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In this paper we study the multiple ergodic averages1n∑k=1nφ(xk,xkq,⋯,xkqℓ−1),(xn)∈Σm on the symbolic space Σm={0,1,⋯,m−1}N⁎ where m≥2,ℓ≥2,q≥2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study...
Let p⩾2 be a prime number and let Zp be the ring of all p-adic integers. For all α,β,z∈Zp, define Tα,β(z)=αz+β. It is shown that the dynamical system (Zp,Tα,β) is minimal if and only if α∈1+prpZp and β is a unit, here rp=1 (respectively rp=2) if p⩾3 (respectively if p=2), and that when it is minimal, it is strictly ergodic and topologically conjugate to (Zp,T1,1) with an analytic and isometric conjugacy...
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