrow (0,1)\) given by

    $$\begin{aligned} g(x,y)=\frac{1}{R-1}\,\frac{1}{\sqrt{x^2+y^2}}\,\big (R-\sqrt{x^2+y^2}\big )\,, \end{aligned}$$

    and then let \(\psi _{U}:W_0{\setminus }{\overline{U}} \rightarrow U{\setminus }\{p\}\) be given by  \(\psi _{U}(x,y)=g(x,y)\,\big (x,y\big )\).

  9. A point \(x\in M\) is said to have historic behaviour if the sequence \(\frac{1}{n}\sum \limits _{j=0}^{n-1}\delta _{f^{j}(x)}\) does not converge in the weak-* topology.

  10. It may be that the diffeomorphism \(h_1\) is useless to get the desired result; however it simplifies significantly the proofs.

  11. This phenomenon has already been shown in Lemma 4.5.

  12. For boxes \(s_v(S_n)\) of the exterior \({\mathcal {L}}^e\) of the attractor, one has to change the powers of 16 by powers of 4, as the return map of \(f_1\) to the union of such boxes is, after renormalization, a rotation of angle \(\pi \).

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H - Campus do Gragoatá, Niterói, RJ, CEP 24.210-201, Brazil

    Pablo Guarino & Bruno Santiago

  2. Sorbonne Université, Université de Paris, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, 75005, Paris, France

    Pierre-Antoine Guihéneuf

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  1. Pablo Guarino
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Correspondence to Pablo Guarino.

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We warmly thank Sylvain Crovisier for his ideas concerning Theorem A, Christian Bonatti for the attention he paid to this work and Davi Obata for his reading of a first draft. We express our very great appreciation to the referee for very keen remarks leading to a considerable improvement of our exposition. P.G. and P.A.G were partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES) Grant 23038.009189/2013-05. P.A.G. was partially funded by the France-Brazil network (RFBM). P.A.G and B.S. were partially supported by a PEPS (CNRS) Grant. B.S. was partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and also acknowledges the support of Fondation Louis D—Institut de France (Project coordinated by M. Viana).

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Guarino, P., Guihéneuf, PA. & Santiago, B. Dirac Physical Measures on Saddle-Type Fixed Points. J Dyn Diff Equat 34, 983–1048 (2022). https://doi.org/10.1007/s10884-020-09911-x

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