Uniform hyperbolicity for curve graphs of non-orientable surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

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Kupka–Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

Nonlinearity ◽

10.1088/1361-6544/aa7efc ◽

2017 ◽

Vol 30 (10) ◽

pp. 3895-3931

Author(s):

Renaud Leplaideur ◽

Isabel Lugão Rios

Keyword(s):

Uniform Hyperbolicity

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Finite fields and the 1-chromatic number of orientable surfaces

Journal of Graph Theory ◽

10.1002/jgt.20417 ◽

2009 ◽

pp. n/a-n/a

Author(s):

Vladimir P. Korzhik

Keyword(s):

Chromatic Number ◽

Finite Fields ◽

Orientable Surfaces

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Graphs of stable maps between closed orientable surfaces

Computational and Applied Mathematics ◽

10.1007/s40314-016-0317-9 ◽

2016 ◽

Vol 36 (3) ◽

pp. 1185-1194 ◽

Cited By ~ 4

Author(s):

C. Mendes de Jesus

Keyword(s):

Stable Maps ◽

Orientable Surfaces

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Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.61 ◽

2014 ◽

Vol 36 (1) ◽

pp. 215-255 ◽

Cited By ~ 9

Author(s):

SAMUEL SENTI ◽

HIROKI TAKAHASI

Keyword(s):

Existence And Uniqueness ◽

Thermodynamic Formalism ◽

Invariant Probability Measure ◽

Statistical Properties ◽

Limit Set ◽

Bifurcation Parameter ◽

Large Interval ◽

Equilibrium Measures ◽

Uniform Hyperbolicity ◽

Unstable Direction

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

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