Equilibrium measures for the Hénon map at the first bifurcation (Q2839492)

The authors study the family of Hénon maps, NEWLINE\[NEWLINE f(x,y) = (1-ax^2+\sqrt{b}y, \pm \sqrt{b}x). NEWLINE\]NEWLINE For small values of \(b > 0\), there is a tangency for \(f\) for a particular value \(a^* = a^* (b) \approx 2\) of the parameter \(a\). For this parameter, the potential \(\phi_t = - t \log J^u\), is not continuous and this complicates the study of equilibrium measures for this potential.NEWLINENEWLINEThe result of this paper is that for any sufficiently small \(\varepsilon > 0\) there is a \(b_0 > 0\) such that for the parameters \(a\) and \(b\) with \(a = a^*\) and \(0 < b < b_0\), there exists a unique equilibrium measure for the potential \(\phi_t\) provided \(t < t_0 (\varepsilon, b) = \inf \{\, t : P(t) \leq - (t/2) \log (4 - \varepsilon) \,\}\).NEWLINENEWLINEIt is also shown that \(t_0\) can be made arbitrarily large by taking \(\varepsilon\) and \(b\) small.