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Regularity and linear response formula of the SRB measures for solenoidal attractors

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  06 February 2024

CARLOS BOCKER ,
RICARDO BORTOLOTTI Open the ORCID record for RICARDO BORTOLOTTI [Opens in a new window]  and
ARMANDO CASTRO Open the ORCID record for ARMANDO CASTRO [Opens in a new window]
CARLOS BOCKER
Affiliation:
Department of Mathematics, UFPB, João Pessoa, PB, Brazil (e-mail: [email protected])
RICARDO BORTOLOTTI*
Affiliation:
Department of Mathematics, UFPE, Recife, PE, Brazil
ARMANDO CASTRO
Affiliation:
Department of Mathematics, UFBA, Salvador, BA, Brazil (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$, where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$, $r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$-generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When $s> {u}/{2}$, it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

Keywords

Solenoidal attractorstransversality conditionlinear response formulathermodynamical formalism

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 10 , October 2024 , pp. 2782 - 2831
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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